43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), ICALP 2016, July 11-15, 2016, Rome, Italy
ICALP 2016
July 11-15, 2016
Rome, Italy
International Colloquium on Automata, Languages, and Programming
ICALP
http://eatcs.org/index.php/international-colloquium
https://dblp.org/db/conf/icalp
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Ioannis
Chatzigiannakis
Ioannis Chatzigiannakis
Michael
Mitzenmacher
Michael Mitzenmacher
Yuval
Rabani
Yuval Rabani
Davide
Sangiorgi
Davide Sangiorgi
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
55
2016
978-3-95977-013-2
https://www.dagstuhl.de/dagpub/978-3-95977-013-2
Front Matter, Table of Contents, Preface, Organization, List of Authors
Front Matter, Table of Contents, Preface, Organization, List of Authors
Front Matter
Table of Contents
Preface
Organization
List of Authors
0:i-0:xliv
Front Matter
Ioannis
Chatzigiannakis
Ioannis Chatzigiannakis
Michael
Mitzenmacher
Michael Mitzenmacher
Yuval
Rabani
Yuval Rabani
Davide
Sangiorgi
Davide Sangiorgi
10.4230/LIPIcs.ICALP.2016.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Compute Choice (Invited Talk)
In this talk, we shall discuss the question of learning distribution over permutations of n choices based on partial observations. This is central to capturing the so called "choice" in a variety of contexts: understanding preferences of consumers over a collection of products based on purchasing and browsing data in the setting of retail and e-commerce, learning public opinion amongst a collection of socio-economic issues based on sparse polling data, and deciding a ranking of teams or players based on outcomes of games. The talk will primarily discuss the relationship between the ability to learn, nature of partial information and number of available observations. Connections to the classical theory of social choice and behavioral psychology, as well as modern literature in Statistics, learning theory and operations research will be discussed.
Decision Systems
Learning Distributions
Partial observations
1:1-1:1
Invited Talk
Devavrat
Shah
Devavrat Shah
10.4230/LIPIcs.ICALP.2016.1
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Formally Verifying a Compiler: What Does It Mean, Exactly? (Invited Talk)
Compilers, and especially optimizing compilers, are complicated programs. Bugs in compilers happen, and can lead to miscompilation: the production of wrong executable code from a correct source program. Miscompilation is documented in the literature and a concern for high-assurance software, as it endangers the guarantees obtained by source-level formal verification of programs.
Compiler verification is a radical solution to the miscompilation problem: by applying program proof to the compiler itself, we can obtain mathematically strong guarantees that the generated executable code is faithful to the semantics of the source program. The state of the art in this line of research is arguably the CompCert verified compiler. This talk will give an overview of this optimizing C compiler and of its formal verification, conducted with the Coq proof assistant.
A formal verification is as good as the specifications it uses. In other words, verification reduces the problem of trusting a large implementation to that of ensuring that its formal specification enforce the intended correctness properties. In the case of CompCert, the correctness statement that is proved is rather complex, as it involves large operational semantics (for the C language and for the assembly languages of the target architectures) and simulations between these semantics that support both choice refinement and behavior refinement. The talk will review and discuss these elements of the specification, along with some of the accompanying proof principles.
Compilers
Compiler Optimization
Compiler Verification
2:1-2:1
Invited Talk
Xavier
Leroy
Xavier Leroy
10.4230/LIPIcs.ICALP.2016.2
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Hardness of Approximation (Invited Talk)
The talk will present connections between approximability of NP-complete problems, analysis, and geometry, and the role played by the Unique Games Conjecture in facilitating these connections.
NP-completeness
Approximation algorithms
Inapproximability
Probabilistically Checkable Proofs
Discrete Fourier analysis
3:1-3:1
Invited Talk
Subhash
Khot
Subhash Khot
10.4230/LIPIcs.ICALP.2016.3
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Model Checking and Strategy Synthesis for Stochastic Games: From Theory to Practice (Invited Talk)
Probabilistic model checking is an automatic procedure for establishing if a desired property holds in a probabilistic model, aimed at verifying quantitative probabilistic specifications such as the probability of a critical failure occurring or expected time to termination.
Much progress has been made in recent years in algorithms, tools and applications of probabilistic model checking, as exemplified by the probabilistic model checker PRISM (http://www.prismmodelchecker.org). However, the unstoppable rise of autonomous systems, from robotic assistants to self-driving cars, is placing greater and greater demands on quantitative modelling and verification technologies. To address the challenges of autonomy we need to consider collaborative, competitive and adversarial behaviour, which is naturally modelled using game-theoretic abstractions, enhanced with stochasticity arising from randomisation and uncertainty. This paper gives an overview of quantitative verification and strategy synthesis techniques developed for turn-based stochastic multi-player games, summarising recent advances concerning multi-objective properties and compositional strategy synthesis. The techniques have been implemented in the PRISM-games model checker built as an extension of PRISM.
Quantitative verification
Stochastic games
Temporal logic
Model checking
Strategy synthesis
4:1-4:18
Invited Talk
Marta Z.
Kwiatkowska
Marta Z. Kwiatkowska
10.4230/LIPIcs.ICALP.2016.4
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Taolue Chen, Vojtech Forejt, Marta Z. Kwiatkowska, Aistis Simaitis, and Clemens Wiltsche. On Stochastic Games with Multiple Objectives. In Proc. of Mathematical Foundations of Computer Science MFCS, pages 266-277, 2013.
Taolue Chen, Vojtěch Forejt, Marta Kwiatkowska, Aistis Simaitis, Ashutosh Trivedi, and Michael Ummels. Playing Stochastic Games Precisely. In Proc. of Concurrency Theory CONCUR, volume 7454 of LNCS, pages 348-363. 2012.
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Fine-Grained Complexity Analysis of Two Classic TSP Variants
We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity.
Our first set of results is motivated by the Bitonic tsp problem: given a set of n points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamicprogramming exercise to solve this problem in O(n^2) time. While the near-quadratic dependency of similar dynamic programs for Longest Common Subsequence and Discrete Fréchet Distance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic tsp in O(n*log^2(n)) time and its bottleneck version in O(n*log^3(n)) time. In the more general pyramidal tsp problem, the points to be visited are labeled 1, ..., n and the sequence of labels in the solution is required to have at most one local maximum. Our algorithms for the bitonic (bottleneck) tsp problem also work for the pyramidal tsp problem in the plane.
Our second set of results concerns the popular k-opt heuristic for tsp in the graph setting. More precisely, we study the k-opt decision problem, which asks whether a given tour can be improved by a k-opt move that replaces k edges in the tour by k new edges. A simple algorithm solves k-opt in O(n^k) time for fixed k. For 2-opt, this is easily seen to be optimal. For k = 3 we prove that an algorithm with a runtime of the form ~O(n^{3-epsilon}) exists if and only if All-Pairs Shortest Paths in weighted digraphs has such an algorithm. For general k-opt, it is known that a runtime of f(k)*n^{o(k/log(k))} would contradict the Exponential Time Hypothesis. The results for k = 2, 3 may suggest that the actual time complexity of k-opt is Theta(n^k). We show that this is not the case, by presenting an algorithm that finds the best k-move in O(n^{lfoor 2k/3 rfloor +1}) time for fixed k >= 3. This implies that 4-opt can be solved in O(n^3) time, matching the best-known algorithm for 3-opt. Finally, we show how to beat the quadratic barrier for k = 2 in two important settings, namely for points in the plane and when we want to solve 2-opt repeatedly
Traveling salesman problem
fine-grained complexity
bitonic tours
k-opt
5:1-5:14
Regular Paper
Mark
de Berg
Mark de Berg
Kevin
Buchin
Kevin Buchin
Bart M. P.
Jansen
Bart M. P. Jansen
Gerhard
Woeginger
Gerhard Woeginger
10.4230/LIPIcs.ICALP.2016.5
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Bicovering: Covering Edges With Two Small Subsets of Vertices
We study the following basic problem called Bi-Covering. Given a graph G(V, E), find two (not necessarily disjoint) sets A subseteq V and B subseteq V such that A union B = V and that every edge e belongs to either the graph induced by A or to the graph induced by B. The goal is to minimize max{|A|, |B|}. This is the most simple case of the Channel Allocation problem [Gandhi et al., Networks, 2006]. A solution that outputs V,emptyset gives ratio at most 2. We show that under the similar Strong Unique Game Conjecture by [Bansal-Khot, FOCS, 2009] there is no 2 - epsilon ratio algorithm for the problem, for any constant epsilon > 0.
Given a bipartite graph, Max-bi-clique is a problem of finding largest k*k complete bipartite sub graph. For Max-bi-clique problem, a constant factor hardness was known under random 3-SAT hypothesis of Feige [Feige, STOC, 2002] and also under the assumption that NP !subseteq intersection_{epsilon > 0} BPTIME(2^{n^{epsilon}}) [Khot, SIAM J. on Comp., 2011]. It was an open problem in [Ambühl et. al., SIAM J. on Comp., 2011] to prove inapproximability of Max-bi-clique assuming weaker conjecture. Our result implies similar hardness result assuming the Strong Unique Games Conjecture.
On the algorithmic side, we also give better than 2 approximation for Bi-Covering on numerous special graph classes. In particular, we get 1.876 approximation for Chordal graphs, exact algorithm for Interval Graphs, 1 + o(1) for Minor Free Graph, 2 - 4*delta/3 for graphs with minimum degree delta*n, 2/(1+delta^2/8) for delta-vertex expander, 8/5 for Split Graphs, 2 - (6/5)*1/d for graphs with minimum constant degree d etc. Our algorithmic results are quite non-trivial. In achieving these results, we use various known structural results about the graphs, combined with the techniques that we develop tailored to getting better than 2 approximation.
Bi-covering
Unique Games
Max Bi-clique
6:1-6:12
Regular Paper
Amey
Bhangale
Amey Bhangale
Rajiv
Gandhi
Rajiv Gandhi
Mohammad Taghi
Hajiaghayi
Mohammad Taghi Hajiaghayi
Rohit
Khandekar
Rohit Khandekar
Guy
Kortsarz
Guy Kortsarz
10.4230/LIPIcs.ICALP.2016.6
Noga Alon, Uriel Feige, Avi Wigderson, and David Zuckerman. Derandomized graph products. Computational Complexity, 5(1):60-75, 1995.
Christoph Ambühl, Monaldo Mastrolilli, and Ola Svensson. Inapproximability results for maximum edge biclique, minimum linear arrangement, and sparsest cut. SIAM Journal on Computing, 40(2):567-596, 2011.
Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for unique games and related problems. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 563-572. IEEE, 2010.
Sanjeev Arora, Subhash A Khot, Alexandra Kolla, David Steurer, Madhur Tulsiani, and Nisheeth K Vishnoi. Unique games on expanding constraint graphs are easy. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 21-28. ACM, 2008.
Nikhil Bansal and Subhash Khot. Optimal long code test with one free bit. In Foundations of Computer Science, 2009. FOCS'09. 50th Annual IEEE Symposium on, pages 453-462. IEEE, 2009.
Thang Nguyen Bui and Lisa C. Strite. An ant system algorithm for graph bisection. In Proceedings of the Genetic and Evolutionary Computation Conference, GECCO'02, pages 43-51, 2002.
Irit Dinur, Elchanan Mossel, and Oded Regev. Conditional hardness for approximate coloring. SIAM Journal on Computing, 39(3):843-873, 2009.
Uriel Feige. Relations between average case complexity and approximation complexity. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 534-543. ACM, 2002.
Uriel Feige and Shimon Kogan. Hardness of approximation of the balanced complete bipartite subgraph problem, 2004.
Rajiv Gandhi, Samir Khuller, Aravind Srinivasan, and Nan Wang. Approximation algorithms for channel allocation problems in broadcast networks. Networks, 47(4):225-236, 2006.
Venkatesan Guruswami, Johan Håstad, Rajsekar Manokaran, Prasad Raghavendra, and Moses Charikar. Beating the random ordering is hard: Every ordering csp is approximation resistant. SIAM Journal on Computing, 40(3):878-914, 2011.
Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 767-775. ACM, 2002.
Subhash Khot. Ruling out ptas for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM Journal on Computing, 36(4):1025-1071, 2006.
Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2- ε. Journal of Computer and System Sciences, 74(3):335-349, 2008.
Subhash Khot, Madhur Tulsiani, and Pratik Worah. A characterization of strong approximation resistance. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 634-643. ACM, 2014.
Prasad Raghavendra. Optimal algorithms and inapproximability results for every csp? In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 245-254. ACM, 2008.
Ola Svensson. Conditional hardness of precedence constrained scheduling on identical machines. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 745-754. ACM, 2010.
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Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs
We study the problem of routing symmetric demand pairs in planar digraphs. The input consists of a directed planar graph G = (V, E) and a collection of k source-destination pairs M = {s_1t_1, ..., s_kt_k}. The goal is to maximize the number of pairs that are routed along disjoint paths. A pair s_it_i is routed in the symmetric setting if there is a directed path connecting s_i to t_i and a directed path connecting t_i to s_i. In this paper we obtain a randomized poly-logarithmic approximation with constant congestion for this problem in planar digraphs. The main technical contribution is to show that a planar digraph with directed treewidth h contains a constant congestion crossbar of size Omega(h/polylog(h)).
Disjoint paths
symmetric demands
planar directed graph
7:1-7:14
Regular Paper
Chandra
Chekuri
Chandra Chekuri
Alina
Ene
Alina Ene
Marcin
Pilipczuk
Marcin Pilipczuk
10.4230/LIPIcs.ICALP.2016.7
M. Andrews. Approximation algorithms for the edge-disjoint paths problem via Raecke decompositions. In Proc. of IEEE FOCS, pages 277-286, 2010.
M. Andrews, J. Chuzhoy, V. Guruswami, S. Khanna, K. Talwar, and L. Zhang. Inapproximability of edge-disjoint paths and low congestion routing on undirected graphs. Combinatorica, 30(5):485-520, 2010.
C. Chekuri and J. Chuzhoy. Large-treewidth graph decompositions and applications. In Proc. of ACM STOC, pages 291-300, 2013.
C. Chekuri and J. Chuzhoy. Polynomial bounds for the grid-minor theorem. In Proc. of ACM STOC, pages 60-69, 2014.
C. Chekuri and J. Chuzhoy. Degree-3 treewidth sparsifiers. In Proc. of ACM-SIAM SODA, pages 242-255, 2015.
C. Chekuri and A. Ene. Poly-logarithmic approximation for maximum node disjoint paths with constant congestion. In Proc. of ACM-SIAM SODA, pages 326-341, 2013.
C. Chekuri and A. Ene. The all-or-nothing flow problem in directed graphs with symmetric demand pairs. Mathematical Programming, pages 1-24, 2014.
C. Chekuri, S. Khanna, and F.B. Shepherd. Multicommodity flow, well-linked terminals, and routing problems. In Proc. of ACM STOC, pages 183-192, 2005.
C. Chekuri, S. Khanna, and F.B. Shepherd. An O(√n) approximation and integrality gap for disjoint paths and unsplittable flow. Theory of Computing, 2(7):137-146, 2006.
Chandra Chekuri, Sreeram Kannan, Adnan Raja, and Pramod Viswanath. Multicommodity flows and cuts in polymatroidal networks. SIAM Journal on Computing, 44(4):912-943, 2015. Preliminary version in Proc. of ITCS, 2012.
Chandra Chekuri, Sanjeev Khanna, and F Bruce Shepherd. The all-or-nothing multicommodity flow problem. SIAM Journal on Computing, 42(4):1467-1493, 2013. Preliminary version in Proc. of ACM STOC, 2004.
J. Chen, Y. Liu, S. Lu, B. O'Sullivan, and I. Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. Journal of the ACM, 55(5), 2008.
J. Chuzhoy. Routing in undirected graphs with constant congestion. In Proc. of ACM STOC, pages 855-874, 2012.
J. Chuzhoy. Excluded grid theorem: Improved and simplified. In Proc. of ACM STOC, pages 645-654, 2015.
J. Chuzhoy, V. Guruswami, S. Khanna, and K. Talwar. Hardness of routing with congestion in directed graphs. In Proc. of ACM STOC, pages 165-178, 2007.
J. Chuzhoy, D.H.K. Kim, and S. Li. Improved approximation for node-disjoint paths in planar graphs. In Proc. of ACM STOC, 2016. To appear.
J. Chuzhoy and S. Li. A polylogarithimic approximation algorithm for edge-disjoint paths with congestion 2. In Proc. of IEEE FOCS, 2012.
M. Cygan, F. Fomin, L. Kowalik, D. Loksthanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2014. In print.
E. D. Demaine and M. Hajiaghayi. Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica, 28(1):19-36, 2008.
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T. Johnson, N. Robertson, P. D. Seymour, and R. Thomas. Excluding a grid minor in digraphs. Manuscript, http://arxiv.org/abs/1510.00473, 2001.
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Quasi-4-Connected Components
We introduce a new decomposition of a graphs into quasi-4-connected components, where we call a graph quasi-4-connected if it is 3-connected and it only has separations of order 3 that separate a single vertex from the rest of the graph. Moreover, we give a cubic time algorithm computing the decomposition of a given graph.
Our decomposition into quasi-4-connected components refines the well-known decompositions of graphs into biconnected and triconnected components. We relate our decomposition to Robertson and Seymour's theory of tangles by establishing a correspondence between the quasi-4-connected components of a graph and its tangles of order 4.
decompositions
connectivity
tangles
8:1-8:13
Regular Paper
Martin
Grohe
Martin Grohe
10.4230/LIPIcs.ICALP.2016.8
J. Carmesin, R. Diestel, M. Hamann, and F. Hundertmark. Canonical tree-decompositions of finite graphs I. Existence and algorithms. ArXiv, arXiv:1305.4668v3 [math.CO], 2013.
J. Carmesin, R. Diestel, M. Hamann, and F. Hundertmark. Canonical tree-decompositions of finite graphs II. Essential parts. ArXiv, arXiv:1305.4909v2 [math.CO], 2013.
J. Carmesin, R. Diestel, F. Hundertmark, and M. Stein. Connectivity and tree structure in finite graphs. Combinatorica, 34(1):11-46, 2014.
R. Diestel. Graph Theory. Springer-Verlag, 3rd edition, 2005.
M. Grohe. Descriptive complexity, canonisation, and definable graph structure theory. Manuscript available at URL: http://www.lii.rwth-aachen.de/de/mitarbeiter/13-mitarbeiter/professoren/39-book-descriptive-complexity.html.
http://www.lii.rwth-aachen.de/de/mitarbeiter/13-mitarbeiter/professoren/39-book-descriptive-complexity.html
M. Grohe. Fixed-point definability and polynomial time on graphs with excluded minors. Journal of the ACM, 59(5), 2012.
M. Grohe. Quasi-4-connected components. ArXiv, arXiv:1602.04505 [cs.DM], 2016. Full version of this paper.
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M. Grohe and P. Schweitzer. Computing with tangles. In Proceedings of the 47th ACM Symposium on Theory of Computing, pages 683-692, 2015.
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Creative Commons Attribution 3.0 Unported license
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Subexponential Time Algorithms for Embedding H-Minor Free Graphs
We establish the complexity of several graph embedding problems: Subgraph Isomorphism, Graph Minor, Induced Subgraph and Induced Minor, when restricted to H-minor free graphs. In each of these problems, we are given a pattern graph P and a host graph G, and want to determine whether P is a subgraph (minor, induced subgraph or induced minor) of G. We show that, for any fixed graph H and epsilon > 0, if P is H-Minor Free and G has treewidth tw, (induced) subgraph can be solved 2^{O(k^{epsilon}*tw+k/log(k))}*n^{O(1)} time and (induced) minor can be solved in 2^{O(k^{epsilon}*tw+tw*log(tw)+k/log(k))}*n^{O(1)} time, where k = |V(P)|.
We also show that this is optimal, in the sense that the existence of an algorithm for one of these problems running in 2^{o(n/log(n))} time would contradict the Exponential Time Hypothesis. This solves an open problem on the complexity of Subgraph Isomorphism for planar graphs.
The key algorithmic insight is that dynamic programming approaches can be sped up by identifying isomorphic connected components in the pattern graph. This technique seems widely applicable, and it appears that there is a relatively unexplored class of problems that share a similar upper and lower bound.
subgraph isomorphism
graph minors
subexponential time
9:1-9:14
Regular Paper
Hans L.
Bodlaender
Hans L. Bodlaender
Jesper
Nederlof
Jesper Nederlof
Tom C.
van der Zanden
Tom C. van der Zanden
10.4230/LIPIcs.ICALP.2016.9
Isolde Adler, Frederic Dorn, Fedor V. Fomin, Ignasi Sau, and Dimitrios M. Thilikos. Fast minor testing in planar graphs. Algorithmica, 64(1):69-84, 2012.
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http://dx.doi.org/10.1145/195058.195179
Omid Amini, Fedor V. Fomin, and Saket Saurabh. Counting subgraphs via homomorphisms. SIAM Journal on Discrete Mathematics, 26(2):695-717, 2012.
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http://dx.doi.org/10.1016/0304-3975(89)90011-X
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Dániel Marx. What is next? Future directions in parameterized complexity. In The Multivariate Algorithmic Revolution and Beyond, pages 469-496. Springer, 2012.
Dániel Marx and Michał Pilipczuk. Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask). arXiv preprint arXiv:1307.2187, 2013.
Jiří Matoušek and Robin Thomas. On the complexity of finding iso-and other morphisms for partial k-trees. Discrete Mathematics, 108(1):343-364, 1992.
Neil Robertson and Paul D. Seymour. Graph minors. XIII. The Disjoint Paths Problem. Journal of Combinatorial Theory, Series B, 63(1):65-110, 1995. URL: http://dx.doi.org/10.1006/jctb.1995.1006.
http://dx.doi.org/10.1006/jctb.1995.1006
Maciej M. Sysło. The subgraph isomorphism problem for outerplanar graphs. Theoretical Computer Science, 17(1):91-97, 1982. URL: http://dx.doi.org/10.1016/0304-3975(82)90133-5.
http://dx.doi.org/10.1016/0304-3975(82)90133-5
Dimitrios M. Thilikos. The Multivariate Algorithmic Revolution and Beyond. chapter Graph Minors and Parameterized Algorithm Design, pages 228-256. Springer-Verlag, Berlin, Heidelberg, 2012.
Julian R. Ullmann. An algorithm for subgraph isomorphism. Journal of the ACM (JACM), 23(1):31-42, 1976.
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Relating Graph Thickness to Planar Layers and Bend Complexity
The thickness of a graph G = (V, E) with n vertices is the minimum number of planar subgraphs of G whose union is G. A polyline drawing of G in R^2 is a drawing Gamma of G, where each vertex is mapped to a point and each edge is mapped to a polygonal chain. Bend and layer complexities are two important aesthetics of such a drawing. The bend complexity of Gamma is the maximum number of bends per edge in Gamma, and the layer complexity of Gamma is the minimum integer r such that the set of polygonal chains in Gamma can be partitioned into r disjoint sets, where each set corresponds to a planar polyline drawing. Let G be a graph of thickness t. By Fáry’s theorem, if t = 1, then G can be drawn on a single layer with bend complexity 0. A few extensions to higher thickness are known, e.g., if t = 2 (resp., t > 2), then G can be drawn on t layers with bend complexity 2 (resp., 3n + O(1)).
In this paper we present an elegant extension of Fáry's theorem to draw graphs of thickness t > 2. We first prove that thickness-t graphs can be drawn on t layers with 2.25n + O(1) bends per edge. We then develop another technique to draw thickness-t graphs on t layers with reduced bend complexity for small values of t, e.g., for t in {3, 4}, the bend complexity decreases to O(sqrt(n)).
Previously, the bend complexity was not known to be sublinear for t > 2. Finally, we show that graphs with linear arboricity k can be drawn on k layers with bend complexity 3*(k-1)*n/(4k-2).
Graph Drawing
Thickness
Geometric Thickness
Layers; Bends
10:1-10:13
Regular Paper
Stephane
Durocher
Stephane Durocher
Debajyoti
Mondal
Debajyoti Mondal
10.4230/LIPIcs.ICALP.2016.10
Melanie Badent, Emilio Di Giacomo, and Giuseppe Liotta. Drawing colored graphs on colored points. Theoretical Computer Science, 408(2-3):129-142, 2008.
Reuven Bar-Yehuda and Sergio Fogel. Partitioning a sequence into few monotone subsequences. Acta Informatica, 35(5):421-440, 1998.
János Barát, Jiří Matoušek, and David R. Wood. Bounded-degree graphs have arbitrarily large geometric thickness. Electronic Journal of Combinatorics, 13(R3), 2006.
Thomas Bläsius, Stephen G. Kobourov, and Ignaz Rutter. Simultaneous embedding of planar graphs. In Roberto Tamassia, editor, Handbook of Graph Drawing and Visualization, chapter 11, pages 349-380. CRC Press, August 2013.
Peter Braß, Eowyn Cenek, Christian A. Duncan, Alon Efrat, Cesim Erten, Dan Ismailescu, Stephen G. Kobourov, Anna Lubiw, and Joseph S. B. Mitchell. On simultaneous planar graph embeddings. Computational Geometry, 36(2):117-130, 2007.
Michael B. Dillencourt, David Eppstein, and Daniel S. Hirschberg. Geometric thickness of complete graphs. Journal of Graph Algorithms and Applications, 4(3):5-17, 2000.
Vida Dujmović and David R. Wood. Graph treewidth and geometric thickness parameters. Discrete & Computational Geometry, 37(4):641-670, 2007.
Christian A. Duncan. On graph thickness, geometric thickness, and separator theorems. Computational Geometry, 44(2):95-99, 2011.
Christian A. Duncan, David Eppstein, and Stephen G. Kobourov. The geometric thickness of low degree graphs. In Proceedings of the 20th ACM Symposium on Computational Geometry (SoCG), pages 340-346. ACM, 2004.
Stephane Durocher, Ellen Gethner, and Debajyoti Mondal. Thickness and colorability of geometric graphs. Computational Geometry: Theory and Applications, 56:1-18, 2016.
Stephane Durocher and Debajyoti Mondal. Relating graph thickness to planar layers and bend complexity, 2016. URL: http://arxiv.org/abs/1602.07816.
http://arxiv.org/abs/1602.07816
Hikoe Enomoto and Miki Shimabara Miyauchi. Embedding graphs into a three page book with O(m log n) crossings of edges over the spine. SIAM Journal on Discrete Mathematics, 12(3):337-341, 1999.
David Eppstein. Separating thickness from geometric thickness. In János Pach, editor, Towards a Theory of Geometric Graphs. American Mathematical Society, 2004.
Paul Erdös and George Szekeres. A combinatorial theorem in geometry. Compositio Math., 2:463-470, 1935.
Cesim Erten and Stephen G. Kobourov. Simultaneous embedding of planar graphs with few bends. Journal of Graph Algorithms and Applications, 9(3):347-364, 2005.
István Fáry. On straight-line representation of planar graphs. Acta Sci. Math. (Szeged), 11:229-233, 1948.
Emilio Di Giacomo and Giuseppe Liotta. Simultaneous embedding of outerplanar graphs, paths, and cycles. International Journal of Computational Geometry &Applications, 17(2):139-160, 2007.
Taylor Gordon. Simultaneous embeddings with vertices mapping to pre-specified points. In Proceedings of the 18th Annual International Conference on Computing and Combinatorics (COCOON), volume 7434 of LNCS, pages 299-310. Springer, 2012.
Joseph B. Kruskal. Monotonic subsequences. Proceedings of the American Mathematical Society, 4:264-274, 1953.
János Pach and Rephael Wenger. Embedding planar graphs at fixed vertex locations. Graphs &Combinatorics, 17(4):717-728, 2001.
David R. Wood. Geometric thickness in a grid. Discrete Mathematics, 273(1-3):221-234, 2003.
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Optimal Approximate Matrix Product in Terms of Stable Rank
We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having m = O(˜r/epsilon^2) rows. Here r˜ is the maximum stable rank, i.e., the squared ratio of Frobenius and operator norms, of the two matrices being multiplied. This is a quantitative improvement over previous work of [Magen and Zouzias, SODA, 2011] and [Kyrillidis et al., arXiv, 2014] and is also optimal for any oblivious dimensionality-reducing map. Furthermore, due to the black box reliance on the subspace embedding property in our proofs, our theorem can be applied to a much more general class of sketching matrices than what was known before, in addition to achieving better bounds. For example, one can apply our theorem to efficient subspace embeddings such as the Subsampled Randomized Hadamard Transform or sparse subspace embeddings, or even with subspace embedding constructions that may be developed in the future.
Our main theorem, via connections with spectral error matrix multiplication proven in previous work, implies quantitative improvements for approximate least squares regression and low rank approximation, and implies faster low rank approximation for popular kernels in machine learning such as the gaussian and Sobolev kernels. Our main result has also already been applied to improve dimensionality reduction guarantees for k-means clustering, and also implies new results for nonparametric regression.
Lastly, we point out that the proof of the "BSS" deterministic row-sampling result of [Batson et al., SICOMP, 2012] can be modified to obtain deterministic row-sampling for approximate matrix product in terms of the stable rank of the matrices. The original "BSS" proof was in terms of the rank rather than the stable rank.
subspace embeddings
approximate matrix multiplication
stable rank
regression
low rank approximation
11:1-11:14
Regular Paper
Michael B.
Cohen
Michael B. Cohen
Jelani
Nelson
Jelani Nelson
David P.
Woodruff
David P. Woodruff
10.4230/LIPIcs.ICALP.2016.11
Nir Ailon and Bernard Chazelle. The fast Johnson-Lindenstrauss transform and approximate nearest neighbors. SIAM J. Comput., 39(1):302-322, 2009.
Nir Ailon and Edo Liberty. An almost optimal unrestricted fast Johnson-Lindenstrauss transform. ACM Transactions on Algorithms, 9(3):21, 2013.
Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-Ramanujan sparsifiers. SIAM J. Comput., 41(6):1704-1721, 2012.
Jean Bourgain. An improved estimate in the restricted isometry problem. Geometric Aspects of Functional Analysis, 2116:65-70, 2014.
Christos Boutsidis, Anastasios Zouzias, Michael W. Mahoney, and Petros Drineas. Randomized dimensionality reduction for k-means clustering. IEEE Transactions on Information Theory, 61(2):1045-1062, 2015.
Moses Charikar, Kevin C. Chen, and Martin Farach-Colton. Finding frequent items in data streams. Theor. Comput. Sci., 312(1):3-15, 2004.
Pei-Chun Chen, Kuang-Yao Lee, Tsung-Ju Lee, Yuh-Jye Lee, and Su-Yun Huang. Multiclass support vector classification via coding and regression. Neurocomputing, 73(7-9):1501-1512, 2010.
Kenneth L. Clarkson and David P. Woodruff. Numerical linear algebra in the streaming model. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), pages 205-214, 2009.
Kenneth L. Clarkson and David P. Woodruff. Low rank approximation and regression in input sparsity time. In Proceedings of the 45th ACM Symposium on Theory of Computing (STOC), pages 81-90, 2013. Full version at URL: http://arxiv.org/abs/1207.6365v4.
http://arxiv.org/abs/1207.6365v4
Michael B. Cohen. Nearly tight oblivious subspace embeddings by trace inequalities. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 278-287, 2016.
Michael B. Cohen, Sam Elder, Cameron Musco, Christopher Musco, and Mădălina Persu. Dimensionality reduction for k-means clustering and low rank approximation. In Proceedings of the 47th ACM Symposium on Theory of Computing (STOC), 2015. Full version at URL: http://arxiv.org/abs/1410.6801v3.
http://arxiv.org/abs/1410.6801v3
Michael B. Cohen, Yin Tat Lee, Cameron Musco, Christopher Musco, Richard Peng, and Aaron Sidford. Uniform sampling for matrix approximation. In Proc. of the 6th Annual Conference on Innovations in Theoretical Computer Science (ITCS), pages 181-190, 2015.
Michael B. Cohen, Jelani Nelson, and David P. Woodruff. Optimal approximate matrix product in terms of stable rank. CoRR, abs/1507.02268, 2015.
Anirban Dasgupta, Ravi Kumar, and Tamás Sarlós. A sparse Johnson-Lindenstrauss transform. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), 2010.
Petros Drineas, Ravi Kannan, and Michael W. Mahoney. Fast Monte Carlo algorithms for matrices I: approximating matrix multiplication. SIAM J. Comput., 36(1):132-157, 2006.
Petros Drineas, Malik Magdon-Ismail, Michael W. Mahoney, and David P. Woodruff. Fast approximation of matrix coherence and statistical leverage. Journal of Machine Learning Research, 13:3475-3506, 2012.
Petros Drineas, Michael W. Mahoney, and S. Muthukrishnan. Sampling algorithms for 𝓁₂ regression and applications. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1127-1136, 2006.
Alex Gittens and Michael W. Mahoney. Revisiting the nystrom method for improved large-scale machine learning. In Proceedings of the 30th International Conference on Machine Learning (ICML), pages 567-575, 2013.
Nathan Halko, Per-Gunnar Martinsson, and Joel A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53(2):217-288, 2011.
Ishay Haviv and Oded Regev. The restricted isometry property of subsampled Fourier matrices. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), to appear, 2016.
William B. Johnson and Joram Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. Contemporary Mathematics, 26:189-206, 1984.
Daniel M. Kane and Jelani Nelson. Sparser Johnson-Lindenstrauss transforms. J. ACM, 61(1):4, 2014.
Alexandra Kolla, Yury Makarychev, Amin Saberi, and Shang-Hua Teng. Subgraph sparsification and nearly optimal ultrasparsifiers. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), pages 57-66, 2010.
Felix Krahmer and Rachel Ward. New and improved Johnson-Lindenstrauss embeddings via the Restricted Isometry Property. SIAM J. Math. Anal., 43(3):1269-1281, 2011.
Anastasios T. Kyrillidis, Michail Vlachos, and Anastasios Zouzias. Approximate matrix multiplication with application to linear embeddings. CoRR, abs/1403.7683, 2014.
Yin Tat Lee and He Sun. Constructing linear sized spectral sparsification in almost linear time. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 250-269, 2015.
Mu Li, Gary L. Miller, and Richard Peng. Iterative row sampling. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2013.
Yingyu Liang, Maria-Florina Balcan, Vandana Kanchanapally, and David P. Woodruff. Improved distributed principal component analysis. In Proceedings of the 27th Annual Conference on Advances in Neural Information Processing Systems (NIPS), 2014.
Edo Liberty, Franco Woolfe, Per-Gunnar Martinsson, Vladimir Rokhlin, and Mark Tygert. Randomized algorithms for the low-rank approximation of matrices. Proceedings of the National Academy of Sciences, 104(51):20167-20172, 2007.
Yichao Lu, Paramveer Dhillon, Dean Foster, and Lyle Ungar. Faster ridge regression via the subsampled randomized Hadamard transform. In Proceedings of the 26th Annual Conference on Advances in Neural Information Processing Systems (NIPS), 2013.
Avner Magen and Anastasios Zouzias. Low rank matrix-valued Chernoff bounds and approximate matrix multiplication. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1422-1436, 2011.
Michael W. Mahoney. Randomized algorithms for matrices and data. Foundations and Trends in Machine Learning, 3(2):123-224, 2011.
Xiangrui Meng and Michael W. Mahoney. Low-distortion subspace embeddings in input-sparsity time and applications to robust linear regression. In Proceedings of the 45th ACM Symposium on Theory of Computing (STOC), pages 91-100, 2013.
Jelani Nelson and Huy L. Nguŷẽn. OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddings. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 117-126, 2013.
Jelani Nelson and Huy L. Nguŷẽn. Lower bounds for oblivious subspace embeddings. In Proceedings of the 41st International Colloquium on Automata, Languages, and Programming (ICALP), pages 883-894, 2014.
Jelani Nelson, Eric Price, and Mary Wootters. New constructions of RIP matrices with fast multiplication and fewer rows. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2014.
Nima Reyhani, Hideitsu Hino, and Ricardo Vigário. New probabilistic bounds on eigenvalues and eigenvectors of random kernel matrices. In Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence (UAI), pages 627-634, 2011.
Tamás Sarlós. Improved approximation algorithms for large matrices via random projections. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 143-152, 2006.
Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. SIAM J. Comput., 40(6):1913-1926, 2011.
Mikkel Thorup and Yin Zhang. Tabulation-based 5-independent hashing with applications to linear probing and second moment estimation. SIAM J. Comput., 41(2):293-331, 2012.
Joel A. Tropp. Improved analysis of the subsampled randomized Hadamard transform. Adv. Adapt. Data Anal., 3(1-2):115-126, 2011.
David P. Woodruff. Sketching as a tool for numerical linear algebra. Foundations and Trends in Theoretical Computer Science, 10(1-2):1-157, 2014.
Yun Yang, Mert Pilanci, and Martin J. Wainwright. Randomized sketches for kernels: Fast and optimal non-parametric regression. CoRR, abs/1501.06195, 2015.
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Approximate Span Programs
Span programs are a model of computation that have been used to design quantum algorithms, mainly in the query model. It is known that for any decision problem, there exists a span program that leads to an algorithm with optimal quantum query complexity, however finding such an algorithm is generally challenging. In this work, we consider new ways of designing quantum algorithms using span programs. We show how any span program that decides a problem f can also be used to decide "property testing" versions of the function f, or more generally, approximate a quantity called the span program witness size, which is some property of the input related to f. For example, using our techniques, the span program for OR, which can be used to design an optimal algorithm for the OR function, can also be used to design optimal algorithms for: threshold functions, in which we want to decide if the Hamming weight of a string is above a threshold, or far below, given the promise that one of these is true; and approximate counting, in which we want to estimate the Hamming weight of the input up to some desired accuracy. We achieve these results by relaxing the requirement that 1-inputs hit some target exactly in the span program, which could potentially make design of span programs significantly easier. In addition, we give an exposition of span program structure, which increases the general understanding of this important model. One implication of this is alternative algorithms for estimating the witness size when the phase gap of a certain unitary can be lower bounded. We show how to lower bound this phase gap in certain cases.
As an application, we give the first upper bounds in the adjacency query model on the quantum time complexity of estimating the effective resistance between s and t, R_{s,t}(G). For this problem we obtain ~O(1/epsilon^{3/2}*n*sqrt(R_{s,t}(G)), using O(log(n)) space. In addition, when mu is a lower bound on lambda_2(G), by our phase gap lower bound, we can obtain an upper bound of ~O(1/epsilon*n*sqrt(R){s,t}(G)/mu)) for estimating effective resistance, also using O(log(n)) space.
Quantum algorithms
span programs
quantum query complexity
effective resistance
12:1-12:14
Regular Paper
Tsuyoshi
Ito
Tsuyoshi Ito
Stacey
Jeffery
Stacey Jeffery
10.4230/LIPIcs.ICALP.2016.12
R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf. Quantum lower bounds by polynomials. Journal of the ACM, 48, 2001.
A. Belovs. Learning-graph-based quantum algorithm for k-distinctness. In Prooceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2012), pages 207-216, 2012.
A. Belovs. Span programs for functions with constant-sized 1-certificates. In Proceedings of the 44th Symposium on Theory of Computing (STOC 2012), pages 77-84, 2012.
A. Belovs, A. M. Childs, S. Jeffery, R. Kothari, and F. Magniez. Time efficient quantum walks for 3-distinctness. In Proceedings of the 40th International Colloquium on Automata, Languages and Programming (ICALP 2013), pages 105-122, 2013.
A. Belovs and B. Reichardt. Span programs and quantum algorithms for st-connectivity and claw detection. In Proceedings of the 20th European Symposium on Algorithms (ESA 2012), pages 193-204, 2012.
C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computing. SIAM Journal on Computing (special issue on quantum computing), 26:1510-1523, 1997.
G. Brassard, P. Høyer, M. Mosca, and A. Tapp. Quantum amplitude amplification and estimation. In S. J. Lomonaca and H. E. Brandt, editors, Quantum Computation and Quantum Information: A Millennium Volume, volume 305 of AMS Contemporary Mathematics Series Millennium Volume, pages 53-74. AMS, 2002. arXiv:quant-ph/0005055v1.
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A. W. Harrow, A. Hassidim, and S. Lloyd. Quantum algorithm for linear systems of equations. Physical Review Letters, 103:150502, Oct 2009.
S. Jeffery. Frameworks for Quantum Algorithms. PhD thesis, University of Waterloo, 2014. Available at URL: http://uwspace.uwaterloo.ca/handle/10012/8710.
http://uwspace.uwaterloo.ca/handle/10012/8710
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D. A. Levin, Y. Peres, and E. L. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, 2009.
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G. Wang. Quantum algorithms for approximating the effective resistances in electrical networks, 2013. arXiv:1311.1851.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Power of Quantum Computation with Few Clean Qubits
This paper investigates the power of polynomial-time quantum computation in which only a very limited number of qubits are initially clean in the |0> state, and all the remaining qubits are initially in the totally mixed state. No initializations of qubits are allowed during the computation, nor are intermediate measurements. The main contribution of this paper is to develop unexpectedly strong error-reduction methods for such quantum computations that simultaneously reduce the number of necessary clean qubits. It is proved that any problem solvable by a polynomialtime quantum computation with one-sided bounded error that uses logarithmically many clean qubits is also solvable with exponentially small one-sided error using just two clean qubits, and with polynomially small one-sided error using just one clean qubit. It is further proved in the twosided-error case that any problem solvable by such a computation with a constant gap between completeness and soundness using logarithmically many clean qubits is also solvable with exponentially small two-sided error using just two clean qubits. If only one clean qubit is available, the problem is again still solvable with exponentially small error in one of the completeness and soundness and with polynomially small error in the other. An immediate consequence is that the Trace Estimation problem defined with fixed constant threshold parameters is complete for BQ_{[1]}P and BQ_{log}P, the classes of problems solvable by polynomial-time quantum computations with completeness 2/3 and soundness 1/3 using just one and logarithmically many clean qubits, respectively. The techniques used for proving the error-reduction results may be of independent interest in themselves, and one of the technical tools can also be used to show the hardness of weak classical simulations of one-clean-qubit computations (i.e., DQC1 computations).
DQC1
quantum computing
complete problems
error reduction
13:1-13:14
Regular Paper
Keisuke
Fujii
Keisuke Fujii
Hirotada
Kobayashi
Hirotada Kobayashi
Tomoyuki
Morimae
Tomoyuki Morimae
Harumichi
Nishimura
Harumichi Nishimura
Shuhei
Tamate
Shuhei Tamate
Seiichiro
Tani
Seiichiro Tani
10.4230/LIPIcs.ICALP.2016.13
Scott Aaronson. Quantum computing, postselection, and probabilistic polynomial-time. Proceedings of the Royal Society A, 461(2063):3473-3482, 2005. URL: http://dx.doi.org/10.1098/rspa.2005.1546.
http://dx.doi.org/10.1098/rspa.2005.1546
Scott Aaronson and Alex Arkhipov. The computational complexity of linear optics. Theory of Computing, 9:143-252, 2013. URL: http://dx.doi.org/10.4086/toc.2013.v009a004.
http://dx.doi.org/10.4086/toc.2013.v009a004
Leonard M. Adleman, Jonathan DeMarrais, and Ming-Deh A. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524-1540, 1997. URL: http://dx.doi.org/10.1137/S0097539795293639.
http://dx.doi.org/10.1137/S0097539795293639
Andris Ambainis and Rūsiņš Freivalds. 1-way quantum finite automata: strengths, weaknesses and generalizations. In 39th Annual Symposium on Foundations of Computer Science, pages 332-341, 1998. URL: http://dx.doi.org/10.1109/SFCS.1998.743469.
http://dx.doi.org/10.1109/SFCS.1998.743469
Andris Ambainis, Leonard J. Schulman, and Umesh Vazirani. Computing with highly mixed states. Journal of the ACM, 53(3):507-531, 2006. URL: http://dx.doi.org/10.1145/1147954.1147962.
http://dx.doi.org/10.1145/1147954.1147962
Elmar Böhler, Christian Glaßer, and Daniel Meister. Error-bounded probabilistic computations between MA and AM. Journal of Computer and System Sciences, 72(6):1043-1076, 2006. URL: http://dx.doi.org/10.1016/j.jcss.2006.05.001.
http://dx.doi.org/10.1016/j.jcss.2006.05.001
Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd. Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy. Proceedings of the Royal Society A, 467(2126):459-472, 2011. URL: http://dx.doi.org/10.1098/rspa.2010.0301.
http://dx.doi.org/10.1098/rspa.2010.0301
Daniel J. Brod. The complexity of simulating constant-depth BosonSampling. Physical Review A, 91(4):article 042316, 2015. URL: http://dx.doi.org/10.1103/PhysRevA.91.042316.
http://dx.doi.org/10.1103/PhysRevA.91.042316
Joseph Emerson, Yaakov S. Weinstein, Seth Lloyd, and D. G. Cory. Fidelity decay as an efficient indicator of quantum chaos. Physical Review Letters, 89(28), 2002. URL: http://dx.doi.org/10.1103/PhysRevLett.89.284102.
http://dx.doi.org/10.1103/PhysRevLett.89.284102
Bill Fefferman, Hirotada Kobayashi, Cedric Yen-Yu Lin, Tomoyuki Morimae, and Harumichi Nishimura. Space-efficient error reduction for unitary quantum computations. In Automata, Languages, and Programming, 43rd International Colloquium, ICALP 2016, Proceedings, Leibniz International Proceedings in Informatics, 2016.
Keisuke Fujii, Hirotada Kobayashi, Tomoyuki Morimae, Harumichi Nishimura, Shuhei Tamate, and Seiichiro Tani. Power of quantum computation with few clean qubits. arXiv.org e-Print archive, arXiv:1509.07276 [quant-ph], 2015. URL: http://arxiv.org/abs/1507.05592.
http://arxiv.org/abs/1507.05592
Stephen P. Jordan and Gorjan Alagic. Approximating the Turaev-Viro invariant of mapping tori is complete for one clean qubit. In Dave Bacon, Miguel Martin-Delgado, and Martin Roetteler, editors, Theory of Quantum Computation, Communication, and Cryptography, 6th Conference, TQC 2011, Madrid, Spain, May 24-26, 2011, Revised Selected Papers, volume 6745 of Lecture Notes in Computer Science, pages 53-72. Springer-Verlag, 2014. URL: http://dx.doi.org/10.1007/978-3-642-54429-3_5.
http://dx.doi.org/10.1007/978-3-642-54429-3_5
Stephen P. Jordan and Pawel Wocjan. Estimating Jones and HOMFLY polynomials with one clean qubit. Quantum Information and Computation, 9(3-4):0264-0289, 2009.
Richard Jozsa and Maarten Van den Nest. Classical simulation complexity of extended Clifford circuits. Quantum Information and Computation, 14(7-8):0633-0648, 2014.
E. Knill and R. Laflamme. Power of one bit of quantum information. Physical Review Letters, 81(25):5672-5675, 1998. URL: http://dx.doi.org/10.1103/PhysRevLett.81.5672.
http://dx.doi.org/10.1103/PhysRevLett.81.5672
Greg Kuperberg. How hard is it to approximate the Jones polynomial? Theory of Computing, 11:183-219 (article 6), 2015. URL: http://dx.doi.org/10.4086/toc.2015.v011a006.
http://dx.doi.org/10.4086/toc.2015.v011a006
Tomoyuki Morimae, Keisuke Fujii, and Joseph F. Fitzsimons. Hardness of classically simulating the one-clean-qubit model. Physical Review Letters, 112(13):article 130502, 2014. URL: http://dx.doi.org/10.1103/PhysRevLett.112.130502.
http://dx.doi.org/10.1103/PhysRevLett.112.130502
Xiaotong Ni and Maarten Van den Nest. Commuting quantum circuits: Efficient classical simulations versus hardness results. Quantum Information and Computation, 13(1-2):0054-0072, 2013.
David Poulin, Robin Blume-Kohout, Raymond Laflamme, and Harold Ollivier. Exponential speedup with a single bit of quantum information: Measuring the average fidelity decay. Physical Review Letters, 92(17), 2004. URL: http://dx.doi.org/10.1103/PhysRevLett.92.177906.
http://dx.doi.org/10.1103/PhysRevLett.92.177906
David Poulin, Raymond Laflamme, G. J. Milburn, and Juan Pablo Paz. Testing integrability with a single bit of quantum information. Physical Review A, 68(2):article 022302, 2003. URL: http://dx.doi.org/10.1103/PhysRevA.68.022302.
http://dx.doi.org/10.1103/PhysRevA.68.022302
D. J. Shepherd. Computation with unitaries and one pure qubit. arXiv.org e-Print archive, arXiv:quant-ph/0608132, 2006. URL: http://arxiv.org/abs/quant-ph/0608132.
http://arxiv.org/abs/quant-ph/0608132
Daniel James Shepherd. Quantum Complexity : restrictions on algorithms and architectures. PhD thesis, Department of Computer Science, Faculty of Engineering, University of Bristol, 2009. arXiv.org e-Print archive, arXiv:1005.1425 [cs.CC]. URL: http://arxiv.org/abs/1005.1425.
http://arxiv.org/abs/1005.1425
Peter W. Shor and Stephen P. Jordan. Estimating Jones polynomials is a complete problem for one clean qubit. Quantum Information and Computation, 8(8-9):0681-0714, 2008.
Yasuhiro Takahashi, Seiichiro Tani, Takeshi Yamazaki, and Kazuyuki Tanaka. Commuting quantum circuits with few outputs are unlikely to be classically simulatable. In Computing and Combinatorics, 21st International Conference, COCOON 2015, volume 9198 of Lecture Notes in Computer Science, pages 223-234, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21398-9_18.
http://dx.doi.org/10.1007/978-3-319-21398-9_18
Yasuhiro Takahashi, Takeshi Yamazaki, and Kazuyuki Tanaka. Hardness of classically simulating quantum circuits with unbounded Toffoli and fan-out gates. Quantum Information and Computation, 14(13-14):1149-1164, 2014.
Barbara M. Terhal and David P. DiVincenzo. Adaptive quantum computation, constant depth quantum circuits and Arthur-Merlin games. Quantum Information and Computation, 4(2):134-145, 2004.
Seinosuke Toda. PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20(5):865-877, 1991. URL: http://dx.doi.org/10.1137/0220053.
http://dx.doi.org/10.1137/0220053
John Watrous. Quantum simulations of classical random walks and undirected graph connectivity. Journal of Computer and System Sciences, 62(2):376-391, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1732.
http://dx.doi.org/10.1006/jcss.2000.1732
Edward Witten. Quantum field theory and the Jones polynomial. Communications in Mathematical Physics, 121(3):351-399, 1989. URL: http://dx.doi.org/10.1007/BF01217730.
http://dx.doi.org/10.1007/BF01217730
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https://creativecommons.org/licenses/by/3.0/legalcode
Space-Efficient Error Reduction for Unitary Quantum Computations
This paper presents a general space-efficient method for error reduction for unitary quantum computation. Consider a polynomial-time quantum computation with completeness c and soundness s, either with or without a witness (corresponding to QMA and BQP, respectively). To convert this computation into a new computation with error at most 2^{-p}, the most space-efficient method known requires extra workspace of O(p*log(1/(c-s))) qubits. This space requirement is too large for scenarios like logarithmic-space quantum computations. This paper shows an errorreduction method for unitary quantum computations (i.e., computations without intermediate measurements) that requires extra workspace of just O(log(p/(c-s))) qubits. This in particular gives the first method of strong amplification for logarithmic-space unitary quantum computations with two-sided bounded error. This also leads to a number of consequences in complexity theory, such as the uselessness of quantum witnesses in bounded-error logarithmic-space unitary quantum computations, the PSPACE upper bound for QMA with exponentially-small completeness-soundness gap, and strong amplification for matchgate computations.
space-bounded computation
quantum Merlin-Arthur proof systems
error reduction
quantum computing
14:1-14:14
Regular Paper
Bill
Fefferman
Bill Fefferman
Hirotada
Kobayashi
Hirotada Kobayashi
Cedric
Yen-Yu Lin
Cedric Yen-Yu Lin
Tomoyuki
Morimae
Tomoyuki Morimae
Harumichi
Nishimura
Harumichi Nishimura
10.4230/LIPIcs.ICALP.2016.14
Andris Ambainis and Rūsiņš Freivalds. 1-way quantum finite automata: strengths, weaknesses and generalizations. In 39th Annual Symposium on Foundations of Computer Science, pages 332-341, 1998. URL: http://dx.doi.org/10.1109/SFCS.1998.743469.
http://dx.doi.org/10.1109/SFCS.1998.743469
Charles H. Bennett. Time/space trade-offs for reversible computation. SIAM Journal on Computing, 18(4):766-776, 1989. URL: http://dx.doi.org/10.1137/0218053.
http://dx.doi.org/10.1137/0218053
Bill Fefferman, Hirotada Kobayashi, Cedric Yen-Yu Lin, Tomoyuki Morimae, and Harumichi Nishimura. Space-efficient error reduction for unitary quantum computations. arXiv.org e-Print archive, arXiv:1604.08192 [quant-ph], 2016. URL: http://arxiv.org/abs/1604.08192.
http://arxiv.org/abs/1604.08192
Bill Fefferman and Cedric Lin. Quantum Merlin Arthur with exponentially small gap. arXiv.org e-Print archive, arXiv:1601.01975 [quant-ph], 2016. URL: http://arxiv.org/abs/1601.01975.
http://arxiv.org/abs/1601.01975
Bill Fefferman and Cedric Yen-Yu Lin. A complete characterization of unitary quantum space. arXiv.org e-Print archive, arXiv:1604.01384 [quant-ph], 2016. URL: http://arxiv.org/abs/1604.01384.
http://arxiv.org/abs/1604.01384
Lov K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, pages 212-219, 1996. URL: http://dx.doi.org/10.1145/237814.237866.
http://dx.doi.org/10.1145/237814.237866
Tsuyoshi Ito, Hirotada Kobayashi, and John Watrous. Quantum interactive proofs with weak error bounds. In ITCS'12, Proceedings of the 2012 ACM Conference on Innovations in Theoretical Computer Science, pages 266-275, 2012. URL: http://dx.doi.org/10.1145/2090236.2090259.
http://dx.doi.org/10.1145/2090236.2090259
Stephen P. Jordan, Hirotada Kobayashi, Daniel Nagaj, and Harumichi Nishimura. Achieving perfect completeness in classical-witness quantum Merlin-Arthur proof systems. Quantum Information and Computation, 12(5-6):0461-0471, 2012.
Richard Jozsa, Barbara Kraus, Akimasa Miyake, and John Watrous. Matchgate and space-bounded quantum computations are equivalent. Proceedings of the Royal Society A, 466(2115):809-830, 2010. URL: http://dx.doi.org/10.1098/rspa.2009.0433.
http://dx.doi.org/10.1098/rspa.2009.0433
Alexei Kitaev and John Watrous. Parallelization, amplification, and exponential time simulation of quantum interactive proof systems. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 608-617, 2000. URL: http://dx.doi.org/10.1145/335305.335387.
http://dx.doi.org/10.1145/335305.335387
Alexei \relaxYu. Kitaev, Alexander H. Shen, and Mikhail N. Vyalyi. Classical and Quantum Computation, volume 47 of Graduate Studies in Mathematics. American Mathematical Society, 2002. URL: http://dx.doi.org/10.1090/gsm/047.
http://dx.doi.org/10.1090/gsm/047
Hirotada Kobayashi, François Le Gall, and Harumichi Nishimura. Stronger methods of making quantum interactive proofs perfectly complete. SIAM Journal on Computing, 44(2):243-289, 2015. URL: http://dx.doi.org/10.1137/140971944.
http://dx.doi.org/10.1137/140971944
Chris Marriott and John Watrous. Quantum Arthur-Merlin games. Computational Complexity, 14(2):122-152, 2005. URL: http://dx.doi.org/10.1007/s00037-005-0194-x.
http://dx.doi.org/10.1007/s00037-005-0194-x
Dieter van Melkebeek and Thomas Watson. Time-space efficient simulations of quantum computations. Theory of Computing, 8:1-51 (Article 1), 2012. URL: http://dx.doi.org/10.4086/toc.2012.v008a001.
http://dx.doi.org/10.4086/toc.2012.v008a001
Daniel Nagaj, Pawel Wocjan, and Yong Zhang. Fast amplification of QMA. Quantum Information and Computation, 9(11-12):1053-1068, 2009.
Anand Natarajan and Xiaodi Wu. Private communication, January 2016.
Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000.
Amnon Ta-Shma. Inverting well conditioned matrices in quantum logspace. In STOC'13, Proceedings of the 2013 ACM Symposium on Theory of Computing, pages 881-890, 2013. URL: http://dx.doi.org/10.1145/2488608.2488720.
http://dx.doi.org/10.1145/2488608.2488720
Barbara M. Terhal and David P. DiVincenzo. Classical simulation of noninteracting-fermion quantum circuits. Physical Review A, 65:article 032325, 2002. URL: http://dx.doi.org/10.1103/PhysRevA.65.032325.
http://dx.doi.org/10.1103/PhysRevA.65.032325
Leslie G. Valiant. Quantum circuits that can be simulated classically in polynomial time. SIAM Journal on Computing, 31(4):1229-1254, 2002. URL: http://dx.doi.org/10.1137/S0097539700377025.
http://dx.doi.org/10.1137/S0097539700377025
John Watrous. Space-bounded quantum complexity. Journal of Computer and System Sciences, 59(2):281-326, 1999. URL: http://dx.doi.org/10.1006/jcss.1999.1655.
http://dx.doi.org/10.1006/jcss.1999.1655
John Watrous. Quantum simulations of classical random walks and undirected graph connectivity. Journal of Computer and System Sciences, 62(2):376-391, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1732.
http://dx.doi.org/10.1006/jcss.2000.1732
John Watrous. On the complexity of simulating space-bounded quantum computations. Computational Complexity, 12(1-2):48-84, 2003. URL: http://dx.doi.org/10.1007/s00037-003-0177-8.
http://dx.doi.org/10.1007/s00037-003-0177-8
John Watrous. Quantum computational complexity. In Robert A. Meyers, editor, Encyclopedia of Complexity and Systems Science, pages 7174-7201. Springer New York, 2009. URL: http://dx.doi.org/10.1007/978-0-387-30440-3_428.
http://dx.doi.org/10.1007/978-0-387-30440-3_428
John Watrous. Zero-knowledge against quantum attacks. SIAM Journal on Computing, 39(1):25-58, 2009. URL: http://dx.doi.org/10.1137/060670997.
http://dx.doi.org/10.1137/060670997
Mark M. Wilde. Quantum Information Theory. Cambridge University Press, 2013. URL: http://dx.doi.org/10.1017/CBO9781139525343.
http://dx.doi.org/10.1017/CBO9781139525343
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Linear Time Algorithm for Quantum 2SAT
A canonical result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors Q_{ij} on a system of n qubits, and the task is to decide whether the Hamiltonian H = sum Q_{ij} has a 0-eigenvalue, or it is larger than 1/n^c for some c = O(1). The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding the ground state of 2-local frustration-free Hamiltonians of spin 1/2, a well-studied model believed to capture certain key properties in modern condensed matter physics. While Bravyi has shown that the quantum 2-SAT problem has a classical polynomial-time algorithm, the running time of his algorithm is O(n^4). In this paper we give a classical algorithm with linear running time in the number of local projectors, therefore achieving the best possible complexity.
Quantum SAT
Davis-Putnam Procedure
Linear Time Algorithm
15:1-15:14
Regular Paper
Itai
Arad
Itai Arad
Miklos
Santha
Miklos Santha
Aarthi
Sundaram
Aarthi Sundaram
Shengyu
Zhang
Shengyu Zhang
10.4230/LIPIcs.ICALP.2016.15
B. Aspvall, M. Plass, and R. E. Tarjan. A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inf. Process. Lett., 8(3):121-123, 1979. Erratum: Information Processing Letters 14(4): 195 (1982). URL: http://dx.doi.org/10.1016/0020-0190(79)90002-4.
http://dx.doi.org/10.1016/0020-0190(79)90002-4
S. Bravyi. Efficient algorithm for a quantum analogue of 2-SAT. In K. Mahdavi, D. Koslover, and L. L. Brown, editors, Contemporary Mathematics, volume 536. American Mathematical Society, 2011. URL: http://arxiv.org/abs/quant-ph/0602108.
http://arxiv.org/abs/quant-ph/0602108
J. Chen, X. Chen, R. Duan, Z. Ji, and B. Zeng. No-go theorem for one-way quantum computing on naturally occurring two-level systems. Physical Review A, 83(5):050301, 2011.
S. Cook. The complexity of theorem proving procedures. In Proceedings of the Third Annual ACM Symposium, pages 151-158, New York, 1971. ACM.
M. Davis, G. Logemann, and D. Loveland. A machine program for theorem-proving. Commun. ACM, 5(7):394-397, July 1962. URL: http://dx.doi.org/10.1145/368273.368557.
http://dx.doi.org/10.1145/368273.368557
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http://dx.doi.org/10.1145/321033.321034
N. de Beaudrap and S. Gharibian. A linear time algorithm for quantum 2-SAT. CoRR, abs/1508.07338, 2015. To appear in 31st Conference on Computational Complexity. URL: http://arxiv.org/abs/1508.07338.
http://arxiv.org/abs/1508.07338
J. Eisert, M. Cramer, and M. Plenio. Area laws for the entanglement entropy - a review. Reviews of Modern Physics, 82(277), 2010.
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http://dx.doi.org/10.1137/0205048
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http://dx.doi.org/10.1109/FOCS.2013.86
Z. Ji, Z. Wei, and B. Zeng. Complete characterization of the ground-space structure of two-body frustration-free hamiltonians for qubits. Physical Review A, 84:042338, 2011.
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A. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1):2-30, 2003. URL: http://dx.doi.org/10.1016/S0003-4916(02)00018-0.
http://dx.doi.org/10.1016/S0003-4916(02)00018-0
A. Kitaev, A. Shen, and M. Vyalyi. Classical and Quantum Computation. American Mathematical Society, Boston, MA, USA, 2002.
M. Krom. The decision problem for a class of first-order formulas in which all disjunctions are binary. Mathematical Logic Quarterly, 13(1-2):15-20, 1967. URL: http://dx.doi.org/10.1002/malq.19670130104.
http://dx.doi.org/10.1002/malq.19670130104
C. Laumann, R. Moessner, A. Scardicchio, and S. Sondhi. Phase transitions and random quantum satisfiability. Quantum Information &Computation, 10(1), 2010.
L. Levin. Universal sequential search problems. Problems of Information Transmission, 9(3):265-266, 1973.
C. Papadimitriou. On selecting a satisfying truth assignment (extended abstract). In 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 1-4 October 1991, pages 163-169, 1991. URL: http://dx.doi.org/10.1109/SFCS.1991.185365.
http://dx.doi.org/10.1109/SFCS.1991.185365
S. Sachdev. Quantum phase transitions. Wiley Online Library, 2007.
G. Vidal, J.-I. Latorre, E. Rico, and A. Kitaev. Entanglement in quantum critical phenomena. Phys. Rev. Lett., 90:227902, Jun 2003. URL: http://dx.doi.org/10.1103/PhysRevLett.90.227902.
http://dx.doi.org/10.1103/PhysRevLett.90.227902
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Optimal Quantum Algorithm for Polynomial Interpolation
We consider the number of quantum queries required to determine the coefficients of a degree-d polynomial over F_q. A lower bound shown independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2 + 1/2 quantum queries are needed to solve this problem with bounded error, whereas an algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We show that the lower bound is achievable: d/2 + 1/2 quantum queries suffice to determine the polynomial with bounded error. Furthermore, we show that d/2 + 1 queries suffice to achieve probability approaching 1 for large q. These upper bounds improve results of Boneh and Zhandry on the insecurity of cryptographic protocols against quantum attacks. We also show that our algorithm’s success probability as a function of the number of queries is precisely optimal. Furthermore, the algorithm can be implemented with gate complexity poly(log(q)) with negligible decrease in the success probability. We end with a conjecture about the quantum query complexity of multivariate polynomial interpolation.
Quantum algorithms
query complexity
polynomial interpolation
finite fields
16:1-16:13
Regular Paper
Andrew M.
Childs
Andrew M. Childs
Wim
van Dam
Wim van Dam
Shih-Han
Hung
Shih-Han Hung
Igor E.
Shparlinski
Igor E. Shparlinski
10.4230/LIPIcs.ICALP.2016.16
Dave Bacon, Childs, Andrew M., and Wim van Dam. From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups. In Proceedings of the 46th IEEE Symposium on Foundations of Computer Science, pages 469-478, 2005. URL: http://arxiv.org/abs/arXiv:quant-ph/0504083.
http://arxiv.org/abs/arXiv:quant-ph/0504083
Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. Journal of the ACM, 48(4):778-797, 2001. URL: http://arxiv.org/abs/arXiv:quant-ph/9802049.
http://arxiv.org/abs/arXiv:quant-ph/9802049
Dan Boneh and Mark Zhandry. Quantum-secure message authentication codes. In Proceedings of Eurocrypt, pages 592-608, 2013.
François Charles and Bjorn Poonen. Bertini irreducibility theorems over finite fields. Journal of the American Mathematical Society, 29:81-94, 2016.
Andrew M. Childs and Wim van Dam. Quantum algorithm for a generalized hidden shift problem. In Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms, pages 1225-1234, 2007. URL: http://arxiv.org/abs/arXiv:quant-ph/0507190.
http://arxiv.org/abs/arXiv:quant-ph/0507190
Andrew M. Childs, Wim van Dam, Shih-Han Hung, and Igor E. Shparlinski. Optimal quantum algorithm for polynomial interpolation, 2015. URL: http://arxiv.org/abs/arXiv:1509.09271v2.
http://arxiv.org/abs/arXiv:1509.09271v2
Thomas Decker, Jan Draisma, and Pawel Wocjan. Efficient quantum algorithm for identifying hidden polynomials. Quantum Information and Computation, 9(3):215-230, 2009. URL: http://arxiv.org/abs/arXiv:0706.1219.
http://arxiv.org/abs/arXiv:0706.1219
Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser. Limit on the speed of quantum computation in determining parity. Physical Review Letters, 81(24):5442-5444, 1998. URL: http://arxiv.org/abs/arXiv:quant-ph/9802045.
http://arxiv.org/abs/arXiv:quant-ph/9802045
Daniel M. Kane and Samuel A. Kutin. Quantum interpolation of polynomials. Quantum Information and Computation, 11(1):95-103, 2011. URL: http://arxiv.org/abs/arXiv:0909.5683.
http://arxiv.org/abs/arXiv:0909.5683
Arnold Knopfmacher and John Knopfmacher. Counting polynomials with a given number of zeros in a finite field. Linear and Multilinear Algebra, 26(4):287-292, 1990.
Serge Lang and André Weil. Number of points of varieties in finite fields. American Journal of Mathematics, 76:819-827, 1954.
David A. Meyer and James Pommersheim. On the uselessness of quantum queries. Theoretical Computer Science, 412(51):7068-7074, 2011. URL: http://arxiv.org/abs/arXiv:1004.1434.
http://arxiv.org/abs/arXiv:1004.1434
Ashley Montanaro. The quantum query complexity of learning multilinear polynomials. Information Processing Letters, 112(11):438-442, 2012. URL: http://arxiv.org/abs/arXiv:1105.3310.
http://arxiv.org/abs/arXiv:1105.3310
Gerald M. Pitstick, João R. Cruz, and Robert J. Mulholland. A novel interpretation of Prony’s method. Proceedings of the IEEE, 76(8):1052-1053, 1988.
Bjorn Poonen. Bertini theorems over finite fields. Annals of Mathematics, 160:1099-1127, 2004.
Jaikumar Radhakrishnan, Pranab Sen, and S. Venkatesh. The quantum complexity of set membership. Algorithmica, pages 462-479, 2002. URL: http://arxiv.org/abs/arXiv:quant-ph/0007021.
http://arxiv.org/abs/arXiv:quant-ph/0007021
Adi Shamir. How to share a secret. Communications of the ACM, 22(11):612-613, 1979.
Wim van Dam. Quantum oracle interrogation: Getting all information for almost half the price. In Proceedings of the 39th IEEE Symposium on Foundations of Computer Science, pages 362-367, 1998. URL: http://arxiv.org/abs/arXiv:quant-ph/9805006.
http://arxiv.org/abs/arXiv:quant-ph/9805006
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Lower Bounds for the Approximate Degree of Block-Composed Functions
We describe a new hardness amplification result for point-wise approximation of Boolean functions by low-degree polynomials.
Specifically, for any function f on N bits, define F(x_1,...,x_M) = OMB(f(x_1),...,f(x_M)) to be the function on M*N bits obtained by block-composing f with a function known as ODD-MAX-BIT. We show that, if f requires large degree to approximate to error 2/3 in a certain one-sided sense (captured by a complexity measure known as positive one-sided approximate degree), then F requires large degree to approximate even to error 1-2^{-M}. This generalizes a result of Beigel (Computational Complexity, 1994), who proved an identical result for the special case f=OR.
Unlike related prior work, our result implies strong approximate degree lower bounds even for many functions F that have low threshold degree. Our proof is constructive: we exhibit a solution to the dual of an appropriate linear program capturing the approximate degree of any function. We describe several applications, including improved separations between the complexity classes P^{NP} and PP in both the query and communication complexity settings. Our separations improve on work of Beigel (1994) and Buhrman, Vereshchagin, and de Wolf (CCC, 2007).
approximate degree
one-sided approximate degree
polynomial approx- imations
threshold degree
communication complexity
17:1-17:15
Regular Paper
Justin
Thaler
Justin Thaler
10.4230/LIPIcs.ICALP.2016.17
S. Aaronson and A. Wigderson. Algebrization: A new barrier in complexity theory. TOCT, 1(1), 2009.
Scott Aaronson and Yaoyun Shi. Quantum lower bounds for the collision and the element distinctness problems. J. ACM, 51(4):595-605, 2004.
Andris Ambainis. Polynomial degree and lower bounds in quantum complexity: Collision and element distinctness with small range. Theory of Computing, 1(1):37-46, 2005.
László Babai, Peter Frankl, and Janos Simon. Complexity classes in communication complexity theory (preliminary version). In FOCS, pages 337-347, 1986.
Theodore P. Baker, John Gill, and Robert Solovay. Relativizations of the P =? NP question. SIAM J. Comput., 4(4):431-442, 1975.
Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. J. ACM, 48(4):778-797, 2001.
Richard Beigel. Perceptrons, PP, and the Polynomial Hierarchy. Computational Complexity, 4:339-349, 1994.
Harry Buhrman, Nikolai K. Vereshchagin, and Ronald de Wolf. On computation and communication with small bias. In CCC, pages 24-32, 2007.
Mark Bun and Justin Thaler. Dual lower bounds for approximate degree and markov-bernstein inequalities. In ICALP (1), pages 303-314, 2013.
Mark Bun and Justin Thaler. Hardness amplification and the approximate degree of constant-depth circuits. In ICALP, Part I, pages 268-280, 2015.
Karthekeyan Chandrasekaran, Justin Thaler, Jonathan Ullman, and Andrew Wan. Faster private release of marginals on small databases. In ITCS, pages 387-402, 2014.
Arkadev Chattopadhyay and Anil Ada. Multiparty communication complexity of disjointness. Electronic Colloquium on Computational Complexity (ECCC), 15(002), 2008.
Matei David and Toniann Pitassi. Separating NOF communication complexity classes RP and NP. Electronic Colloquium on Computational Complexity (ECCC), 15(014), 2008.
Matei David, Toniann Pitassi, and Emanuele Viola. Improved separations between nondeterministic and randomized multiparty communication. TOCT, 1(2), 2009.
R. de Wolf. A note on quantum algorithms and the minimal degree of ε-error polynomials for symmetric functions. Quantum Information & Computation, 8(10):943-950, 2010.
Dmitry Gavinsky and A. A. Sherstov. A separation of NP and conp in multiparty communication complexity. Theory of Computing, 6(1):227-245, 2010.
Dmitry Gavinsky and A. A. Sherstov. A separation of NP and coNP in multiparty communication complexity. Theory of Computing, 6(1):227-245, 2010.
Mika Göös, Toniann Pitassi, and Thomas Watson. The landscape of communication complexity classes. Electronic Colloquium on Computational Complexity (ECCC), 22:49, 2015. To appear in ICALP, 2016.
Russell Impagliazzo and Ryan Williams. Communication complexity with synchronized clocks. In CCC, pages 259-269, 2010.
Adam Tauman Kalai, Adam R. Klivans, Yishay Mansour, and Rocco A. Servedio. Agnostically learning halfspaces. SIAM J. Comput., 37(6):1777-1805, 2008.
Varun Kanade and Justin Thaler. Distribution-independent reliable learning. In COLT, pages 3-24, 2014.
Hartmut Klauck. Lower bounds for quantum communication complexity. SIAM J. Comput., 37(1):20-46, 2007.
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Troy Lee and Adi Shraibman. Disjointness is hard in the multiparty number-on-the-forehead model. Computational Complexity, 18(2):309-336, 2009.
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Nati Linial and Adi Shraibman. Lower bounds in communication complexity based on factorization norms. Random Struct. Algorithms, 34(3):368-394, 2009.
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Periklis A. Papakonstantinou, Dominik Scheder, and Hao Song. Overlays and limited memory communication. In CCC, pages 298-308, 2014.
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Rocco A. Servedio, Li-Yang Tan, and Justin Thaler. Attribute-efficient learning and weight-degree tradeoffs for polynomial threshold functions. In COLT, pages 14.1-14.19, 2012.
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Dynamic Graph Stream Algorithms in o(n) Space
In this paper we study graph problems in dynamic streaming model, where the input is defined by a sequence of edge insertions and deletions. As many natural problems require Omega(n) space, where n is the number of vertices, existing works mainly focused on designing ~O(n) space algorithms. Although sublinear in the number of edges for dense graphs, it could still be too large for many applications (e.g. n is huge or the graph is sparse). In this work, we give single-pass algorithms beating this space barrier for two classes of problems. We present o(n) space algorithms for estimating the number of connected components with additive error epsilon*n and (1 + epsilon)-approximating the weight of minimum spanning tree. The latter improves previous ~O(n) space algorithm given by Ahn et al. (SODA 2012) for connected graphs with bounded edge weights. We initiate the study of approximate graph property testing in the dynamic streaming model, where we want to distinguish graphs satisfying the property from graphs that are epsilon-far from having the property. We consider the problem of testing k-edge connectivity, k-vertex connectivity, cycle-freeness and bipartiteness (of planar graphs), for which, we provide algorithms using roughly ~O(n^{1-epsilon}) space, which is o(n) for any constant epsilon. To complement our algorithms, we present Omega(n^{1-O(epsilon)}) space lower bounds for these problems, which show that such a dependence on epsilon is necessary.
dynamic graph streams
sketching
property testing
minimum spanning tree
18:1-18:16
Regular Paper
Zengfeng
Huang
Zengfeng Huang
Pan
Peng
Pan Peng
10.4230/LIPIcs.ICALP.2016.18
Kook Jin Ahn, Graham Cormode, Sudipto Guha, Andrew McGregor, and Anthony Wirth. Correlation clustering in data streams. In Proceedings of the 32nd International Conference on Machine Learning, ICML, pages 6-11, 2015.
Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 459-467. SIAM, 2012.
Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Graph sketches: sparsification, spanners, and subgraphs. In Proceedings of the 31st symposium on Principles of Database Systems, pages 5-14. ACM, 2012.
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Sepelir Assadi, Sanjeev Khanna, Yang Li, and Grigory Yaroslavtsev. Maximum matchings in dynamic graph streams and the simultaneous communication model. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 1345-1364. SIAM, 2016. URL: http://dl.acm.org/citation.cfm?id=2884435.2884528.
http://dl.acm.org/citation.cfm?id=2884435.2884528
Sayan Bhattacharya, Monika Henzinger, Danupon Nanongkai, and Charalampos E Tsourakakis. Space-and time-efficient algorithm for maintaining dense subgraphs on one-pass dynamic streams. In ACM Symposium on Theory of Computing, 2015.
Marc Bury and Chris Schwiegelshohn. Sublinear estimation of weighted matchings in dynamic data streams. ESA, 2015.
Bernard Chazelle, Ronitt Rubinfeld, and Luca Trevisan. Approximating the minimum spanning tree weight in sublinear time. SIAM Journal on computing, 34(6):1370-1379, 2005.
Rajesh Chitnis, Graham Cormode, Hossein Esfandiari, MohammadTaghi Hajiaghayi, Andrew McGregor, Morteza Monemizadeh, and Sofya Vorotnikova. Kernelization via sampling with applications to dynamic graph streams. SODA, 2016.
Rajesh Chitnis, Graham Cormode, MohammadTaghi Hajiaghayi, and Morteza Monemizadeh. Parameterized streaming: maximal matching and vertex cover. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1234-1251. SIAM, 2015.
Artur Czumaj, Funda Ergün, Lance Fortnow, Avner Magen, Ilan Newman, Ronitt Rubinfeld, and Christian Sohler. Approximating the weight of the euclidean minimum spanning tree in sublinear time. SIAM Journal on Computing, 35(1):91-109, 2005.
Artur Czumaj, Morteza Monemizadeh, Krzysztof Onak, and Christian Sohler. Planar graphs: Random walks and bipartiteness testing. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 423-432. IEEE, 2011.
Artur Czumaj and Christian Sohler. Estimating the weight of metric minimum spanning trees in sublinear time. SIAM Journal on Computing, 39(3):904-922, 2009.
Hossein Esfandiari, Mohammad T Hajiaghayi, Vahid Liaghat, Morteza Monemizadeh, and Krzysztof Onak. Streaming algorithms for estimating the matching size in planar graphs and beyond. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1217-1233. SIAM, 2015.
Hossein Esfandiari, MohammadTaghi Hajiaghayi, and David P Woodruff. Applications of uniform sampling: Densest subgraph and beyond. arXiv preprint arXiv:1506.04505, 2015.
Stefan Fafianie and Stefan Kratsch. Streaming kernelization. In Mathematical Foundations of Computer Science 2014, pages 275-286. Springer, 2014.
Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Theoretical Computer Science, 348(2):207-216, 2005.
Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. Graph distances in the data-stream model. SIAM Journal on Computing, 38(5):1709-1727, 2008.
Gereon Frahling, Piotr Indyk, and Christian Sohler. Sampling in dynamic data streams and applications. In Proceedings of the twenty-first annual symposium on Computational geometry, pages 142-149. ACM, 2005.
Oded Goldreich. Introduction to testing graph properties. In Property testing, pages 105-141. Springer, 2011.
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Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. Algorithmica, 32:302-343, 2002.
Sudipto Guha, Andrew McGregor, and David Tench. Vertex and hyperedge connectivity in dynamic graph streams. In Proceedings of the 34th ACM Symposium on Principles of Database Systems, pages 241-247. ACM, 2015.
Monika Rauch Henzinger, Prabhakar Raghavan, and Sridhar Rajagopalan. Computing on data streams. In External Memory Algorithms, Proceedings of a DIMACS Workshop, New Brunswick, New Jersey, USA, May 20-22, 1998, pages 107-118, 1998.
Hossein Jowhari. Estimating the number of connected components in graph streams. Personal communication.
Michael Kapralov, Sanjeev Khanna, and Madhu Sudan. Approximating matching size from random streams. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 734-751. SIAM, 2014.
Michael Kapralov, Sanjeev Khanna, and Madhu Sudan. Streaming lower bounds for approximating max-cut. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1263-1282. SIAM, 2015.
Michael Kapralov, Yin Tat Lee, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 561-570. IEEE, 2014.
Michael Kapralov and David Woodruff. Spanners and sparsifiers in dynamic streams. In Proceedings of the 2014 ACM symposium on Principles of distributed computing, pages 272-281. ACM, 2014.
Dmitry Kogan and Robert Krauthgamer. Sketching cuts in graphs and hypergraphs. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, pages 367-376. ACM, 2015.
Christian Konrad. Maximum matching in turnstile streams. ESA, 2015.
Andrew McGregor. Graph stream algorithms: A survey. ACM SIGMOD Record, 43(1):9-20, 2014.
Andrew McGregor, David Tench, Sofya Vorotnikova, and Hoa T Vu. Densest subgraph in dynamic graph streams. MFCS, 2015.
S Muthukrishnan. Data streams: Algorithms and applications. Theoretical Computer Science, 1(2):117-236, 2005.
Yaron Orenstein and Dana Ron. Testing eulerianity and connectivity in directed sparse graphs. Theoretical Computer Science, 412(45):6390-6408, 2011.
Michal Parnas and Dana Ron. Testing the diameter of graphs. Random Structures &Algorithms, 20(2):165-183, 2002.
Eric Price. Efficient sketches for the set query problem. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 41-56. SIAM, 2011.
Dana Ron. Algorithmic and analysis techniques in property testing. Foundations and Trendsregistered in Theoretical Computer Science, 5(2):73-205, 2010.
Ronitt Rubinfeld and Asaf Shapira. Sublinear time algorithms. SIAM Journal on Discrete Mathematics, 25(4):1562-1588, 2011.
Xiaoming Sun and David P. Woodruff. Tight bounds for graph problems in insertion streams. In The 18th. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX'2015), 2015.
Elad Verbin and Wei Yu. The streaming complexity of cycle counting, sorting by reversals, and other problems. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 11-25. SIAM, 2011.
Yuichi Yoshida and Hiro Ito. Property testing on k-vertex-connectivity of graphs. In Automata, Languages and Programming, pages 539-550. Springer, 2008.
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Diameter and k-Center in Sliding Windows
In this paper we develop streaming algorithms for the diameter problem and the k-center clustering problem in the sliding window model. In this model we are interested in maintaining a solution for the N most recent points of the stream. In the diameter problem we would like to maintain two points whose distance approximates the diameter of the point set in the window. Our algorithm computes a (3 + epsilon)-approximation and uses O(1/epsilon*ln(alpha)) memory cells, where alpha is the ratio of the largest and smallest distance and is assumed to be known in advance. We also prove that under reasonable assumptions obtaining a (3 - epsilon)-approximation requires Omega(N1/3) space.
For the k-center problem, where the goal is to find k centers that minimize the maximum distance of a point to its nearest center, we obtain a (6 + epsilon)-approximation using O(k/epsilon*ln(alpha)) memory cells and a (4 + epsilon)-approximation for the special case k = 2. We also prove that any algorithm for the 2-center problem that achieves an approximation ratio of less than 4 requires Omega(N^{1/3}) space.
Streaming
k-Center
Diameter
Sliding Windows
19:1-19:12
Regular Paper
Vincent
Cohen-Addad
Vincent Cohen-Addad
Chris
Schwiegelshohn
Chris Schwiegelshohn
Christian
Sohler
Christian Sohler
10.4230/LIPIcs.ICALP.2016.19
Pankaj K. Agarwal, Jirí Matousek, and Subhash Suri. Farthest neighbors, maximum spanning trees and related problems in higher dimensions. Comput. Geom., 1:189-201, 1991. URL: http://dx.doi.org/10.1016/0925-7721(92)90001-9.
http://dx.doi.org/10.1016/0925-7721(92)90001-9
Pankaj K. Agarwal and R. Sharathkumar. Streaming algorithms for extent problems in high dimensions. Algorithmica, 72(1):83-98, 2015. URL: http://dx.doi.org/10.1007/s00453-013-9846-4.
http://dx.doi.org/10.1007/s00453-013-9846-4
Brian Babcock, Mayur Datar, Rajeev Motwani, and Liadan O'Callaghan. Maintaining variance and k-medians over data stream windows. In Proceedings of the Twenty-Second ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, June 9-12, 2003, San Diego, CA, USA, pages 234-243, 2003. URL: http://dx.doi.org/10.1145/773153.773176.
http://dx.doi.org/10.1145/773153.773176
Vladimir Braverman, Harry Lang, Keith Levin, and Morteza Monemizadeh. Clustering on sliding windows in polylogarithmic space. In 35th IARCS Annual Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2015, December 16-18, 2015, Bangalore, India, pages 350-364, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2015.350.
http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2015.350
Vladimir Braverman, Harry Lang, Keith Levin, and Morteza Monemizadeh. Clustering problems on sliding windows. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1374-1390, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch95.
http://dx.doi.org/10.1137/1.9781611974331.ch95
Vladimir Braverman and Rafail Ostrovsky. Effective computations on sliding windows. SIAM J. Comput., 39(6):2113-2131, 2010. URL: http://dx.doi.org/10.1137/090749281.
http://dx.doi.org/10.1137/090749281
Timothy M. Chan and Vinayak Pathak. Streaming and dynamic algorithms for minimum enclosing balls in high dimensions. Comput. Geom., 47(2):240-247, 2014. URL: http://dx.doi.org/10.1016/j.comgeo.2013.05.007.
http://dx.doi.org/10.1016/j.comgeo.2013.05.007
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http://dx.doi.org/10.1142/S0218195906001975
Moses Charikar, Chandra Chekuri, Tomás Feder, and Rajeev Motwani. Incremental clustering and dynamic information retrieval. In Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 626-635, 1997. URL: http://dx.doi.org/10.1145/258533.258657.
http://dx.doi.org/10.1145/258533.258657
Moses Charikar, Liadan O'Callaghan, and Rina Panigrahy. Better streaming algorithms for clustering problems. In Proc. of the 35th Annual ACM Symp. on Theory of Computing, June 9-11, 2003, San Diego, CA, USA, pages 30-39, 2003. URL: http://dx.doi.org/10.1145/780542.780548.
http://dx.doi.org/10.1145/780542.780548
Michael S. Crouch, Andrew McGregor, and Daniel Stubbs. Dynamic graphs in the sliding-window model. In Algorithms - ESA 2013 - 21st Annual European Symposium, Sophia Antipolis, France, September 2-4, 2013. Proceedings, pages 337-348, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40450-4_29.
http://dx.doi.org/10.1007/978-3-642-40450-4_29
Mayur Datar, Aristides Gionis, Piotr Indyk, and Rajeev Motwani. Maintaining stream statistics over sliding windows. SIAM J. Comput., 31(6):1794-1813, 2002. URL: http://dx.doi.org/10.1137/S0097539701398363.
http://dx.doi.org/10.1137/S0097539701398363
Joan Feigenbaum, Sampath Kannan, and Jian Zhang. Computing diameter in the streaming and sliding-window models. Algorithmica, 41(1):25-41, 2004. URL: http://dx.doi.org/10.1007/s00453-004-1105-2.
http://dx.doi.org/10.1007/s00453-004-1105-2
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http://dx.doi.org/10.1145/5925.5933
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Sang-Sub Kim and Hee-Kap Ahn. An improved data stream algorithm for clustering. Comput. Geom., 48(9):635-645, 2015. URL: http://dx.doi.org/10.1016/j.comgeo.2015.06.003.
http://dx.doi.org/10.1016/j.comgeo.2015.06.003
Richard Matthew McCutchen and Samir Khuller. Streaming algorithms for k-center clustering with outliers and with anonymity. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, 11th International Workshop, APPROX 2008, and 12th International Workshop, RANDOM 2008, Boston, MA, USA, August 25-27, 2008. Proceedings, pages 165-178, 2008. URL: http://dx.doi.org/10.1007/978-3-540-85363-3_14.
http://dx.doi.org/10.1007/978-3-540-85363-3_14
Hamid Zarrabi-Zadeh. Core-preserving algorithms. In Proc. of the 20th Annual Canadian Conf. on Computational Geometry, Montréal, Canada, August 13-15, 2008, 2008.
Hamid Zarrabi-Zadeh. An almost space-optimal streaming algorithm for coresets in fixed dimensions. Algorithmica, 60(1):46-59, 2011. URL: http://dx.doi.org/10.1007/s00453-010-9392-2.
http://dx.doi.org/10.1007/s00453-010-9392-2
Creative Commons Attribution 3.0 Unported license
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Approximate Hamming Distance in a Stream
We consider the problem of computing a (1+epsilon)-approximation of the Hamming distance between a pattern of length n and successive substrings of a stream. We first look at the one-way randomised communication complexity of this problem. We show the following:
- If Alice and Bob both share the pattern and Alice has the first half of the stream and Bob the second half, then there is an O(epsilon^{-4}*log^2(n)) bit randomised one-way communication protocol.
- If Alice has the pattern, Bob the first half of the stream and Charlie the second half, then there is an O(epsilon^{-2}*sqrt(n)*log(n)) bit randomised one-way communication protocol. We then go on to develop small space streaming algorithms for (1 + epsilon)-approximate Hamming distance which give worst case running time guarantees per arriving symbol.
- For binary input alphabets there is an O(epsilon^{-3}*sqrt(n)*log^2(n)) space and O(epsilon^{-2}*log(n)) time streaming
(1 + epsilon)-approximate Hamming distance algorithm.
- For general input alphabets there is an O(epsilon^{-5}*sqrt(n)*log^4(n)) space and O(epsilon^{-4}*log^3(n)) time streaming
(1 + epsilon)-approximate Hamming distance algorithm.
Hamming distance
communication complexity
data stream model
20:1-20:14
Regular Paper
Raphaël
Clifford
Raphaël Clifford
Tatiana
Starikovskaya
Tatiana Starikovskaya
10.4230/LIPIcs.ICALP.2016.20
Karl Abrahamson. Generalized string matching. SIAM Journal on Computing, 16(6):1039-1051, 1987.
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Tsvi Kopelowitz and Ely Porat. Breaking the variance: Approximating the Hamming distance in 1/ε time per alignment. In FOCS'15: Proc. 56superscriptth Annual Symp. Foundations of Computer Science, pages 601-613, 2015.
S. Rao Kosaraju. Efficient string matching. Manuscript, 1987.
Gad M. Landau and Uzi Vishkin. Fast string matching with k differences. Journal of Computer System Sciences, 37(1):63-78, 1988.
Ely Porat. Personal communication, 2016.
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Price of Competition and Dueling Games
We study competition in a general framework introduced by Immorlica, Kalai, Lucier, Moitra, Postlewaite, and Tennenholtz and answer their main open question. Immorlica et al. considered classic optimization problems in terms of competition and introduced a general class of games called dueling games. They model this competition as a zero-sum game, where two players are competing for a user’s satisfaction. In their main and most natural game, the ranking duel, a user requests a webpage by submitting a query and players output an ordering over all possible webpages based on the submitted query. The user tends to choose the ordering which displays her requested webpage in a higher rank. The goal of both players is to maximize the probability that her ordering beats that of her opponent and gets the user's attention. Immorlica et al. show this game directs both players to provide suboptimal search results. However, they leave the following as their main open question: "does competition between algorithms improve or degrade expected performance?" (see the introduction for more quotes) In this paper, we resolve this question for the ranking duel and a more general class of dueling games.
More precisely, we study the quality of orderings in a competition between two players. This game is a zero-sum game, and thus any Nash equilibrium of the game can be described by minimax strategies. Let the value of the user for an ordering be a function of the position of her requested item in the corresponding ordering, and the social welfare for an ordering be the expected value of the corresponding ordering for the user. We propose the price of competition which is the ratio of the social welfare for the worst minimax strategy to the social welfare obtained by asocial planner. Finding the price of competition is another approach to obtain structural results of Nash equilibria. We use this criterion for analyzing the quality of orderings in the ranking duel. Although Immorlica et al. show that the competition leads to suboptimal strategies, we prove the quality of minimax results is surprisingly close to that of the optimum solution. In particular, via a novel factor-revealing LP for computing price of anarchy, we prove if the value of the user for an ordering is a linear function of its position, then the price of competition is at least 0.612 and bounded above by 0.833. Moreover we consider the cost minimization version of the problem. We prove, the social cost of the worst minimax strategy is at most 3 times the optimal social cost.
Last but not least, we go beyond linear valuation functions and capture the main challenge for bounding the price of competition for any arbitrary valuation function. We present a principle which states that the lower bound for the price of competition for all 0-1 valuation functions is the same as the lower bound for the price of competition for all possible valuation functions. It is worth mentioning that this principle not only works for the ranking duel but also for all dueling games. This principle says, in any dueling game, the most challenging part of bounding the price of competition is finding a lower bound for 0-1 valuation functions. We leverage this principle to show that the price of competition is at least 0.25 for the generalized ranking duel.
POC
POA
Dueling games
Nash equilibria
sponsored search
21:1-21:14
Regular Paper
Sina
Dehghani
Sina Dehghani
Mohammad Taghi
Hajiaghayi
Mohammad Taghi Hajiaghayi
Hamid
Mahini
Hamid Mahini
Saeed
Seddighin
Saeed Seddighin
10.4230/LIPIcs.ICALP.2016.21
Gagan Aggarwal, Jon Feldman, S. Muthukrishnan, and Martin Pál. Sponsored search auctions with markovian users. In WINE, pages 621-628. 2008.
AmirMahdi Ahmadinejad, Sina Dehghani, MohammadTaghi Hajiaghayi, Hamid Mahini, Saeed Seddighin, and Sadra Yazdanbod. Forming external behaviors by leveraging internal opinions. In Computer Communications (INFOCOM), 2015 IEEE Conference on, pages 1849-1857. IEEE, 2015.
Mahdi Ahmadinejad, Sina Dehghani, MohammadTaghi Hajiaghayi, Brendan Lucier, Hamid Mahini, and Saeed Seddighin. From duels to battefields: Computing equilibria of blotto and other games. AAAI 2016, 2016.
Susanne Albers, Stefan Eilts, Eyal Even-Dar, Yishay Mansour, and Liam Roditty. On nash equilibria for a network creation game. In SODA, pages 89-98, 2006.
Noga Alon, Erik D. Demaine, Mohammad T. Hajiaghayi, and Tom Leighton. Basic network creation games. SIAM Journal on Discrete Mathematics, 27(2):656-668, 2013.
Nir Andelman, Michal Feldman, and Yishay Mansour. Strong price of anarchy. In SODA, pages 189-198, 2007.
Itai Ashlagi, Piotr Krysta, and Moshe Tennenholtz. Social context games. In Internet and Network Economics, pages 675-683. Springer, 2008.
Claude Aspremont, J. Jaskold Gabszewicz, and J.-F. Thisse. On hotelling’s" stability in competition". Econometrica: Journal of the Econometric Society, pages 1145-1150, 1979.
Susan Athey and Glenn Ellison. Position auctions with consumer search. Technical Report 15253, National Bureau of Economic Research, 2009.
J. Bertrand. Book review of théorie mathématique de la richesse sociale and of recherches sur les principes mathématiques de la théorie des richesses. Journal de Savants, 67:499-508, 1883.
Felix Brandt, Felix Fischer, Paul Harrenstein, and Yoav Shoham. Ranking games. Artificial Intelligence, 173(2):221-239, 2009.
George Christodoulou and Elias Koutsoupias. The price of anarchy of finite congestion games. In STOC, pages 67-73, 2005.
Thomas H. Cormen, Clifford Stein, Ronald L. Rivest, and Charles E. Leiserson. Introduction to Algorithms. McGraw-Hill Higher Education, 2nd edition, 2001.
Erik D Demaine, MohammadTaghi Hajiaghayi, Hamid Mahini, and Morteza Zadimoghaddam. The price of anarchy in network creation games. In PODC, pages 292-298, 2007.
Benjamin Edelman, Michael Ostrovsky, and Michael Schwarz. Internet advertising and the generalized second-price auction: Selling billions of dollars worth of keywords. American Economic Review, 97(1):242-259, 2007.
Alex Fabrikant, Ankur Luthra, Elitza Maneva, Christos H. Papadimitriou, and Scott Shenker. On a network creation game. In PODC, pages 347-351, 2003.
Anindya Ghose and Sha Yang. An empirical analysis of search engine advertising: Sponsored search in electronic markets. Management Science, 55(10):1605-1622, 2009.
Harold Hotelling. Stability in competition. The Economic Journal, 39(153):41-57, 1929.
Nicole Immorlica, Adam Tauman Kalai, Brendan Lucier, Ankur Moitra, Andrew Postlewaite, and Moshe Tennenholtz. Dueling algorithms. In STOC, pages 215-224, 2011.
Nicole Immorlica, Li Erran Li, Vahab S. Mirrokni, and Andreas S. Schulz. Coordination mechanisms for selfish scheduling. Theoretical Computer Science, 410(17):1589-1598, 2009.
David Kempe and Brendan Lucier. User satisfaction in competitive sponsored search. In WWW, pages 699-710, 2014.
David Kempe and Mohammad Mahdian. A cascade model for externalities in sponsored search. In Internet and Network Economics, pages 585-596. 2008.
Elias Koutsoupias and Christos Papadimitriou. Worst-case equilibria. In STACS, pages 404-413, 1999.
David M Kreps. A course in microeconomic theory. Harvester Wheatsheaf New York, 1990.
Andreu Mas-Colell, Michael Dennis Whinston, and Jerry R. Green. Microeconomic theory. Oxford university press New York, 1995.
Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani. Algorithmic Game Theory. Cambridge University Press, 2007.
Tim Roughgarden. Selfish routing and the price of anarchy, volume 174. MIT press Cambridge, 2005.
Rahul Telang, Uday Rajan, and Tridas Mukhopadhyay. The market structure for internet search engines. Journal of Management Information Systems, 21(2):137-160, 2004.
Hal R Varian and Jack Repcheck. Intermediate microeconomics: a modern approach. WW Norton &Company New York, NY, 8th edition, 2010.
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https://creativecommons.org/licenses/by/3.0/legalcode
Popular Half-Integral Matchings
In an instance G = (A union B, E) of the stable marriage problem with strict and possibly incomplete preference lists, a matching M is popular if there is no matching M0 where the vertices that prefer M' to M outnumber those that prefer M to M'. All stable matchings are popular and there is a simple linear time algorithm to compute a maximum-size popular matching. More generally, what we seek is a min-cost popular matching where we assume there is a cost function c : E -> Q. However there is no polynomial time algorithm currently known for solving this problem. Here we consider the following generalization of a popular matching called a popular half-integral matching: this is a fractional matching ~x = (M_1 + M_2)/2, where M1 and M2 are the 0-1 edge incidence vectors of matchings in G, such that ~x satisfies popularity constraints. We show that every popular half-integral matching is equivalent to a stable matching in a larger graph G^*. This allows us to solve the min-cost popular half-integral matching problem in polynomial time.
bipartite graphs
stable matchings
fractional matchings
polytopes
22:1-22:13
Regular Paper
Telikepalli
Kavitha
Telikepalli Kavitha
10.4230/LIPIcs.ICALP.2016.22
D. J. Abraham, R. W. Irving, T. Kavitha, and K. Mehlhorn. Popular matchings. SIAM Journal on Computing, 37(4):1030-1045, 2007.
P. Biró, R. W. Irving, and D. F. Manlove. Popular matchings in the marriage and roommates problems. In Proceedings of 7th International Conference on Algorithms and Complexity (CIAC), pages 97-108, 2010.
Á. Cseh, C.-C. Huang, and T. Kavitha. Popular matchings with two-sided preferences and one-sided ties. In Proceedings of 42nd International Colloquium on Automata, Languages, and Programming (ICALP), pages 367-379, 2015.
Á. Cseh and T. Kavitha. Popular edges and dominant matchings. To appear in the Proceedings of the 18th Conference on Integer Programming and Combinatorial Optimization (IPCO), 2016.
T. Feder. A new fixed point approach for stable networks and stable marriages. Journal of Computer and System Sciences, 45:233-284, 1992.
T. Feder. Network flow and 2-satisfiability. Algorithmica, 11(3):291-319, 1994.
D. Gale and L.S. Shapley. College admissions and the stability of marriage. American Mathematical Monthly, 69:9-15, 1962.
D. Gale and M. Sotomayor. Some remarks on the stable matching problem. Discrete Applied Mathematics, 11:223-232, 1985.
P. Gärdenfors. Match making: assignments based on bilateral preferences. Behavioural Sciences, 20:166-173, 1975.
C.-C. Huang and T. Kavitha. Popular matchings in the stable marriage problem. Information and Computation, 222:180-194, 2013.
R. W. Irving, P. Leather, and D. Gusfield. An efficient algorithm for the "optimal" stable marriage. Journal of the ACM, 38(3):532-543, 1987.
T. Kavitha. A size-popularity tradeoff in the stable marriage problem. SIAM Journal on Computing, 43(1):52-71, 2014.
T. Kavitha, J. Mestre, and M. Nasre. Popular mixed matchings. Theoretical Computer Science, 412:2679-2690, 2011.
U. Rothblum. Characterization of stable matchings as extreme points of a polytope. Mathematical Programming, 54:57-67, 1992.
C.-P. Teo and J. Sethuraman. The geometry of fractional stable matchings and its applications. Mathematics of Operations Research, 23(4):874-891, 1998.
J. H. Vande Vate. Linear programming brings marital bliss. Operations Research Letters, 8(3):147-153, 1989.
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https://creativecommons.org/licenses/by/3.0/legalcode
Voronoi Choice Games
We study novel variations of Voronoi games and associated random processes that we call Voronoi choice games. These games provide a rich framework for studying questions regarding the power of small numbers of choices in multi-player, competitive scenarios, and they further lead to many interesting, non-trivial random processes that appear worthy of study.
As an example of the type of problem we study, suppose a group of n miners (or players) are staking land claims through the following process: each miner has m associated points independently and uniformly distributed on an underlying space (such as the unit circle, the unit square, or the unit torus), so the kth miner will have associated points p_{k1}, p_{k2}, ..., p_{km}. We generally here think of m as being a small constant, such as 2. Each miner chooses one of these points as the base point for their claim. Each miner obtains mining rights for the area of the square that is closest to their chosen base; that is, they obtain the Voronoi cell corresponding to their chosen point in the Voronoi diagram of the n chosen points. Each player's goal is simply to maximize the amount of land under their control. What can we say about the players’ strategy and the equilibria of such games?
In our main result, we derive bounds on the expected number of pure Nash equilibria for a variation of the 1-dimensional game on the circle where a player owns the arc starting from their point and moving clockwise to the next point. This result uses interesting properties of random arc lengths on circles, and demonstrates the challenges in analyzing these kinds of problems. We also provide several other related results. In particular, for the 1-dimensional game on the circle, we show that a pure Nash equilibrium always exists when each player owns the part of the circle nearest to their point, but it is NP-hard to determine whether a pure Nash equilibrium exists in the variant when each player owns the arc starting from their point clockwise to the next point. This last result, in part, motivates our examination of the random setting.
Voronoi games
correlated equilibria
power of two choices
Hotelling model
23:1-23:13
Regular Paper
Meena
Boppana
Meena Boppana
Rani
Hod
Rani Hod
Michael
Mitzenmacher
Michael Mitzenmacher
Tom
Morgan
Tom Morgan
10.4230/LIPIcs.ICALP.2016.23
Dimitris Achlioptas, Raissa M D'Souza, and Joel Spencer. Explosive percolation in random networks. Science, 323(5920):1453-1455, 2009.
Robert J Aumann. Subjectivity and correlation in randomized strategies. Journal of Mathematical Economics, 1(1):67-96, 1974.
Yossi Azar, Andrei Z Broder, Anna R Karlin, and Eli Upfal. Balanced allocations. SIAM journal on computing, 29(1):180-200, 1999.
Avrim Blum, MohammadTaghi Hajiaghayi, Katrina Ligett, and Aaron Roth. Regret minimization and the price of total anarchy. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pages 373-382. ACM, 2008.
Meena Boppana, Rani Hod, Michael Mitzenmacher, and Tom Morgan. Voronoi choice games, 2016. URL: http://arxiv.org/abs/1604.07084.
http://arxiv.org/abs/1604.07084
John Byers, Jeffrey Considine, and Michael Mitzenmacher. Simple load balancing for distributed hash tables. In Peer-to-peer Systems II, pages 80-87. Springer, 2003.
Christoph Dürr and Nguyen Kim Thang. Nash equilibria in voronoi games on graphs. In Proceedings of the 5th Annual European Symposium on Algorithms, pages 17-28. Springer, 2007.
Gaëtan Fournier and Marco Scarsini. Hotelling games on networks: efficiency of equilibria. Available at SSRN 2423345, 2014.
J. J. Gabszewicz and J.-F. Thisse. Location. In Handbook of Game Theory with Economic Applications. Volume 1, Chapter 9. R. Aumann and S. Hart, editors. Elsevier Science Publishers, 1992.
Lars Holst. On the lengths of the pieces of a stick broken at random. Journal of Applied Probability, pages 623-634, 1980.
Harold Hotelling. Stability in competition. The Economic Journal, 39(153):41-57, 1929.
Albert Xin Jiang and Kevin Leyton-Brown. Polynomial-time computation of exact correlated equilibrium in compact games. Games and Economic Behavior, 21(1-2):183-202, 2013.
Masashi Kiyomi, Toshiki Saitoh, and Ryuhei Uehara. Voronoi game on a path. IEICE TRANSACTIONS on Information and Systems, 94(6):1185-1189, 2011.
Marios Mavronicolas, Burkhard Monien, Vicky G Papadopoulou, and Florian Schoppmann. Voronoi games on cycle graphs. In Proceedings of Mathematical Foundations of Computer Science, pages 503-514. Springer, 2008.
M Mitzenmacher, Andrea W Richa, and R Sitaraman. The power of two random choices: A survey of techniques and results. In Handbook of Randomized Computing, pages 255-312, 2000.
John Nash. Non-cooperative games. The Annals of Mathematics, 54(2):286-295, 1951.
Christos H Papadimitriou and Tim Roughgarden. Computing correlated equilibria in multi-player games. Journal of the ACM, 55(3):14, 2008.
Sachio Teramoto, Erik D Demaine, and Ryuhei Uehara. Voronoi game on graphs and its complexity. In IEEE Symposium on Computational Intelligence and Games, pages 265-271. IEEE, 2006.
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The Complexity of Hex and the Jordan Curve Theorem
The Jordan curve theorem and Brouwer's fixed-point theorem are fundamental problems in topology. We study their computational relationship, showing that a stylized computational version of Jordan’s theorem is PPAD-complete, and therefore in a sense computationally equivalent to Brouwer’s theorem. As a corollary, our computational result implies that these two theorems directly imply each other mathematically, complementing Maehara's proof that Brouwer implies Jordan [Maehara, 1984]. We then turn to the combinatorial game of Hex which is related to Jordan's theorem, and where the existence of a winner can be used to show Brouwer's theorem [Gale,1979]. We establish that determining who won an (implicitly encoded) play of Hex is PSPACE-complete by adapting a reduction (due to Goldberg [Goldberg,2015]) from Quantified Boolean Formula (QBF). As this problem is analogous to evaluating the output of a canonical path-following algorithm for finding a Brouwer fixed point - and which is known to be PSPACE-complete [Goldberg/Papadimitriou/Savani, 2013] - we thereby establish a connection between Brouwer, Jordan and Hex higher in the complexity hierarchy.
Jordan
Brouwer
Hex
PPAD
PSPACE
24:1-24:14
Regular Paper
Aviv
Adler
Aviv Adler
Constantinos
Daskalakis
Constantinos Daskalakis
Erik D.
Demaine
Erik D. Demaine
10.4230/LIPIcs.ICALP.2016.24
Aviv Adler, Constantinos Daskalakis, and Erik Demaine. The Complexity of Hex and the Jordan Curve Theorem. Arxiv, 2016.
Xi Chen and Xiaotie Deng. On the Complexity of 2D Discrete Fixed Point Problem. In the 33rd International Colloquium on Automata, Languages and Programming (ICALP), 2006.
Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. The Complexity of Computing a Nash Equilibrium. In the 38th Annual ACM Symposium on Theory of Computing (STOC), 2006.
Constantinos Daskalakis and Christos H. Papadimitriou. Continuous local search. In the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2011.
Kousha Etessami and Mihalis Yannakakis. On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract). In the 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2007.
David Gale. The game of Hex and the Brouwer fixed-point theorem. American Mathematical Monthly, pages 818-827, 1979.
Paul Goldberg. The Complexity of the Path-following Solutions of Two-dimensional Sperner/Brouwer Functions. arXiv, 2015.
Paul W Goldberg, Christos H Papadimitriou, and Rahul Savani. The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions. ACM Transactions on Economics and Computation, 1(2):9, 2013.
Camille Jordan. Cours d'analyse de l'École polytechnique, volume 1. Gauthier-Villars et fils, 1893.
Ryuji Maehara. The Jordan curve theorem via the Brouwer fixed point theorem. American Mathematical Monthly, pages 641-643, 1984.
Christos H. Papadimitriou. On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence. Journal of Computer and System Sciences, 48(3):498-532, 1994.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Fractals for Kernelization Lower Bounds, With an Application to Length-Bounded Cut Problems
Bodlaender et al.'s [Bodlaender/Jansen/Kratsch,2014] cross-composition technique is a popular method for excluding polynomial-size problem kernels for NP-hard parameterized problems. We present a new technique exploiting triangle-based fractal structures for extending the range of applicability of cross-compositions. Our technique makes it possible to prove new no-polynomial-kernel results for a number of problems dealing with length-bounded cuts. Roughly speaking, our new technique combines the advantages of serial and parallel composition. In particular, answering an open question of Golovach and Thilikos [Golovach/Thilikos,2011], we show that, unless NP subseteq coNP/poly, the NP-hard Length-Bounded Edge-Cut problem (delete at most k edges such that the resulting graph has no s-t path of length shorter than l) parameterized by the combination of k and l has no polynomial-size problem kernel. Our framework applies to planar as well as directed variants of the basic problems and also applies to both edge and vertex deletion problems.
Parameterized complexity
polynomial-time data reduction
cross-compositions
lower bounds
graph modification problems
interdiction problems
25:1-25:14
Regular Paper
Till
Fluschnik
Till Fluschnik
Danny
Hermelin
Danny Hermelin
André
Nichterlein
André Nichterlein
Rolf
Niedermeier
Rolf Niedermeier
10.4230/LIPIcs.ICALP.2016.25
Georg Baier, Thomas Erlebach, Alexander Hall, Ekkehard Köhler, Petr Kolman, Ondrej Pangrác, Heiko Schilling, and Martin Skutella. Length-bounded cuts and flows. ACM Transactions on Algorithms, 7(1):4, 2010. URL: http://dx.doi.org/10.1145/1868237.1868241.
http://dx.doi.org/10.1145/1868237.1868241
Cristina Bazgan, Morgan Chopin, Marek Cygan, Michael R. Fellows, Fedor V. Fomin, and Erik Jan van Leeuwen. Parameterized complexity of firefighting. Journal of Computer and System Sciences, 80(7):1285-1297, 2014.
Cristina Bazgan, André Nichterlein, and Rolf Niedermeier. A refined complexity analysis of finding the most vital edges for undirected shortest paths. In Proc. of the 9th International Conference on Algorithms and Complexity (CIAC 2015), volume 9079 of LNCS, pages 47-60. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-18173-8_3.
http://dx.doi.org/10.1007/978-3-319-18173-8_3
René van Bevern, Robert Bredereck, Morgan Chopin, Sepp Hartung, Falk Hüffner, André Nichterlein, and Ondřej Suchý. Parameterized complexity of dag partitioning. In Proc. of the 8th International Conference on Algorithms and Complexity (CIAC'13), volume 7878 of LNCS, pages 49-60. Springer, 2013.
Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. Journal of Computer and System Sciences, 75(8):423-434, 2009. URL: http://dx.doi.org/10.1016/j.jcss.2009.04.001.
http://dx.doi.org/10.1016/j.jcss.2009.04.001
Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM Journal on Discrete Mathematics, 28(1):277-305, 2014. URL: http://dx.doi.org/10.1137/120880240.
http://dx.doi.org/10.1137/120880240
Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theoretical Computer Science, 412(35):4570-4578, 2011.
Liming Cai, Jianer Chen, Rodney G. Downey, and Michael R. Fellows. Advice classes of parameterized tractability. Annals of Pure and Applied Logic, 84(1):119-138, 1997.
Reinhard Diestel. Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer, 4th edition, 2010.
Michael Dom, Daniel Lokshtanov, and Saket Saurabh. Kernelization lower bounds through colors and ids. ACM Transactions on Algorithms, 11(2):13, 2014. URL: http://dx.doi.org/10.1145/2650261.
http://dx.doi.org/10.1145/2650261
Pavel Dvořák and Dušan Knop. Parametrized complexity of length-bounded cuts and multi-cuts. In Proc. of the 12th Annual Conference on Theory and Applications of Models of Computation (TAMC 2015), volume 9076 of LNCS, pages 441-452. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-17142-5_37.
http://dx.doi.org/10.1007/978-3-319-17142-5_37
Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. Journal of Computer and System Sciences, 77(1):91-106, 2011. URL: http://dx.doi.org/10.1016/j.jcss.2010.06.007.
http://dx.doi.org/10.1016/j.jcss.2010.06.007
Petr A. Golovach and Dimitrios M. Thilikos. Paths of bounded length and their cuts: Parameterized complexity and algorithms. Discrete Optimization, 8(1):72-86, 2011. URL: http://dx.doi.org/10.1016/j.disopt.2010.09.009.
http://dx.doi.org/10.1016/j.disopt.2010.09.009
Jiong Guo and Rolf Niedermeier. Invitation to data reduction and problem kernelization. ACM SIGACT News, 38(1):31-45, 2007.
Venkatesan Guruswami and Euiwoong Lee. Inapproximability of feedback vertex set for bounded length cycles. Electronic Colloquium on Computational Complexity (ECCC), 21:6, 2014.
Alon Itai, Yehoshua Perl, and Yossi Shiloach. The complexity of finding maximum disjoint paths with length constraints. Networks, 12(3):277-286, 1982. URL: http://dx.doi.org/10.1002/net.3230120306.
http://dx.doi.org/10.1002/net.3230120306
Stefan Kratsch. Recent developments in kernelization: A survey. Bulletin of the EATCS, 113:58-97, 2014.
Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS, 105:41-72, 2011.
Kavindra Malik, Ashok K. Mittal, and Santosh K. Gupta. The k most vital arcs in the shortest path problem. Operations Research Letters, 8(4):223-227, 1989.
Dániel Marx. Parameterized complexity and approximation algorithms. The Computer Journal, 51(1):60-78, 2008.
Karl Menger. Über reguläre Baumkurven. Mathematische Annalen, 96(1):572-582, 1927.
Feng Pan and Aaron Schild. Interdiction problems on planar graphs. Discrete Applied Mathematics, 198:215-231, 2016.
Anneke A. Schoone, Hans L. Bodlaender, and Jan van Leeuwen. Diameter increase caused by edge deletion. Journal of Graph Theory, 11(3):409-427, 1987. URL: http://dx.doi.org/10.1002/jgt.3190110315.
http://dx.doi.org/10.1002/jgt.3190110315
David P. Williamson and David B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, 2011.
Ge Xia and Yong Zhang. On the small cycle transversal of planar graphs. Theoretical Computer Science, 412(29):3501-3509, 2011.
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Kernelization of Cycle Packing with Relaxed Disjointness Constraints
A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e. finding k vertex disjoint cycles in a given graph G, admits no polynomial kernel unless NP subseteq coNP/poly. However, very little is known about this problem beyond the aforementioned kernelization lower bound (within the parameterized complexity framework). In the hope of clarifying the picture and better understanding the types of "constraints" that separate "kernelizable" from "non-kernelizable" variants of Disjoint Cycle Packing, we investigate two relaxations of the problem. The first variant, which we call Almost Disjoint Cycle Packing, introduces a "global" relaxation parameter t. That is, given a graph G and integers k and t, the goal is to find at least k distinct cycles such that every vertex of G appears in at most t of the cycles. The second variant, Pairwise Disjoint Cycle Packing, introduces a "local" relaxation parameter and we seek at least k distinct cycles such that every two cycles intersect in at most t vertices. While the Pairwise Disjoint Cycle Packing problem admits a polynomial kernel for all t >= 1, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower bounds. In particular, for t = k/c, where c could be a function of k, we obtain a kernel of size O(2^{c^{2}}*k^{7+c}*log^3(k)) whenever c in o(sqrt(k))). Thus the kernel size varies from being sub-exponential when c in o(sqrt(k)), to quasipolynomial when c in o(log^l(k)), l in R_+, and polynomial when c in O(1). We complement these results for Almost Disjoint Cycle Packing by showing that the problem does not admit a polynomial kernel whenever t in O(k^{epsilon}), for any 0 <= epsilon < 1.
parameterized complexity
cycle packing
kernelization
relaxation
26:1-26:14
Regular Paper
Akanksha
Agrawal
Akanksha Agrawal
Daniel
Lokshtanov
Daniel Lokshtanov
Diptapriyo
Majumdar
Diptapriyo Majumdar
Amer E.
Mouawad
Amer E. Mouawad
Saket
Saurabh
Saket Saurabh
10.4230/LIPIcs.ICALP.2016.26
Hasan Abasi, Nader H. Bshouty, Ariel Gabizon, and Elad Haramaty. On r-simple k-path. In Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part II, pages 1-12, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44465-8_1.
http://dx.doi.org/10.1007/978-3-662-44465-8_1
Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, December 1996.
Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009. URL: http://dx.doi.org/10.1016/j.jcss.2009.04.001.
http://dx.doi.org/10.1016/j.jcss.2009.04.001
Hans L. Bodlaender and Arie M. C. A. Koster. Combinatorial optimization on graphs of bounded treewidth. Comput. J., 51(3):255-269, May 2008.
Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci., 412(35):4570-4578, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2011.04.039.
http://dx.doi.org/10.1016/j.tcs.2011.04.039
Bruno Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. URL: http://dx.doi.org/10.1016/0890-5401(90)90043-H.
http://dx.doi.org/10.1016/0890-5401(90)90043-H
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
http://dx.doi.org/10.1007/978-3-319-21275-3
Holger Dell and Dániel Marx. Kernelization of packing problems. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 68-81, 2012. URL: http://portal.acm.org/citation.cfm?id=2095122&CFID=63838676&CFTOKEN=79617016.
http://portal.acm.org/citation.cfm?id=2095122&CFID=63838676&CFTOKEN=79617016
Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014. URL: http://dx.doi.org/10.1145/2629620.
http://dx.doi.org/10.1145/2629620
Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
Rod G. Downey and Michael R. Fellows. Parameterized complexity. Springer-Verlag, 1997.
Andrew Drucker. New limits to classical and quantum instance compression. SIAM J. Comput., 44(5):1443-1479, 2015. URL: http://dx.doi.org/10.1137/130927115.
http://dx.doi.org/10.1137/130927115
Paul Erdős and Lajos Pósa. On independent circuits contained in a graph. Canad. Journ. Math, 17(0):347-352, 1965.
Henning Fernau, Alejandro López-Ortiz, and Jazmín Romero. Kernelization algorithms for packing problems allowing overlaps. In Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Singapore, May 18-20, 2015, Proceedings, pages 415-427, 2015. URL: http://dx.doi.org/10.1007/978-3-319-17142-5_35.
http://dx.doi.org/10.1007/978-3-319-17142-5_35
J. Flum and M. Grohe. Parameterized Complexity Theory. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2006.
Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci., 77(1):91-106, 2011. URL: http://dx.doi.org/10.1016/j.jcss.2010.06.007.
http://dx.doi.org/10.1016/j.jcss.2010.06.007
Ariel Gabizon, Daniel Lokshtanov, and Michal Pilipczuk. Fast algorithms for parameterized problems with relaxed disjointness constraints. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 545-556, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_46.
http://dx.doi.org/10.1007/978-3-662-48350-3_46
Danny Hermelin, Stefan Kratsch, Karolina Soltys, Magnus Wahlström, and Xi Wu. A completeness theory for polynomial (turing) kernelization. Algorithmica, 71(3):702-730, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9910-8.
http://dx.doi.org/10.1007/s00453-014-9910-8
Danny Hermelin and Xi Wu. Weak compositions and their applications to polynomial lower bounds for kernelization. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 104-113, 2012. URL: http://portal.acm.org/citation.cfm?id=2095125&CFID=63838676&CFTOKEN=79617016.
http://portal.acm.org/citation.cfm?id=2095125&CFID=63838676&CFTOKEN=79617016
Stefan Kratsch. Recent developments in kernelization: A survey. Bulletin of the EATCS, 113, 2014. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/285.
http://eatcs.org/beatcs/index.php/beatcs/article/view/285
Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Kernelization - preprocessing with a guarantee. In The Multivariate Algorithmic Revolution and Beyond - Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, pages 129-161, 2012. URL: http://dx.doi.org/10.1007/978-3-642-30891-8_10.
http://dx.doi.org/10.1007/978-3-642-30891-8_10
Daniel Lokshtanov, Fahad Panolan, M.S. Ramanujan, and Saket Saurabh. Lossy kernelization. arXiv:1604.04111, 2016.
Rolf Niedermeier. Invitation to fixed-parameter algorithms. Oxford University Press, Oxford, 2006.
Jan Ramon and Siegfried Nijssen. Polynomial-delay enumeration of monotonic graph classes. The Journal of Machine Learning Research, 10:907-929, 2009.
Daniel Ratner and Manfred K. Warmuth. NxN puzzle and related relocation problem. J. Symb. Comput., 10(2):111-138, 1990. URL: http://dx.doi.org/10.1016/S0747-7171(08)80001-6.
http://dx.doi.org/10.1016/S0747-7171(08)80001-6
Jazmín Romero and Alejandro López-Ortiz. The 𝒢-packing with t-overlap problem. In Algorithms and Computation - 8th International Workshop, WALCOM 2014, Chennai, India, February 13-15, 2014, Proceedings, pages 114-124, 2014. URL: http://dx.doi.org/10.1007/978-3-319-04657-0_13.
http://dx.doi.org/10.1007/978-3-319-04657-0_13
Jazmín Romero and Alejandro López-Ortiz. A parameterized algorithm for packing overlapping subgraphs. In Computer Science - Theory and Applications - 9th International Computer Science Symposium in Russia, CSR 2014, Moscow, Russia, June 7-11, 2014. Proceedings, pages 325-336, 2014. URL: http://dx.doi.org/10.1007/978-3-319-06686-8_25.
http://dx.doi.org/10.1007/978-3-319-06686-8_25
Stéphan Thomassé. A 4k² kernel for feedback vertex set. ACM Trans. Algorithms, 6(2):32:1-32:8, April 2010. URL: http://dx.doi.org/10.1145/1721837.1721848.
http://dx.doi.org/10.1145/1721837.1721848
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The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems
Given a directed graph G and a list (s_1, t_1), ..., (s_k, t_k) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed s_i -> t_i path for every 1 <= i <= k. The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t_1, . . . , t_k) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every t_i to every other t_j ) is known to be W[1]-hard parameterized by the number of terminals. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if H is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list (s_1, t_1), ..., (s_k, t_k) of requests form a directed graph that is a member of H. Our main result is a complete characterization of the classes H resulting in fixed-parameter tractable special cases: we show that if every pattern in H has the combinatorial property of being "transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges," then the problem is FPT, and it is W[1]-hard for every recursively enumerable H not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], Strongly Connected Steiner Subgraph is W[1]-hard [Guo et al., SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before.
Directed Steiner Tree
Directed Steiner Network
fixed-parameter tractability
dichotomy
27:1-27:14
Regular Paper
Andreas Emil
Feldmann
Andreas Emil Feldmann
Dániel
Marx
Dániel Marx
10.4230/LIPIcs.ICALP.2016.27
Ajit Agrawal, Philip N. Klein, and R. Ravi. When trees collide: An approximation algorithm for the generalized Steiner problem on networks. SIAM J. Comput., 24(3):440-456, 1995. URL: http://dx.doi.org/10.1137/S0097539792236237.
http://dx.doi.org/10.1137/S0097539792236237
MohammadHossein Bateni, Mohammad Taghi Hajiaghayi, and Dániel Marx. Approximation schemes for Steiner forest on planar graphs and graphs of bounded treewidth. J. ACM, 58(5):21, 2011. URL: http://dx.doi.org/10.1145/2027216.2027219.
http://dx.doi.org/10.1145/2027216.2027219
MohammadHossein Bateni and MohammadTaghi Hajiaghayi. Euclidean prize-collecting Steiner forest. Algorithmica, 62(3-4):906-929, 2012. URL: http://dx.doi.org/10.1007/s00453-011-9491-8.
http://dx.doi.org/10.1007/s00453-011-9491-8
MohammadHossein Bateni, MohammadTaghi Hajiaghayi, and Vahid Liaghat. Improved approximation algorithms for (budgeted) node-weighted Steiner problems. In 40th International Colloquium on Automata, Languages, and Programming, pages 81-92, 2013. URL: http://dx.doi.org/10.1007/978-3-642-39206-1_8.
http://dx.doi.org/10.1007/978-3-642-39206-1_8
Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets Möbius: fast subset convolution. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 67-74, 2007.
Hans L. Bodlaender. Some classes of graphs with bounded treewidth. Bulletin of the EATCS, 36:116-125, 1988.
Glencora Borradaile, Philip N. Klein, and Claire Mathieu. An O(n log n) approximation scheme for Steiner tree in planar graphs. ACM Transactions on Algorithms, 5(3), 2009. URL: http://dx.doi.org/10.1145/1541885.1541892.
http://dx.doi.org/10.1145/1541885.1541892
Glencora Borradaile, Philip N. Klein, and Claire Mathieu. A polynomial-time approximation scheme for Euclidean Steiner forest. ACM Transactions on Algorithms, 11(3):19:1-19:20, 2015. URL: http://dx.doi.org/10.1145/2629654.
http://dx.doi.org/10.1145/2629654
Jaroslaw Byrka, Fabrizio Grandoni, Thomas Rothvoß, and Laura Sanità. Steiner tree approximation via iterative randomized rounding. J. ACM, 60(1):6, 2013. URL: http://dx.doi.org/10.1145/2432622.2432628.
http://dx.doi.org/10.1145/2432622.2432628
Moses Charikar, Chandra Chekuri, To-Yat Cheung, Zuo Dai, Ashish Goel, Sudipto Guha, and Ming Li. Approximation algorithms for directed Steiner problems. J. Algorithms, 33(1):73-91, 1999. URL: http://dx.doi.org/10.1006/jagm.1999.1042.
http://dx.doi.org/10.1006/jagm.1999.1042
Chandra Chekuri, Guy Even, Anupam Gupta, and Danny Segev. Set connectivity problems in undirected graphs and the directed steiner network problem. ACM Transactions on Algorithms, 7(2):18, 2011. URL: http://dx.doi.org/10.1145/1921659.1921664.
http://dx.doi.org/10.1145/1921659.1921664
Chandra Chekuri, Mohammad Taghi Hajiaghayi, Guy Kortsarz, and Mohammad R. Salavatipour. Approximation algorithms for node-weighted buy-at-bulk network design. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1265-1274, 2007. URL: http://dl.acm.org/citation.cfm?id=1283383.1283519.
http://dl.acm.org/citation.cfm?id=1283383.1283519
Rajesh Hemant Chitnis, Hossein Esfandiari, MohammadTaghi Hajiaghayi, Rohit Khandekar, Guy Kortsarz, and Saeed Seddighin. A tight algorithm for strongly connected Steiner subgraph on two terminals with demands (extended abstract). In 9th International Symposium on Parameterized and Exact Computation, pages 159-171, 2014. URL: http://dx.doi.org/10.1007/978-3-319-13524-3_14.
http://dx.doi.org/10.1007/978-3-319-13524-3_14
Rajesh Hemant Chitnis, MohammadTaghi Hajiaghayi, and Dániel Marx. Tight bounds for planar strongly connected Steiner subgraph with fixed number of terminals (and extensions). In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1782-1801, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.129.
http://dx.doi.org/10.1137/1.9781611973402.129
Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Philip N. Klein. Node-weighted Steiner tree and group Steiner tree in planar graphs. ACM Transactions on Algorithms, 10(3):13:1-13:20, 2014. URL: http://dx.doi.org/10.1145/2601070.
http://dx.doi.org/10.1145/2601070
Reinhard Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, third edition, 2005.
S. E. Dreyfus and R. A. Wagner. The Steiner problem in graphs. Networks, 1(3):195-207, 1971. URL: http://dx.doi.org/10.1002/net.3230010302.
http://dx.doi.org/10.1002/net.3230010302
Jon Feldman and Matthias Ruhl. The directed Steiner network problem is tractable for a constant number of terminals. SIAM J. Comput., 36(2):543-561, 2006. URL: http://dx.doi.org/10.1137/S0097539704441241.
http://dx.doi.org/10.1137/S0097539704441241
Michael R. Fellows, Danny Hermelin, Frances A. Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci., 410(1):53-61, 2009. URL: http://dx.doi.org/10.1016/j.tcs.2008.09.065.
http://dx.doi.org/10.1016/j.tcs.2008.09.065
Martin Grohe and Dániel Marx. On tree width, bramble size, and expansion. J. Comb. Theory, Ser. B, 99(1):218-228, 2009. URL: http://dx.doi.org/10.1016/j.jctb.2008.06.004.
http://dx.doi.org/10.1016/j.jctb.2008.06.004
Jiong Guo, Rolf Niedermeier, and Ondrej Suchý. Parameterized complexity of arc-weighted directed Steiner problems. SIAM J. Discrete Math., 25(2):583-599, 2011. URL: http://dx.doi.org/10.1137/100794560.
http://dx.doi.org/10.1137/100794560
Richard M. Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Plenum, 1972.
Philip N. Klein and R. Ravi. A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithms, 19(1):104-115, 1995. URL: http://dx.doi.org/10.1006/jagm.1995.1029.
http://dx.doi.org/10.1006/jagm.1995.1029
Sridhar Rajagopalan and Vijay V. Vazirani. On the bidirected cut relaxation for the metric Steiner tree problem. In Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 742-751, 1999. URL: http://dl.acm.org/citation.cfm?id=314500.314909.
http://dl.acm.org/citation.cfm?id=314500.314909
Ondřej Suchý. On directed steiner trees with multiple roots. To appear in WG 2016. arXiv:1604.05103.
Gabriel Robins and Alexander Zelikovsky. Tighter bounds for graph Steiner tree approximation. SIAM J. Discrete Math., 19(1):122-134, 2005. URL: http://dx.doi.org/10.1137/S0895480101393155.
http://dx.doi.org/10.1137/S0895480101393155
Alexander Zelikovsky. A series of approximation algorithms for the acyclic directed Steiner tree problem. Algorithmica, 18(1):99-110, 1997. URL: http://dx.doi.org/10.1007/BF02523690.
http://dx.doi.org/10.1007/BF02523690
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Double-Exponential and Triple-Exponential Bounds for Choosability Problems Parameterized by Treewidth
Choosability, introduced by Erdös, Rubin, and Taylor [Congr. Number. 1979], is a well-studied concept in graph theory: we say that a graph is c-choosable if for any assignment of a list of c colors to each vertex, there is a proper coloring where each vertex uses a color from its list. We study the complexity of deciding choosability on graphs of bounded treewidth. It follows from earlier work that 3-choosability can be decided in time 2^(2^(O(w)))*n^(O(1)) on graphs of treewidth w. We complement this result by a matching lower bound giving evidence that double-exponential dependence on treewidth may be necessary for the problem: we show that an algorithm with running time 2^(2^(o(w)))*n^(O(1)) would violate the Exponential-Time Hypothesis (ETH). We consider also the optimization problem where the task is to delete the minimum number of vertices to make the graph 4-choosable, and demonstrate that dependence on treewidth becomes tripleexponential for this problem: it can be solved in time 2^(2^(2^(O(w))))*n^(O(1)) on graphs of treewidth w, but an algorithm with running time 2^(2^(2^(o(w))))*n^(O(1)) would violate ETH.
Parameterized Complexity
List coloring
Treewidth
Lower bounds under ETH
28:1-28:15
Regular Paper
Dániel
Marx
Dániel Marx
Valia
Mitsou
Valia Mitsou
10.4230/LIPIcs.ICALP.2016.28
Mohammad Hossein Bateni, Mohammad Taghi Hajiaghayi, and Dániel Marx. Approximation schemes for Steiner forest on planar graphs and graphs of bounded treewidth. J. ACM, 58(5):21, 2011. URL: http://dx.doi.org/10.1145/2027216.2027219.
http://dx.doi.org/10.1145/2027216.2027219
M. Biró, Mihály Hujter, and Zsolt Tuza. Precoloring extension. I. interval graphs. Discrete Mathematics, 100(1-3):267-279, 1992. URL: http://dx.doi.org/10.1016/0012-365X(92)90646-W.
http://dx.doi.org/10.1016/0012-365X(92)90646-W
Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Fourier meets Möbius: fast subset convolution. In David S. Johnson and Uriel Feige, editors, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 67-74. ACM, 2007. URL: http://dx.doi.org/10.1145/1250790.1250801.
http://dx.doi.org/10.1145/1250790.1250801
Ivan Bliznets, Fedor V. Fomin, Marcin Pilipczuk, and Michal Pilipczuk. Subexponential parameterized algorithm for interval completion. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1116-1131. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch78.
http://dx.doi.org/10.1137/1.9781611974331.ch78
Leizhen Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett., 58(4):171-176, 1996. URL: http://dx.doi.org/10.1016/0020-0190(96)00050-6.
http://dx.doi.org/10.1016/0020-0190(96)00050-6
Yixin Cao. Linear recognition of almost interval graphs. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1096-1115. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch77.
http://dx.doi.org/10.1137/1.9781611974331.ch77
Yixin Cao and Dániel Marx. Chordal editing is fixed-parameter tractable. In Ernst W. Mayr and Natacha Portier, editors, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), STACS 2014, March 5-8, 2014, Lyon, France, volume 25 of LIPIcs, pages 214-225. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2014. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2014.214.
http://dx.doi.org/10.4230/LIPIcs.STACS.2014.214
Yixin Cao and Dániel Marx. Interval deletion is fixed-parameter tractable. ACM Transactions on Algorithms, 11(3):21:1-21:35, 2015. URL: http://dx.doi.org/10.1145/2629595.
http://dx.doi.org/10.1145/2629595
Jianer Chen, Fedor V. Fomin, Yang Liu, Songjian Lu, and Yngve Villanger. Improved algorithms for feedback vertex set problems. J. Comput. Syst. Sci., 74(7):1188-1198, 2008. URL: http://dx.doi.org/10.1016/j.jcss.2008.05.002.
http://dx.doi.org/10.1016/j.jcss.2008.05.002
Janka Chlebíková and Klaus Jansen. The d-precoloring problem for k-degenerate graphs. Discrete Mathematics, 307(16):2042-2052, 2007. URL: http://dx.doi.org/10.1016/j.disc.2005.12.049.
http://dx.doi.org/10.1016/j.disc.2005.12.049
Bruno Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. URL: http://dx.doi.org/10.1016/0890-5401(90)90043-H.
http://dx.doi.org/10.1016/0890-5401(90)90043-H
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21275-3.
http://dx.doi.org/10.1007/978-3-319-21275-3
Marek Cygan, Dániel Marx, Marcin Pilipczuk, and Michal Pilipczuk. Hitting forbidden subgraphs in graphs of bounded treewidth. In Erzsébet Csuhaj-Varjú, Martin Dietzfelbinger, and Zoltán Ésik, editors, Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part II, volume 8635 of Lecture Notes in Computer Science, pages 189-200. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44465-8_17.
http://dx.doi.org/10.1007/978-3-662-44465-8_17
Marek Cygan, Jesper Nederlof, Marcin Pilipczuk, Michal Pilipczuk, Johan M. M. van Rooij, and Jakub Onufry Wojtaszczyk. Solving connectivity problems parameterized by treewidth in single exponential time. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 150-159. IEEE Computer Society, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.23.
http://dx.doi.org/10.1109/FOCS.2011.23
Frank K. H. A. Dehne, Michael R. Fellows, Michael A. Langston, Frances A. Rosamond, and Kim Stevens. An o(2^O(k)n³ FPT algorithm for the undirected feedback vertex set problem. Theory Comput. Syst., 41(3):479-492, 2007. URL: http://dx.doi.org/10.1007/s00224-007-1345-z.
http://dx.doi.org/10.1007/s00224-007-1345-z
Paul Erdős, Arthur L Rubin, and Herbert Taylor. Choosability in graphs. Congr. Numer, 26:125-157, 1979.
Michael R Fellows, Fedor V Fomin, Daniel Lokshtanov, Frances Rosamond, Saket Saurabh, Stefan Szeider, and Carsten Thomassen. On the complexity of some colorful problems parameterized by treewidth. Information and Computation, 209(2):143-153, 2011.
Fedor V. Fomin and Yngve Villanger. Subexponential parameterized algorithm for minimum fill-in. SIAM J. Comput., 42(6):2197-2216, 2013. URL: http://dx.doi.org/10.1137/11085390X.
http://dx.doi.org/10.1137/11085390X
Markus Frick and Martin Grohe. The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Logic, 130(1-3):3-31, 2004. URL: http://dx.doi.org/10.1016/j.apal.2004.01.007.
http://dx.doi.org/10.1016/j.apal.2004.01.007
Elisabeth Gassner. The Steiner forest problem revisited. J. Discrete Algorithms, 8(2):154-163, 2010. URL: http://dx.doi.org/10.1016/j.jda.2009.05.002.
http://dx.doi.org/10.1016/j.jda.2009.05.002
Shai Gutner and Michael Tarsi. Some results on (a:b)-choosability. Discrete Mathematics, 309(8):2260-2270, 2009.
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http://dx.doi.org/10.1007/978-3-642-04128-0_51
Creative Commons Attribution 3.0 Unported license
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Do Distributed Differentially-Private Protocols Require Oblivious Transfer?
We study the cryptographic complexity of two-party differentially-private protocols for a large natural class of boolean functionalities. Information theoretically, McGregor et al. [FOCS 2010] and Goyal et al. [Crypto 2013] demonstrated several functionalities for which the maximal possible accuracy in the distributed setting is significantly lower than that in the client-server setting. Goyal et al. [Crypto 2013] further showed that "highly accurate" protocols in the distributed setting for any non-trivial functionality in fact imply the existence of one-way functions. However, it has remained an open problem to characterize the exact cryptographic complexity of this class. In particular, we know that semi-honest oblivious transfer helps obtain optimally accurate distributed differential privacy. But we do not know whether the reverse is true. We study the following question: Does the existence of optimally accurate distributed differentially private protocols for any class of functionalities imply the existence of oblivious transfer (or equivalently secure multi-party computation)? We resolve this question in the affirmative for the class of boolean functionalities that contain an XOR embedded on adjacent inputs. We give a reduction from oblivious transfer to:
- Any distributed optimally accurate epsilon-differentially private protocol with epsilon > 0 computing a functionality with a boolean XOR embedded on adjacent inputs.
- Any distributed non-optimally accurate epsilon-differentially private protocol with epsilon > 0, for a constant range of non-optimal accuracies and constant range of values of epsilon, computing a functionality with a boolean XOR embedded on adjacent inputs.
Enroute to proving these results, we demonstrate a connection between optimally-accurate twoparty differentially-private protocols for functions with a boolean XOR embedded on adjacent inputs, and noisy channels, which were shown by Crépeau and Kilian [FOCS 1988] to be sufficient for oblivious transfer.
Oblivious Transfer
Distributed Differential Privacy
Noisy Channels
Weak Noisy Channels
29:1-29:15
Regular Paper
Vipul
Goyal
Vipul Goyal
Dakshita
Khurana
Dakshita Khurana
Ilya
Mironov
Ilya Mironov
Omkant
Pandey
Omkant Pandey
Amit
Sahai
Amit Sahai
10.4230/LIPIcs.ICALP.2016.29
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http://dx.doi.org/10.1007/978-3-540-85174-5_25
Hai Brenner and Kobbi Nissim. Impossibility of differentially private universally optimal mechanisms. In focs2010 [16], pages 71-80. URL: http://dx.doi.org/10.1109/FOCS.2010.13.
http://dx.doi.org/10.1109/FOCS.2010.13
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http://dx.doi.org/10.1145/73007.73013
Benny Chor and Eyal Kushilevitz. A zero-one law for Boolean privacy. SIAM J. Discrete Math., 4(1):36-47, 1991. URL: http://dx.doi.org/10.1137/0404004.
http://dx.doi.org/10.1137/0404004
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http://dx.doi.org/10.1145/1250790.1250804
Cynthia Dwork, Moni Naor, Omer Reingold, Guy N. Rothblum, and Salil P. Vadhan. On the complexity of differentially private data release: efficient algorithms and hardness results. In Mitzenmacher [38], pages 381-390. URL: http://dx.doi.org/10.1145/1536414.1536467.
http://dx.doi.org/10.1145/1536414.1536467
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51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA. IEEE Computer Society, 2010.
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http://dx.doi.org/10.1145/1536414.1536464
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http://dx.doi.org/10.1007/978-3-642-40041-4_17
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http://dx.doi.org/10.1145/1806689.1806795
Dakshita Khurana, Daniel Kraschewski, Hemanta K. Maji, Manoj Prabhakaran, and Amit Sahai. All complete functionalities are reversible, 2015.
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http://dx.doi.org/10.1007/978-3-662-45608-8_21
Joe Kilian. Founding cryptography on oblivious transfer. In Janos Simon, editor, STOC, pages 20-31. ACM, 1988. URL: http://dx.doi.org/10.1145/62212.62215.
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http://dx.doi.org/10.1145/335305.335342
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http://dx.doi.org/10.1007/978-3-642-34931-7_4
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http://dx.doi.org/10.1109/FOCS.2010.14
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http://dx.doi.org/10.1007/978-3-642-03356-8_8
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Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Functional Commitment Schemes: From Polynomial Commitments to Pairing-Based Accumulators from Simple Assumptions
We formalize a cryptographic primitive called functional commitment (FC) which can be viewed as a generalization of vector commitments (VCs), polynomial commitments and many other special kinds of commitment schemes. A non-interactive functional commitment allows committing to a message in such a way that the committer has the flexibility of only revealing a function of the committed message during the opening phase. We provide constructions for the functionality of linear functions, where messages consist of vectors over some domain and commitments can later be opened to a specific linear function of the vector coordinates. An opening for a function thus generates a witness for the fact that the function indeed evaluates to a given value for the committed message. One security requirement is called function binding and requires that no adversary be able to open a commitment to two different evaluations for the same function.
We propose a construction of functional commitment for linear functions based on constantsize assumptions in composite order groups endowed with a bilinear map. The construction has commitments and openings of constant size (i.e., independent of n or function description) and is perfectly hiding - the underlying message is information theoretically hidden. Our security proofs build on the Déjà Q framework of Chase and Meiklejohn (Eurocrypt 2014) and its extension by Wee (TCC 2016) to encryption primitives, thus relying on constant-size subgroup decisional assumptions. We show that FC for linear functions are sufficiently powerful to solve four open problems. They, first, imply polynomial commitments, and, then, give cryptographic accumulators (i.e., an algebraic hash function which makes it possible to efficiently prove that some input belongs to a hashed set). In particular, specializing our FC construction leads to the first pairing-based polynomial commitments and accumulators for large universes known to achieve security under simple assumptions. We also substantially extend our pairing-based accumulator to handle subset queries which requires a non-trivial extension of the Déjà Q framework.
Cryptography
commitment schemes
functional commitments
accumulators
provable security
pairing-based
simple assumptions.
30:1-30:14
Regular Paper
Benoît
Libert
Benoît Libert
Somindu C.
Ramanna
Somindu C. Ramanna
Moti
Yung
Moti Yung
10.4230/LIPIcs.ICALP.2016.30
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Jan Camenisch and Anna Lysyanskaya. Dynamic Accumulators and Application to Efficient Revocation of Anonymous Credentials. In CRYPTO 2002, volume 2442 of LNCS, pages 61-76. Springer, 2002.
Dario Catalano and Dario Fiore. Vector Commitments and Their Applications. In PKC 2013, volume 7778 of LNCS, pages 55-72. Springer, 2013.
Dario Catalano, Dario Fiore, and Mariagrazia Messina. Zero-Knowledge Sets with Short Proofs. In EUROCRYPT 2008, volume 4965 of LNCS, pages 433-450. Springer, 2008.
Melissa Chase, Alexander Healy, Anna Lysyanskaya, Tal Malkin, and Leonid Reyzin. Mercurial Commitments with Applications to Zero-Knowledge Sets. In EUROCRYPT 2005, Proceedings, volume 3494 of LNCS, pages 422-439. Springer, 2005.
Melissa Chase and Sarah Meiklejohn. Déjà Q: Using Dual Systems to Revisit q-Type Assumptions. In EUROCRYPT 2014, volume 8441 of LNCS, pages 622-639. Springer, 2014.
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David Derler, Christian Hanser, and Daniel Slamanig. Revisiting Cryptographic Accumulators, Additional Properties and Relations to Other Primitives. In CT-RSA 2015, volume 9048 of LNCS, pages 127-144. Springer, 2015.
Cynthia Dwork, Moni Naor, Omer Reingold, and Larry J. Stockmeyer. Magic Functions. J. ACM, 50(6):852-921, 2003.
Sergey Gorbunov, Vinod Vaikuntanathan, and Daniel Wichs. Leveled Fully Homomorphic Signatures from Standard Lattices. In STOC 2015, pages 469-477. ACM, 2015.
Malika Izabachène, Benoît Libert, and Damien Vergnaud. Block-Wise P-Signatures and Non-interactive Anonymous Credentials with Efficient Attributes. In IMACC 2011, volume 7089 of LNCS, pages 431-450. Springer, 2011.
Aniket Kate, Gregory M. Zaverucha, and Ian Goldberg. Constant-Size Commitments to Polynomials and Their Applications. In ASIACRYPT 2010, volume 6477 of LNCS, pages 177-194. Springer, 2010.
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Jiangtao Li, Ninghui Li, and Rui Xue. Universal Accumulators with Efficient Nonmembership Proofs. In ACNS 2007, volume 4521 of LNCS, pages 253-269. Springer, 2007.
Benoît Libert and Moti Yung. Concise Mercurial Vector Commitments and Independent Zero-Knowledge Sets with Short Proofs. In TCC 2010, Proceedings, volume 5978 of LNCS, pages 499-517. Springer, 2010.
Helger Lipmaa. Secure Accumulators from Euclidean Rings without Trusted Setup. In ACNS 2012, volume 7341 of LNCS, pages 224-240. Springer, 2012.
Silvio Micali, Michael O. Rabin, and Joe Kilian. Zero-Knowledge Sets. In FOCS 2003, pages 80-91. IEEE Computer Society, 2003.
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Lan Nguyen. Accumulators from Bilinear Pairings and Applications. In CT-RSA 2005, volume 3376 of LNCS, pages 275-292. Springer, 2005.
Rafail Ostrovsky, Charles Rackoff, and Adam D. Smith. Efficient Consistency Proofs for Generalized Queries on a Committed Database. In ICALP 2004, volume 3142 of LNCS, pages 1041-1053. Springer, 2004.
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Charalampos Papamanthou, Roberto Tamassia, and Nikos Triandopoulos. Optimal Verification of Operations on Dynamic Sets. In CRYPTO 2011, volume 6841 of LNCS, pages 91-110. Springer, 2011.
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Amit Sahai and Brent Waters. Fuzzy Identity-Based Encryption. In EUROCRYPT 2005, Proceedings, volume 3494 of LNCS, pages 457-473. Springer, 2005.
Hoeteck Wee. Déjà Q: Encore! Un Petit IBE. In TCC 2016-A, volume 9563 of LNCS, pages 237-258. Springer, 2016.
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Block-Wise Non-Malleable Codes
Non-malleable codes, introduced by Dziembowski, Pietrzak, and Wichs (ICS'10) provide the guarantee that if a codeword c of a message m, is modified by a tampering function f to c', then c' either decodes to m or to "something unrelated" to m. In recent literature, a lot of focus has been on explicitly constructing such codes against a large and natural class of tampering functions such as split-state model in which the tampering function operates on different parts of the codeword independently.
In this work, we consider a stronger adversarial model called block-wise tampering model, in which we allow tampering to depend on more than one block: if a codeword consists of two blocks c = (c1, c2), then the first tampering function f1 could produce a tampered part c'_1 = f1(c1) and the second tampering function f2 could produce c'_2 = f2(c1, c2) depending on both c2 and c1. The notion similarly extends to multiple blocks where tampering of block ci could happen with the knowledge of all cj for j <= i. We argue this is a natural notion where, for example, the blocks are sent one by one and the adversary must send the tampered block before it gets the next block.
A little thought reveals that it is impossible to construct such codes that are non-malleable (in the standard sense) against such a powerful adversary: indeed, upon receiving the last block, an adversary could decode the entire codeword and then can tamper depending on the message. In light of this impossibility, we consider a natural relaxation called non-malleable codes with replacement which requires the adversary to produce not only related but also a valid codeword in order to succeed. Unfortunately, we show that even this relaxed definition is not achievable in the information-theoretic setting (i.e., when the tampering functions can be unbounded) which implies that we must turn our attention towards computationally bounded adversaries.
As our main result, we show how to construct a block-wise non-malleable code (BNMC) from sub-exponentially hard one-way permutations. We provide an interesting connection between BNMC and non-malleable commitments. We show that any BNMC can be converted into a nonmalleable
(w.r.t. opening) commitment scheme. Our techniques, quite surprisingly, give rise to a non-malleable commitment scheme (secure against so-called synchronizing adversaries), in which only the committer sends messages. We believe this result to be of independent interest. In the other direction, we show that any non-interactive non-malleable (w.r.t. opening) commitment can be used to construct BNMC only with 2 blocks. Unfortunately, such commitment scheme exists only under highly non-standard assumptions (adaptive one-way functions) and hence can not substitute our main construction.
Non-malleable codes
Non-malleable commitments
Block-wise Tampering
Complexity-leveraging
31:1-31:14
Regular Paper
Nishanth
Chandran
Nishanth Chandran
Vipul
Goyal
Vipul Goyal
Pratyay
Mukherjee
Pratyay Mukherjee
Omkant
Pandey
Omkant Pandey
Jalaj
Upadhyay
Jalaj Upadhyay
10.4230/LIPIcs.ICALP.2016.31
Divesh Aggarwal, Yevgeniy Dodis, Tomasz Kazana, and Maciej Obremski. Non-malleable reductions and applications. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 459-468. ACM, 2015.
Divesh Aggarwal, Yevgeniy Dodis, and Shachar Lovett. Non-malleable codes from additive combinatorics. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 774-783. ACM, 2014.
Shashank Agrawal, Divya Gupta, Hemanta K. Maji, Omkant Pandey, and Manoj Prabhakaran. Explicit non-malleable codes against bit-wise tampering and permutations. In Advances in Cryptology - CRYPTO 2015 - 35th Annual Cryptology Conference, Santa Barbara, CA, USA, August 16-20, 2015, Proceedings, Part I, pages 538-557, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47989-6_26.
http://dx.doi.org/10.1007/978-3-662-47989-6_26
Nishanth Chandran, Vipul Goyal, Pratyay Mukherjee, Omkant Pandey, and Jalaj Upadhyay. Block-wise non-malleable codes. IACR Cryptology ePrint Archive, 2015:129, 2015. URL: http://eprint.iacr.org/2015/129.
http://eprint.iacr.org/2015/129
Eshan Chattopadhyay, Vipul Goyal, and Xin Li. Non-malleable extractors and codes, with their many tampered extensions. To Appear in STOC (full version available at arXiv:1505.00107), 2016.
Eshan Chattopadhyay and David Zuckerman. Non-malleable codes against constant split-state tampering. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 306-315. IEEE, 2014.
Mahdi Cheraghchi and Venkatesan Guruswami. Non-malleable coding against bit-wise and split-state tampering. In Theory of Cryptography, pages 440-464. Springer, 2014.
Ivan Damgård, Sebastian Faust, Pratyay Mukherjee, and Daniele Venturi. Bounded tamper resilience: How to go beyond the algebraic barrier. In ASIACRYPT (2), pages 140-160, 2013.
Ivan Damgard and Jens Groth. Non-interactive and reusable non-malleable commitment schemes. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 426-437. ACM, 2003.
Danny Dolev, Cynthia Dwork, and Moni Naor. Nonmalleable cryptography. SIAM review, 45(4):727-784, 2003.
Stefan Dziembowski, Tomasz Kazana, and Maciej Obremski. Non-malleable codes from two-source extractors. In CRYPTO (2), pages 239-257, 2013.
Stefan Dziembowski, Krzysztof Pietrzak, and Daniel Wichs. Non-malleable codes. In ICS, pages 434-452, 2010.
Sebastian Faust, Pratyay Mukherjee, Jesper Buus Nielsen, and Daniele Venturi. Continuous non-malleable codes. In Theory of Cryptography, pages 465-488. Springer, 2014.
Sebastian Faust, Pratyay Mukherjee, Daniele Venturi, and Daniel Wichs. Efficient non-malleable codes and key-derivation for poly-size tampering circuits. In Advances in Cryptology - EUROCRYPT 2014, pages 111-128. Springer, 2014.
Rosario Gennaro, Anna Lysyanskaya, Tal Malkin, Silvio Micali, and Tal Rabin. Algorithmic tamper-proof (atp) security: Theoretical foundations for security against hardware tampering. In Theory of Cryptography, pages 258-277. Springer, 2004.
Vipul Goyal. Constant round non-malleable protocols using one way functions. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 695-704. ACM, 2011.
Vipul Goyal, Omkant Pandey, and Silas Richelson. Textbook non-malleable commitments. IACR Cryptology ePrint Archive, 2015:1178, To appear at STOC 2016. URL: http://eprint.iacr.org/2015/1178.
http://eprint.iacr.org/2015/1178
Yael Tauman Kalai, Bhavana Kanukurthi, and Amit Sahai. Cryptography with tamperable and leaky memory. In CRYPTO, pages 373-390, 2011.
Leslie Lamport. Constructing digital signatures from a one-way function. In Technical Report SRI-CSL-98. SRI International Computer Science Laboratory, 1979.
Huijia Lin and Rafael Pass. Non-malleability amplification. In Proceedings of the forty-first annual ACM symposium on Theory of computing, pages 189-198. ACM, 2009.
Feng-Hao Liu and Anna Lysyanskaya. Tamper and leakage resilience in the split-state model. In CRYPTO, pages 517-532, 2012.
Omkant Pandey, Rafael Pass, and Vinod Vaikuntanathan. Adaptive one-way functions and applications. In Advances in Cryptology - CRYPTO 2008, pages 57-74. Springer, 2008.
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Provably Secure Virus Detection: Using The Observer Effect Against Malware
Protecting software from malware injection is one of the biggest challenges of modern computer science. Despite intensive efforts by the scientific and engineering community, the number of successful attacks continues to increase.
This work sets first footsteps towards a provably secure investigation of malware detection. We provide a formal model and cryptographic security definitions of attestation for systems with dynamic memory, and suggest novel provably secure attestation schemes. The key idea underlying our schemes is to use the very insertion of the malware itself to allow for the systems to detect it. This is, in our opinion, close in spirit to the quantum Observer Effect. The attackers, no matter how clever, no matter when they insert their malware, change the state of the system they are attacking. This fundamental idea can be a game changer. And our system does not rely on heuristics; instead, our scheme enjoys the unique property that it is proved secure in a formal and precise mathematical sense and with minimal and realistic CPU modification achieves strong provable security guarantees. We envision such systems with a formal mathematical security treatment as a venue for new directions in software protection.
Cryptography
Software Attestation
Provable Security
32:1-32:14
Regular Paper
Richard J.
Lipton
Richard J. Lipton
Rafail
Ostrovsky
Rafail Ostrovsky
Vassilis
Zikas
Vassilis Zikas
10.4230/LIPIcs.ICALP.2016.32
T. AbuHmed, N. Nyamaa, and D. Nyang. Software-based remote code attestation in wireless sensor network. In GLOBECOM'09, pages 4680-4687. IEEE Press, 2009.
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http://dl.acm.org/citation.cfm?id=882493.884371
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A. Boldyreva, T. Kim, R. J. Lipton, and B. Warinschi. Provably-secure remote memory attestation for heap overflow protection. Cryptology ePrint Archive, Report 2015/729, 2015.
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R. J. Lipton, R. Ostrovsky, and V. Zikas. Provably secure virus detection. U.S. Patent (pending), Application No. 62/054,160, 2014.
R. J. Lipton, R. Ostrovsky, and V. Zikas. Provably secure virus detection: Using the observer effect against malware. Cryptology ePrint Archive, Report 2015/728, 2015.
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An Almost Cubic Lower Bound for Depth Three Arithmetic Circuits
We show an almost cubic lower bound on the size of any depth three arithmetic circuit computing an explicit multilinear polynomial in n variables over any field. This improves upon the previously known quadratic lower bound by Shpilka and Wigderson [CCC, 1999].
arithmetic circuits
depth-3 circuits
shifted partials
33:1-33:15
Regular Paper
Neeraj
Kayal
Neeraj Kayal
Chandan
Saha
Chandan Saha
Sébastien
Tavenas
Sébastien Tavenas
10.4230/LIPIcs.ICALP.2016.33
Manindra Agrawal, Chandan Saha, Ramprasad Saptharishi, and Nitin Saxena. Jacobian hits circuits: hitting-sets, lower bounds for depth-d occur-k formulas & depth-3 transcendence degree-k circuits. In STOC, pages 599-614, 2012.
Manindra Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 67-75, 2008.
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http://dx.doi.org/10.4230/LIPIcs.STACS.2014.239
Michael A. Forbes and Amir Shpilka. Quasipolynomial-Time Identity Testing of Non-commutative and Read-Once Oblivious Algebraic Branching Programs. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 243-252, 2013.
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Neeraj Kayal. An exponential lower bound for the sum of powers of bounded degree polynomials. Electronic Colloquium on Computational Complexity (ECCC), 19:81, 2012.
Neeraj Kayal, Nutan Limaye, Chandan Saha, and Srikanth Srinivasan. An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas. In Foundations of Computer Science (FOCS), pages 61-70, 2014.
Neeraj Kayal and Chandan Saha. Lower Bounds for Depth Three Arithmetic Circuits with small bottom fanin. In Conference on Computational Complexity, pages 158-208, 2015.
Neeraj Kayal and Chandan Saha. Multi-k-ic depth three circuit lower bound. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), volume 30, pages 527-539, 2015.
Neeraj Kayal, Chandan Saha, and Ramprasad Saptharishi. A super-polynomial lower bound for regular arithmetic formulas. In STOC, pages 146-153, 2014.
Neeraj Kayal, Chandan Saha, and Sébastien Tavenas. On the size of homogeneous and of depth four formulas with low individual degree. Electronic Colloquium on Computational Complexity (ECCC), 22:181, 2015.
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Mrinal Kumar and Shubhangi Saraf. Sums of products of polynomials in few variables : lower bounds and polynomial identity testing. Electronic Colloquium on Computational Complexity (ECCC), 22:71, 2015.
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Boundaries of VP and VNP
One fundamental question in the context of the geometric complexity theory approach to the VP vs. VNP conjecture is whether VP = !VP, where VP is the class of families of polynomials that can be computed by arithmetic circuits of polynomial degree and size, and VP is the class of families of polynomials that can be approximated infinitesimally closely by arithmetic circuits of polynomial degree and size. The goal of this article is to study the conjecture in (Mulmuley, FOCS 2012) that !VP is not contained in VP.
Towards that end, we introduce three degenerations of VP (i.e., sets of points in VP), namely the stable degeneration Stable-VP, the Newton degeneration Newton-VP, and the p-definable one-parameter degeneration VP*. We also introduce analogous degenerations of VNP. We show that Stable-VP subseteq Newton-VP subseteq VP* subseteq VNP, and Stable-VNP = Newton-VNP = VNP* = VNP. The three notions of degenerations and the proof of this result shed light on the problem of separating VP from VP.
Although we do not yet construct explicit candidates for the polynomial families in !VP\VP, we prove results which tell us where not to look for such families. Specifically, we demonstrate that the families in Newton-VP \VP based on semi-invariants of quivers would have to be nongeneric by showing that, for many finite quivers (including some wild ones), Newton degeneration of any generic semi-invariant can be computed by a circuit of polynomial size. We also show that the Newton degenerations of perfect matching Pfaffians, monotone arithmetic circuits over the reals, and Schur polynomials have polynomial-size circuits.
geometric complexity theory
arithmetic circuit
border complexity
34:1-34:14
Regular Paper
Joshua A.
Grochow
Joshua A. Grochow
Ketan D.
Mulmuley
Ketan D. Mulmuley
Youming
Qiao
Youming Qiao
10.4230/LIPIcs.ICALP.2016.34
Genrich R. Belitskii and Vladimir V. Sergeichuk. Complexity of matrix problems. Linear Algebra Appl., 361:203-222, 2003. Ninth Conference of the International Linear Algebra Society (Haifa, 2001). URL: http://dx.doi.org/10.1016/S0024-3795(02)00391-9.
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P. Bürgisser, M. Clausen, and M.A. Shokrollahi. Algebraic Complexity Theory. A series of comprehensive studies in mathematics. Springer, 1997.
P. Bürgisser, J. Landsberg, L. Manivel, and J. Weyman. An overview of mathematical issues arising in the geometric complexity theory approach to VP ̸ = VNP. SIAM Journal on Computing, 40(4):1179-1209, 2011.
H. Derksen and J. Weyman. Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients. Journal of the American Mathematical Society, 13(3):467-479, 2000.
H. Derksen and J. Weyman. Quiver representations. Notices of the American Mathematical Society, 52(2):200-206, 2005.
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J. A. Grochow. Symmetry and equivalence relations in classical and geometric complexity theory. PhD thesis, University of Chicago, Chicago, IL, 2012.
J. A. Grochow. Unifying known lower bounds via geometric complexity theory. Computational Complexity, 24:393-475, 2015. Special issue on IEEE CCC 2014. URL: http://dx.doi.org/10.1007/s00037-015-0103-x.
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http://dx.doi.org/10.1145/800135.804419
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https://creativecommons.org/licenses/by/3.0/legalcode
AC^0 o MOD_2 Lower Bounds for the Boolean Inner Product
AC^0 o MOD_2 circuits are AC^0 circuits augmented with a layer of parity gates just above the input layer. We study AC^0 o MOD2 circuit lower bounds for computing the Boolean Inner Product functions. Recent works by Servedio and Viola (ECCC TR12-144) and Akavia et al. (ITCS 2014) have highlighted this problem as a frontier problem in circuit complexity that arose both as a first step towards solving natural special cases of the matrix rigidity problem and as a candidate for constructing pseudorandom generators of minimal complexity. We give the first superlinear lower bound for the Boolean Inner Product function against AC^0 o MOD2 of depth four or greater. Specifically, we prove a superlinear lower bound for circuits of arbitrary constant depth, and an ~Omega(n^2) lower bound for the special case of depth-4 AC^0 o MOD_2. Our proof of the depth-4 lower bound employs a new "moment-matching" inequality for bounded, nonnegative integer-valued random variables that may be of independent interest: we prove an optimal bound on the maximum difference between two discrete distributions’ values at 0, given that their first d moments match.
Boolean analysis
circuit complexity
lower bounds
35:1-35:14
Regular Paper
Mahdi
Cheraghchi
Mahdi Cheraghchi
Elena
Grigorescu
Elena Grigorescu
Brendan
Juba
Brendan Juba
Karl
Wimmer
Karl Wimmer
Ning
Xie
Ning Xie
10.4230/LIPIcs.ICALP.2016.35
A. Akavia, A. Bogdanov, S. Guo, A. Kamath, and A. Rosen. Candidate weak pseudorandom functions in AC⁰∘ MOD₂. In Proc. 5th ITCS, pages 251-260, 2014.
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S. Chaudhuri and J. Radhakrishnan. Deterministic restrictions in circuit complexity. In Proc. 28th STOC, pages 30-36, 1996.
G. Cohen and I. Shinkar. The complexity of DNF of parities. To appear in ITCS'16, 2016.
R.E. Curto and L.A. Fialow. Recursiveness, positivity, and truncated moment problems. Houston J. Math., 17:603-635, 1991.
A. Hajnal, W. Maass, P. Pudlák, M. Szegedy, and G. Turán. Threshold circuits of bounded depth. JCSS, 46:129-154, 1993. Earlier version in FOCS'87.
R. Impagliazzo, R. Shaltiel, and A. Wigderson. Reducing the seed length in the Nisan-Wigderson generator. Combinatorica, 26(6):647-681, 2006. Earlier version in FOCS'01.
J.C. Jackson. An efficient membership-query algorithm for learning DNF with respect to the uniform distribution. JCSS, 55(3):414-440, 1997.
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N. Nisan. Pseudorandom bits for constant depth circuits. Combinatorica, 11(1):63-70, 1991.
N. Nisan and M. Szegedy. On the degree of Boolean functions as real polynomials. Computational Complexity, 4:301-313, 1994. Earlier version in STOC'92.
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R. Servedio and E. Viola. On a special case of rigidity. Manuscript, 2012.
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Lower Bounds for Nondeterministic Semantic Read-Once Branching Programs
We prove exponential lower bounds on the size of semantic read-once 3-ary nondeterministic branching programs. Prior to our result the best that was known was for D-ary branching programs with |D| >= 2^{13}.
Branching Programs
Semantic
Non-deterministic
Lower Bounds
36:1-36:13
Regular Paper
Stephen
Cook
Stephen Cook
Jeff
Edmonds
Jeff Edmonds
Venkatesh
Medabalimi
Venkatesh Medabalimi
Toniann
Pitassi
Toniann Pitassi
10.4230/LIPIcs.ICALP.2016.36
M. Ajtai. A non-linear time lower bound for boolean branching programs. In Proceedings 40th FOCS, pages 60-70, 1999.
P. Beame, T.S. Jayram, and M. Saks. Time-space tradeoffs for branching programs. J. Comput. Syst. Sci, 63(4):542-572, 2001.
P. Beame, M. Saks, X. Sun, and E. Vee. Time-space trade-off lower bounds for randomized computation of decision problems. Journal of the ACM, 50(2):154-195, 2003.
Allan Borodin, A Razborov, and Roman Smolensky. On lower bounds for read-k-times branching programs. Computational Complexity, 3(1):1-18, 1993.
Alan Cobham. The recognition problem for the set of perfect squares. In Switching and Automata Theory, 1966., IEEE Conference Record of Seventh Annual Symposium on, pages 78-87. IEEE, 1966.
Scott Diehl and Dieter Van Melkebeek. Time-space lower bounds for the polynomial-time hierarchy on randomized machines. SIAM Journal on Computing, 36(3):563-594, 2006.
L. Fortnow. Nondeterministic polynomial time versus nondeterministic logarithmic space: Time space tradeoffs for satifiability. In Proceedings 12th Conference on Computational Complexity, pages 52-60, 1997.
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Matthias Krause, Christoph Meinel, and Stephan Waack. Separating the eraser turing machine classes le, nle, co-nle and pe. In Mathematical Foundations of Computer Science 1988, pages 405-413. Springer, 1988.
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Ryan Williams. Better time-space lower bounds for sat and related problems. In Computational Complexity, 2005. Proceedings. Twentieth Annual IEEE Conference on, pages 40-49. IEEE, 2005.
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https://creativecommons.org/licenses/by/3.0/legalcode
Improved Bounds on the Sign-Rank of AC^0
The sign-rank of a matrix A with entries in {-1, +1} is the least rank of a real matrix B with A_{ij}*B_{ij} > 0 for all i, j. Razborov and Sherstov (2008) gave the first exponential lower bounds on the sign-rank of a function in AC^0, answering an old question of Babai, Frankl, and Simon (1986). Specifically, they exhibited a matrix A = [F(x,y)]_{x,y} for a specific function F:{-1,1}^n*{-1,1}^n -> {-1,1} in AC^0, such that A has sign-rank exp(Omega(n^{1/3}).
We prove a generalization of Razborov and Sherstov’s result, yielding exponential sign-rank lower bounds for a non-trivial class of functions (that includes the function used by Razborov and Sherstov). As a corollary of our general result, we improve Razborov and Sherstov's lower bound on the sign-rank of AC^0 from exp(Omega(n^{1/3})) to exp(~Omega(n^{2/5})). We also describe several applications to communication complexity, learning theory, and circuit complexity.
Sign-rank
circuit complexity
communication complexity
constant-depth circuits
37:1-37:14
Regular Paper
Mark
Bun
Mark Bun
Justin
Thaler
Justin Thaler
10.4230/LIPIcs.ICALP.2016.37
Scott Aaronson and Yaoyun Shi. Quantum lower bounds for the collision and the element distinctness problems. J. ACM, 51(4):595-605, 2004.
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Noga Alon, Shay Moran, and Amir Yehudayoff. Sign rank, VC dimension and spectral gaps. Electronic Colloquium on Computational Complexity (ECCC), 21:135, 2014.
László Babai, Peter Frankl, and Janos Simon. Complexity classes in communication complexity theory (preliminary version). In 27th Annual Symposium on Foundations of Computer Science, Toronto, Canada, 27-29 October 1986, pages 337-347, 1986.
Mark Bun and Justin Thaler. Dual lower bounds for approximate degree and Markov-Bernstein inequalities. In ICALP (1), pages 303-314, 2013.
Mark Bun and Justin Thaler. Dual polynomials for collision and element distinctness. CoRR, abs/1503.07261, 2015. URL: http://arxiv.org/abs/1503.07261.
http://arxiv.org/abs/1503.07261
Mark Bun and Justin Thaler. Hardness amplification and the approximate degree of constant-depth circuits. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 268-280, 2015.
Jürgen Forster. A linear lower bound on the unbounded error probabilistic communication complexity. J. Comput. Syst. Sci., 65(4):612-625, 2002.
Jürgen Forster, Matthias Krause, Satyanarayana V. Lokam, Rustam Mubarakzjanov, Niels Schmitt, and Hans-Ulrich Simon. Relations between communication complexity, linear arrangements, and computational complexity. In FST TCS 2001: Foundations of Software Technology and Theoretical Computer Science, 21st Conference, Bangalore, India, December 13-15, 2001, Proceedings, pages 171-182, 2001.
Jürgen Forster and Hans-Ulrich Simon. On the smallest possible dimension and the largest possible margin of linear arrangements representing given concept classes uniform distribution. In Algorithmic Learning Theory, 13th International Conference, ALT 2002, Lübeck, Germany, November 24-26, 2002, Proceedings, pages 128-138, 2002.
Mika Göös, Toniann Pitassi, and Thomas Watson. The landscape of communication complexity classes. Electronic Colloquium on Computational Complexity (ECCC), 22:49, 2015. URL: http://eccc.hpi-web.de/report/2015/049.
http://eccc.hpi-web.de/report/2015/049
Lisa Hellerstein and Rocco A. Servedio. On PAC learning algorithms for rich boolean function classes. Theor. Comput. Sci., 384(1):66-76, 2007.
Adam R. Klivans and Rocco A. Servedio. Learning DNF in time 2^õ(n^1/3). J. Comput. Syst. Sci., 68(2):303-318, 2004.
Nati Linial, Shahar Mendelson, Gideon Schechtman, and Adi Shraibman. Complexity measures of sign matrices. Combinatorica, 27(4):439-463, 2007.
Marvin Minsky and Seymour Papert. Perceptrons - an introduction to computational geometry. MIT Press, 1969.
Ramamohan Paturi and Janos Simon. Probabilistic communication complexity. J. Comput. Syst. Sci., 33(1):106-123, 1986.
Alexander A. Razborov and Alexander A. Sherstov. The sign-rank of AC⁰. SIAM J. Comput., 39(5):1833-1855, 2010.
A. A. Sherstov. The power of asymmetry in constant-depth circuits. In Foundations of Computer Science, 2015.
Alexander A. Sherstov. Communication lower bounds using dual polynomials. Bulletin of the EATCS, 95:59-93, 2008.
Alexander A. Sherstov. The unbounded-error communication complexity of symmetric functions. Combinatorica, 31(5):583-614, 2011.
Alexander A. Sherstov. Approximating the AND-OR tree. Theory of Computing, 9(20):653-663, 2013.
Alexander A. Sherstov. The intersection of two halfspaces has high threshold degree. SIAM J. Comput., 42(6):2329-2374, 2013.
Alexander A. Sherstov. Breaking the Minsky-Papert barrier for constant-depth circuits. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 223-232, 2014.
Robert Spalek. A dual polynomial for OR. CoRR, abs/0803.4516, 2008. URL: http://arxiv.org/abs/0803.4516.
http://arxiv.org/abs/0803.4516
Justin Thaler. Lower bounds for the approximate degree of block-composed functions. Electronic Colloquium on Computational Complexity (ECCC), 21:150, 2014. URL: http://eccc.hpi-web.de/report/2014/150.
http://eccc.hpi-web.de/report/2014/150
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On the Sensitivity Conjecture
The sensitivity of a Boolean function f:{0,1}^n -> {0,1} is the maximal number of neighbors a point in the Boolean hypercube has with different f-value. Roughly speaking, the block sensitivity allows to flip a set of bits (called a block) rather than just one bit, in order to change the value of f. The sensitivity conjecture, posed by Nisan and Szegedy (CC, 1994), states that the block sensitivity, bs(f), is at most polynomial in the sensitivity, s(f), for any Boolean function f. A positive answer to the conjecture will have many consequences, as the block sensitivity is polynomially related to many other complexity measures such as the certificate complexity, the decision tree complexity and the degree. The conjecture is far from being understood, as there is an exponential gap between the known upper and lower bounds relating bs(f) and s(f).
We continue a line of work started by Kenyon and Kutin (Inf. Comput., 2004), studying the l-block sensitivity, bs_l(f), where l bounds the size of sensitive blocks. While for bs_2(f) the picture is well understood with almost matching upper and lower bounds, for bs_3(f) it is not. We show that any development in understanding bs_3(f) in terms of s(f) will have great implications on the original question. Namely, we show that either bs(f) is at most sub-exponential in s(f) (which improves the state of the art upper bounds) or that bs_3(f) >= s(f){3-epsilon} for some Boolean functions (which improves the state of the art separations).
We generalize the question of bs(f) versus s(f) to bounded functions f:{0,1}^n -> [0,1] and show an analog result to that of Kenyon and Kutin: bs_l(f) = O(s(f))^l. Surprisingly, in this case, the bounds are close to being tight. In particular, we construct a bounded function f:{0,1}^n -> [0, 1] with bs(f) n/log(n) and s(f) = O(log(n)), a clear counterexample to the sensitivity conjecture for bounded functions.
Finally, we give a new super-quadratic separation between sensitivity and decision tree complexity by constructing Boolean functions with DT(f) >= s(f)^{2.115}. Prior to this work, only quadratic separations, DT(f) = s(f)^2, were known.
sensitivity conjecture
decision tree
block sensitivity
38:1-38:13
Regular Paper
Avishay
Tal
Avishay Tal
10.4230/LIPIcs.ICALP.2016.38
A. Ambainis, M. Bavarian, Y. Gao, J. Mao, X. Sun, and S. Zuo. Tighter relations between sensitivity and other complexity measures. In ICALP (1), pages 101-113, 2014. URL: http://dx.doi.org/10.1007/978-3-662-43948-7_9.
http://dx.doi.org/10.1007/978-3-662-43948-7_9
A. Ambainis and K. Prusis. A tight lower bound on certificate complexity in terms of block sensitivity and sensitivity. In MFCS, pages 33-44, 2014.
A. Ambainis, K. Prusis, and J. Vihrovs. Sensitivity versus certificate complexity of boolean functions. CoRR, abs/1503.07691, 2015.
A. Ambainis and X. Sun. New separation between s(f) and bs(f). Electronic Colloquium on Computational Complexity (ECCC), 18:116, 2011.
A. Ambainis and J. Vihrovs. Size of sets with small sensitivity: A generalization of simon’s lemma. In Theory and Applications of Models of Computation - 12th Annual Conference, TAMC 2015, Singapore, May 18-20, 2015, Proceedings, pages 122-133, 2015.
M. Boppana. Lattice variant of the sensitivity conjecture. Electronic Colloquium on Computational Complexity (ECCC), 19:89, 2012.
H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci., 288(1):21-43, 2002.
S. Chakraborty. On the sensitivity of cyclically-invariant boolean functions. Discrete Mathematics & Theoretical Computer Science, 13(4):51-60, 2011.
J. Gilmer, M. Koucký, and M. E. Saks. A new approach to the sensitivity conjecture. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 247-254, 2015.
P. Gopalan, N. Nisan, R. A. Servedio, K. Talwar, and A. Wigderson. Smooth boolean functions are easy: Efficient algorithms for low-sensitivity functions. In ITCS, pages 59-70, 2016.
P. Hatami, R. Kulkarni, and D. Pankratov. Variations on the sensitivity conjecture. Theory of Computing, Graduate Surveys, 2:1-27, 2011.
C. Kenyon and S. Kutin. Sensitivity, block sensitivity, and l-block sensitivity of boolean functions. Inf. Comput., 189(1):43-53, 2004. URL: http://dx.doi.org/10.1016/j.ic.2002.12.001.
http://dx.doi.org/10.1016/j.ic.2002.12.001
N. Nisan. Crew prams and decision trees. In STOC, pages 327-335, 1989. URL: http://dx.doi.org/10.1145/73007.73038.
http://dx.doi.org/10.1145/73007.73038
N. Nisan and M. Szegedy. On the degree of Boolean functions as real polynomials. Computational Complexity, 4:301-313, 1994.
D. Rubinstein. Sensitivity vs. block sensitivity of boolean functions. Combinatorica, 15(2):297-299, 1995. URL: http://dx.doi.org/10.1007/BF01200762.
http://dx.doi.org/10.1007/BF01200762
H. U. Simon. A tight Ω(log log n)-bound on the time for parallel ram’s to compute nondegenerated boolean functions. In Foundations of computation theory, pages 439-444. Springer, 1983.
M. Szegedy. An O(n^0.4732) upper bound on the complexity of the GKS communication game. Electronic Colloquium on Computational Complexity (ECCC), 22:102, 2015.
A. Tal. Properties and applications of boolean function composition. In ITCS, pages 441-454, 2013.
A. Tal. On the sensitivity conjecture. Electronic Colloquium on Computational Complexity (ECCC), 23:62, 2016.
M. Virza. Sensitivity versus block sensitivity of boolean functions. Inf. Process. Lett., 111(9):433-435, 2011. URL: http://dx.doi.org/10.1016/j.ipl.2011.02.001.
http://dx.doi.org/10.1016/j.ipl.2011.02.001
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Randomization Can Be as Helpful as a Glimpse of the Future in Online Computation
We provide simple but surprisingly useful direct product theorems for proving lower bounds on online algorithms with a limited amount of advice about the future. Intuitively, our direct product theorems say that if b bits of advice are needed to ensure a cost of at most t for some problem, then r*b bits of advice are needed to ensure a total cost of at most r*t when solving r independent instances of the problem. Using our direct product theorems, we are able to translate decades of research on randomized online algorithms to the advice complexity model. Doing so improves significantly on the previous best advice complexity lower bounds for many online problems, or provides the first known lower bounds. For example, we show that
- A paging algorithm needs Omega(n) bits of advice to achieve a competitive ratio better than H_k = Omega(log k), where k is the cache size. Previously, it was only known that Omega(n) bits of advice were necessary to achieve a constant competitive ratio smaller than 5/4.
- Every O(n^{1-epsilon})-competitive vertex coloring algorithm must use Omega(n log n) bits of advice. Previously, it was only known that Omega(n log n) bits of advice were necessary to be optimal.
For certain online problems, including the MTS, k-server, metric matching, paging, list update, and dynamic binary search tree problem, we prove that randomization and sublinear advice are equally powerful (if the underlying metric space or node set is finite). This means that several long-standing open questions regarding randomized online algorithms can be equivalently stated as questions regarding online algorithms with sublinear advice. For example, we show that there exists a deterministic O(log k)-competitive k-server algorithm with sublinear advice if and only if there exists a randomized O(log k)-competitive k-server algorithm without advice. Technically, our main direct product theorem is obtained by extending an information theoretical lower bound technique due to Emek, Fraigniaud, Korman, and Rosén [ICALP'09].
online algorithms
advice complexity
information theory
randomization
39:1-39:14
Regular Paper
Jesper W.
Mikkelsen
Jesper W. Mikkelsen
10.4230/LIPIcs.ICALP.2016.39
Anna Adamaszek, Artur Czumaj, Matthias Englert, and Harald Räcke. Almost tight bounds for reordering buffer management. In STOC, 2011.
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Christoph Ambühl. On the list update problem. PhD thesis, ETH Zürich, 2002.
Spyros Angelopoulos, Christoph Dürr, Shahin Kamali, Marc P. Renault, and Adi Rosén. Online bin packing with advice of small size. In WADS, 2015.
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Maria Paola Bianchi, Hans-Joachim Böckenhauer, Juraj Hromkovič, and Lucia Keller. Online coloring of bipartite graphs with and without advice. Algorithmica, 70(1):92-111, 2014.
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Hans-Joachim Böckenhauer, Richard Dobson, Sacha Krug, and Kathleen Steinhöfel. On energy-efficient computations with advice. In COCOON, 2015.
Hans-Joachim Böckenhauer, Sascha Geulen, Dennis Komm, and Walter Unger. Constructing randomized online algorithms from algorithms with advice. ETH-Zürich, 2015.
Hans-Joachim Böckenhauer, Juraj Hromkovič, Dennis Komm, Sacha Krug, Jasmin Smula, and Andreas Sprock. The string guessing problem as a method to prove lower bounds on the advice complexity. Theor. Comput. Sci., 554:95-108, 2014.
Hans-Joachim Böckenhauer, Dennis Komm, Rastislav Královič, and Richard Královič. On the advice complexity of the k-server problem. In ICALP (1), 2011.
Hans-Joachim Böckenhauer, Dennis Komm, Rastislav Královič, Richard Královič, and Tobias Mömke. On the advice complexity of online problems. In ISAAC, 2009.
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Joan Boyar, Lene M. Favrholdt, Christian Kudahl, and Jesper W. Mikkelsen. Advice complexity for a class of online problems. In STACS, 2015.
Joan Boyar, Lene M. Favrholdt, Christian Kudahl, and Jesper W. Mikkelsen. Weighted online problems with advice. In submission, 2016.
Joan Boyar, Shahin Kamali, Kim S. Larsen, and Alejandro López-Ortiz. On the list update problem with advice. In LATA, 2014.
Joan Boyar, Shahin Kamali, Kim S. Larsen, and Alejandro López-Ortiz. Online bin packing with advice. In STACS, 2014. Full paper to appear in Algorithmica.
Carl Burch. Machine learning in metrical task systems and other on-line problems. PhD thesis, Carnegie Mellon University, 2000. http://cburch.com/pub/thesis.ps.gz.
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Yuval Emek, Pierre Fraigniaud, Amos Korman, and Adi Rosén. Online computation with advice. Theor. Comput. Sci., 412(24):2642-2656, 2011.
Leah Epstein and Rob van Stee. On the online unit clustering problem. ACM Transactions on Algorithms, 7(1):1-18, 2010.
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Michal Forišek, Lucia Keller, and Monika Steinová. Advice complexity of online graph coloring. Unpublished manuscript, 2012.
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https://creativecommons.org/licenses/by/3.0/legalcode
Online Semidefinite Programming
We consider semidefinite programming through the lens of online algorithms - what happens if not all input is given at once, but rather iteratively? In what way does it make sense for a semidefinite program to be revealed? We answer these questions by defining a model for online semidefinite programming. This model can be viewed as a generalization of online coveringpacking linear programs, and it also captures interesting problems from quantum information theory. We design an online algorithm for semidefinite programming, utilizing the online primaldual method, achieving a competitive ratio of O(log(n)), where n is the number of matrices in the primal semidefinite program. We also design an algorithm for semidefinite programming with box constraints, achieving a competitive ratio of O(log F*), where F* is a sparsity measure of the semidefinite program. We conclude with an online randomized rounding procedure.
online algorithms
semidefinite programming
primal-dual
40:1-40:13
Regular Paper
Noa
Elad
Noa Elad
Satyen
Kale
Satyen Kale
Joseph (Seffi)
Naor
Joseph (Seffi) Naor
10.4230/LIPIcs.ICALP.2016.40
Rudolf Ahlswede and Andreas Winter. Strong converse for identification via quantum channels. Information Theory, IEEE Transactions on, 48(3):569-579, 2002.
Allan Borodin and Ran El-Yaniv. Online computation and competitive analysis. cambridge university press, 2005.
Niv Buchbinder and Joseph Naor. Online primal-dual algorithms for covering and packing. Mathematics of Operations Research, 34(2):270-286, 2009.
Michel X Goemans and David P Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42(6):1115-1145, 1995.
Anupam Gupta and Viswanath Nagarajan. Approximating sparse covering integer programs online. In Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part I, pages 436-448, 2012.
Joel A Tropp. User-friendly tail bounds for sums of random matrices. Foundations of Computational Mathematics, 12(4):389-434, 2012.
Avi Wigderson and David Xiao. Derandomizing the AW matrix-valued Chernoff bound using pessimistic estimators and applications, 2006.
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Beating the Harmonic Lower Bound for Online Bin Packing
In the online bin packing problem, items of sizes in (0,1] arrive online to be packed into bins of size 1. The goal is to minimize the number of used bins. Harmonic++ achieves a competitive ratio of 1.58889 and belongs to the Super Harmonic framework [Seiden, J. ACM, 2002]; a lower bound of Ramanan et al. shows that within this framework, no competitive ratio below 1.58333 can be achieved [Ramanan et al., J. Algorithms, 1989]. In this paper, we present an online bin packing algorithm with asymptotic performance ratio of 1.5815, which constitutes the first improvement in fifteen years and reduces the gap to the lower bound by roughly 15%.
We make two crucial changes to the Super Harmonic framework. First, some of the decisions of the algorithm will depend on exact sizes of items, instead of only their types. In particular, for item pairs where the size of one item is in (1/3,1/2] and the other is larger than 1/2 (a large item), when deciding whether to pack such a pair together in one bin, our algorithm does not consider their types, but only checks whether their total size is at most 1.
Second, for items with sizes in (1/3,1/2] (medium items), we try to pack the larger items of every type in pairs, while combining the smallest items with large items whenever possible. To do this, we postpone the coloring of medium items (i.e., the decision which items to pack in pairs and which to pack alone) where possible, and later select the smallest ones to be reserved for combining with large items. Additionally, in case such large items arrive early, we pack medium items with them whenever possible. This is a highly unusual idea in the context of Harmonic-like algorithms, which initially seems to preclude analysis (the ratio of items combined with large items is no longer a fixed constant).
For the analysis, we carefully mark medium items depending on how they end up packed, enabling us to add crucial constraints to the linear program used by Seiden. We consider the dual, eliminate all but one variable and then solve it with the ellipsoid method using a separation oracle. Our implementation uses additional algorithmic ideas to determine previously hand set parameters automatically and gives certificates for easy verification of the results.
We give a lower bound of 1.5766 for algorithms like ours. This shows that fundamentally different ideas will be required to make further improvements
Bin packing
online algorithms
harmonic algorithm
41:1-41:14
Regular Paper
Sandy
Heydrich
Sandy Heydrich
Rob
van Stee
Rob van Stee
10.4230/LIPIcs.ICALP.2016.41
Luitpold Babel, Bo Chen, Hans Kellerer, and Vladimir Kotov. Algorithms for on-line bin-packing problems with cardinality constraints. Discrete Applied Mathematics, 143(1-3):238-251, 2004. URL: http://dx.doi.org/10.1016/j.dam.2003.05.006.
http://dx.doi.org/10.1016/j.dam.2003.05.006
János Balogh, József Békési, György Dósa, Jirí Sgall, and Rob van Stee. The optimal absolute ratio for online bin packing. In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1425-1438. SIAM, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.94.
http://dx.doi.org/10.1137/1.9781611973730.94
János Balogh, József Békési, and Gábor Galambos. New lower bounds for certain classes of bin packing algorithms. In Klaus Jansen and Roberto Solis-Oba, editors, Approximation and Online Algorithms - 8th International Workshop, WAOA 2010, Liverpool, UK, September 9-10, 2010. Revised Papers, volume 6534 of Lecture Notes in Computer Science, pages 25-36. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-18318-8_3.
http://dx.doi.org/10.1007/978-3-642-18318-8_3
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Michel X. Goemans and Thomas Rothvoß. Polynomiality for bin packing with a constant number of item types. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 830-839. SIAM, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.61.
http://dx.doi.org/10.1137/1.9781611973402.61
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http://dx.doi.org/10.1109/FOCS.2013.11
Steve S. Seiden. On the online bin packing problem. Journal of the ACM, 49(5):640-671, 2002.
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Online Weighted Degree-Bounded Steiner Networks via Novel Online Mixed Packing/Covering
We design the first online algorithm with poly-logarithmic competitive ratio for the edge-weighted degree-bounded Steiner forest (EW-DB-SF) problem and its generalized variant. We obtain our result by demonstrating a new generic approach for solving mixed packing/covering integer programs in the online paradigm. In EW-DB-SF, we are given an edge-weighted graph with a degree bound for every vertex. Given a root vertex in advance, we receive a sequence of terminal vertices in an online manner. Upon the arrival of a terminal, we need to augment our solution subgraph to connect the new terminal to the root. The goal is to minimize the total weight of the solution while respecting the degree bounds on the vertices. In the offline setting, edge-weighted degree-bounded Steiner tree (EW-DB-ST) and its many variations have been extensively studied since early eighties. Unfortunately, the recent advancements in the online network design problems are inherently difficult to adapt for degree-bounded problems. In particular, it is not known whether the fractional solution obtained by standard primal-dual techniques for mixed packing/covering LPs can be rounded online. In contrast, in this paper we obtain our result by using structural properties of the optimal solution, and reducing the EW-DB-SF problem to an exponential-size mixed packing/covering integer program in which every variable appears only once in covering constraints. We then design a generic integral algorithm for solving this restricted family of IPs.
As mentioned above, we demonstrate a new technique for solving mixed packing/covering integer programs. Define the covering frequency k of a program as the maximum number of covering constraints in which a variable can participate. Let m denote the number of packing constraints. We design an online deterministic integral algorithm with competitive ratio of O(k*log(m)) for the mixed packing/covering integer programs. We prove the tightness of our result by providing a matching lower bound for any randomized algorithm. We note that our solution solely depends on m and k. Indeed, there can be exponentially many variables. Furthermore, our algorithm directly provides an integral solution, even if the integrality gap of the program is unbounded. We believe this technique can be used as an interesting alternative for the standard primal-dual techniques in solving online problems.
Online
Steiner Tree
Approximation
Competitive ratio
42:1-42:14
Regular Paper
Sina
Dehghani
Sina Dehghani
Soheil
Ehsani
Soheil Ehsani
Mohammad Taghi
Hajiaghayi
Mohammad Taghi Hajiaghayi
Vahid
Liaghat
Vahid Liaghat
Harald
Räcke
Harald Räcke
Saeed
Seddighin
Saeed Seddighin
10.4230/LIPIcs.ICALP.2016.42
Ajit Agrawal, Philip Nathan Klein, and R Ravi. How tough is the minimum-degree steiner tree?: A new approximate min-max equality. Technical Report CS-91-49, Brown University, 1991.
Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Naor. The online set cover problem. SIAM Journal on Computing, 39(2):361-370, 2009.
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MohammadTaghi Hajiaghayi, Vahid Liaghat, and Debmalya Panigrahi. Near-optimal online algorithms for prize-collecting steiner problems. In Automata, Languages, and Programming, pages 576-587. Springer, 2014.
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Rohit Khandekar, Guy Kortsarz, and Zeev Nutov. On some network design problems with degree constraints. Journal of Computer and System Sciences, 79(5):725-736, 2013.
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Carpooling in Social Networks
We consider the online carpool fairness problem of [Fagin and Williams, 1983] in which an online algorithm is presented with a sequence of pairs drawn from a group of n potential drivers. The online algorithm must select one driver from each pair, with the objective of partitioning the driving burden as fairly as possible for all drivers. The unfairness of an online algorithm is a measure of the worst-case deviation between the number of times a person has driven and the number of times they would have driven if life was completely fair.
We introduce a version of the problem in which drivers only carpool with their neighbors in a given social network graph; this is a generalization of the original problem, which corresponds to the social network of the complete graph. We show that for graphs of degree d, the unfairness of deterministic algorithms against adversarial sequences is exactly d/2. For random sequences of edges from planar graph social networks we give a [deterministic] algorithm with logarithmic unfairness (holds more generally for any bounded-genus graph). This does not follow from previous random sequence results in the original model, as we show that restricting the random sequences to sparse social network graphs may increase the unfairness.
A very natural class of randomized online algorithms are so-called static algorithms that preserve the same state distribution over time. Surprisingly, we show that any such algorithm has unfairness ~Theta(sqrt(d)) against oblivious adversaries. This shows that the local random greedy algorithm of [Ajtai et al, 1996] is close to optimal amongst the class of static algorithms. A natural (non-static) algorithm is global random greedy (which acts greedily and breaks ties at random). We improve the lower bound on the competitive ratio from Omega(log^{1/3}(d)) to Omega(log(d)). We also show that the competitive ratio of global random greedy against adaptive adversaries is Omega(d).
Online algorithms
Fairness
Randomized algorithms
Competitive ratio
Carpool problem
43:1-43:13
Regular Paper
Amos
Fiat
Amos Fiat
Anna R.
Karlin
Anna R. Karlin
Elias
Koutsoupias
Elias Koutsoupias
Claire
Mathieu
Claire Mathieu
Rotem
Zach
Rotem Zach
10.4230/LIPIcs.ICALP.2016.43
Miklos Ajtai, James Aspnes, Moni Naor, Yuval Rabani, Leonard J Schulman, and Orli Waarts. Fairness in scheduling. Journal of Algorithms, 29(2):306-357, 1998.
Amihood Amir, Oren Kapah, Tsvi Kopelowitz, Moni Naor, and Ely Porat. The family holiday gathering problem or fair and periodic scheduling of independent sets. CoRR, abs/1408.2279, 2014. URL: http://arxiv.org/abs/1408.2279.
http://arxiv.org/abs/1408.2279
Joao Pedro Boavida, Vikram Kamat, Darshana Nakum, Ryan Nong, Chai Wah Wu, and Xinyi Zhang. Algorithms for the carpool problem. IMA Preprint Series, pages 2133-6, 2006.
Don Coppersmith, Tomasz Nowicki, Giuseppe Paleologo, Charles Tresser, and Chai Wah Wu. The optimality of the online greedy algorithm in carpool and chairman assignment problems. ACM Trans. Algorithms, 7(3):37:1-37:22, July 2011. URL: http://dx.doi.org/10.1145/1978782.1978792.
http://dx.doi.org/10.1145/1978782.1978792
Ronald Fagin and John H Williams. A fair carpool scheduling algorithm. IBM Journal of Research and development, 27(2):133-139, 1983.
Ernst Fehr and Klaus M Schmidt. A theory of fairness, competition, and cooperation. Quarterly journal of Economics, pages 817-868, 1999.
L. Festinger. A Theory of Cognitive Dissonance. Mass communication series. Stanford University Press, 1962.
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http://dx.doi.org/10.1037/0022-3514.57.3.426
Moni Naor. On fairness in the carpool problem. Journal of Algorithms, 55(1):93 - 98, 2005. URL: http://dx.doi.org/10.1016/j.jalgor.2004.05.001.
http://dx.doi.org/10.1016/j.jalgor.2004.05.001
R. Tijdeman. The chairman assignment problem. Discrete Mathematics, 32(3):323 - 330, 1980. URL: http://dx.doi.org/10.1016/0012-365X(80)90269-1.
http://dx.doi.org/10.1016/0012-365X(80)90269-1
Zach and Fiat. Lower bounds for carpooling, 2015. URL: https://www.cs.tau.ac.il/~fiat/rotem_msc_thesis.pdf.
https://www.cs.tau.ac.il/~fiat/rotem_msc_thesis.pdf
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
An Improved Analysis of the ER-SpUD Dictionary Learning Algorithm
In dictionary learning we observe Y = AX + E for some Y in R^{n*p}, A in R^{m*n}, and X in R^{m*p}, where p >= max{n, m}, and typically m >=n. The matrix Y is observed, and A, X, E are unknown. Here E is a "noise" matrix of small norm, and X is column-wise sparse. The matrix A is referred to as a dictionary, and its columns as atoms. Then, given some small number p of samples, i.e. columns of Y , the goal is to learn the dictionary A up to small error, as well as the coefficient matrix X. In applications one could for example think of each column of Y as a distinct image in a database. The motivation is that in many applications data is expected to sparse when represented by atoms in the "right" dictionary A (e.g. images in the Haar wavelet basis), and the goal is to learn A from the data to then use it for other applications.
Recently, the work of [Spielman/Wang/Wright, COLT'12] proposed the dictionary learning algorithm ER-SpUD with provable guarantees when E = 0 and m = n. That work showed that if X has independent entries with an expected Theta n non-zeroes per column for 1/n <~ Theta <~ 1/sqrt(n), and with non-zero entries being subgaussian, then for p >~ n^2 log^2 n with high probability ER-SpUD outputs matrices A', X' which equal A, X up to permuting and scaling columns (resp. rows) of A (resp. X). They conjectured that p >~ n log n suffices, which they showed was information theoretically necessary for any algorithm to succeed when Theta =~ 1/n. Significant progress toward showing that p >~ n log^4 n might suffice was later obtained in [Luh/Vu, FOCS'15].
In this work, we show that for a slight variant of ER-SpUD, p >~ n log(n/delta) samples suffice for successful recovery with probability 1 - delta. We also show that without our slight variation made to ER-SpUD, p >~ n^{1.99} samples are required even to learn A, X with a small success probability of 1/ poly(n). This resolves the main conjecture of [Spielman/Wang/Wright, COLT'12], and contradicts a result of [Luh/Vu, FOCS'15], which claimed that p >~ n log^4 n guarantees high probability of success for the original ER-SpUD algorithm.
dictionary learning
stochastic processes
generic chaining
44:1-44:14
Regular Paper
Jaroslaw
Blasiok
Jaroslaw Blasiok
Jelani
Nelson
Jelani Nelson
10.4230/LIPIcs.ICALP.2016.44
Radosław Adamczak. A note on the sample complexity of the Er-SpUD algorithm by Spielman, Wang and Wright for exact recovery of sparsely used dictionaries. CoRR, abs/1601.02049, 2016.
Alekh Agarwal, Animashree Anandkumar, Prateek Jain, Praneeth Netrapalli, and Rashish Tandon. Learning sparsely used overcomplete dictionaries. In Proceedings of The 27th Conference on Learning Theory (COLT), pages 123-137, 2014.
M. Aharon, M. Elad, and A. Bruckstein. SVDD: An algorithm for designing overcomplete dictionaries for sparse representation. Trans. Sig. Proc., 54(11):4311-4322, November 2006.
Sanjeev Arora, Aditya Bhaskara, Rong Ge, and Tengyu Ma. More algorithms for provable dictionary learning. CoRR, abs/1401.0579, 2014.
Sanjeev Arora, Rong Ge, and Ankur Moitra. New algorithms for learning incoherent and overcomplete dictionaries. In Proceedings of The 27th Conference on Learning Theory (COLT), pages 779-806, 2014.
Sanjeev Arora, Rong Ge, Ankur Moitra, and Sushant Sachdeva. Provable ICA with unknown gaussian noise, and implications for gaussian mixtures and autoencoders. Algorithmica, 72(1):215-236, 2015.
Boaz Barak, Jonathan A. Kelner, and David Steurer. Dictionary learning and tensor decomposition via the sum-of-squares method. In Proceedings of the 47th Annual ACM on Symposium on Theory of Computing (STOC), pages 143-151, 2015.
Mikhail Belkin, Luis Rademacher, and James R. Voss. Blind signal separation in the presence of gaussian noise. In Proceedings of the 26th Annual Conference on Learning Theory (COLT), pages 270-287, 2013.
Stephane Boucheron, Gabor Lugosi, and Pascal Massart. Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, 2013.
Ori Bryt and Michael Elad. Compression of facial images using the K-SVD algorithm. J. Visual Communication and Image Representation, 19(4):270-282, 2008.
Sjoerd Dirksen. Tail bounds via generic chaining. Electron. J. Probab., 20(53):1-29, 2015.
Michael Elad and Michal Aharon. Image denoising via sparse and redundant representations over learned dictionaries. IEEE Transactions on Image Processing, 15(12):3736-3745, 2006.
Alan M. Frieze, Mark Jerrum, and Ravi Kannan. Learning linear transformations. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science (FOCS), pages 359-368, 1996.
Navin Goyal, Santosh Vempala, and Ying Xiao. Fourier PCA and robust tensor decomposition. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 584-593, 2014.
Michel Ledoux and Michel Talagrand. Probability in Banach Spaces: Isoperimetry and Processes. Springer-Verlag, 1991.
Yuanqing Li, Zhu Liang Yu, Ning Bi, Yong Xu, Zhenghui Gu, and S.-I. Amari. Sparse representation for brain signal processing: A tutorial on methods and applications. Signal Processing Magazine, IEEE, 31(3):96-106, May 2014. URL: http://dx.doi.org/10.1109/MSP.2013.2296790.
http://dx.doi.org/10.1109/MSP.2013.2296790
Kyle Luh and Van Vu. Random matrices: l₁ concentration and dictionary learning with few samples. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 1409-1425, 2015.
Julien Mairal, Francis Bach, Jean Ponce, and Guillermo Sapiro. Online learning for matrix factorization and sparse coding. Journal of Machine Learning Research, 11:19-60, 2010.
Julien Mairal, Francis R. Bach, and Jean Ponce. Sparse modeling for image and vision processing. Foundations and Trends in Computer Graphics and Vision, 8(2-3):85-283, 2014.
Julien Mairal, Francis R. Bach, Jean Ponce, Guillermo Sapiro, and Andrew Zisserman. Supervised dictionary learning. In Proceedings of the 22nd Annual Conference on Advances in Neural Information Processing Systems (NIPS), pages 1033-1040, 2008.
Julien Mairal, Francis R. Bach, Jean Ponce, Guillermo Sapiro, and Andrew Zisserman. Non-local sparse models for image restoration. In IEEE 12th International Conference on Computer Vision (ICCV), pages 2272-2279, 2009.
Phong Q. Nguyen and Oded Regev. Learning a parallelepiped: Cryptanalysis of GGH and NTRU signatures. J. Cryptology, 22(2):139-160, 2009.
Rajat Raina, Alexis Battle, Honglak Lee, Benjamin Packer, and Andrew Y. Ng. Self-taught learning: transfer learning from unlabeled data. In Proceedings of the Twenty-Fourth International Conference on Machine Learning (ICML), pages 759-766, 2007.
Daniel A. Spielman, Huan Wang, and John Wright. Exact recovery of sparsely-used dictionaries. In The 25th Annual Conference on Learning Theory (COLT), pages 37.1-37.18, 2012. Full version: URL: http://arxiv.org/abs/1206.5882v1.
http://arxiv.org/abs/1206.5882v1
Ju Sun, Qing Qu, and John Wright. Complete dictionary recovery over the sphere. CoRR, abs/1504.06785, 2015.
Michel Talagrand. Upper and lower bounds for stochastic processes: modern methods and classical problems. Springer, 2014.
Santosh Vempala and Ying Xiao. Max vs Min: Tensor decomposition and ICA with nearly linear sample complexity. In Proceedings of The 28th Conference on Learning Theory (COLT), pages 1710-1723, 2015.
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Approximation via Correlation Decay When Strong Spatial Mixing Fails
Approximate counting via correlation decay is the core algorithmic technique used in the sharp delineation of the computational phase transition that arises in the approximation of the partition function of anti-ferromagnetic two-spin models.
Previous analyses of correlation-decay algorithms implicitly depended on the occurrence of strong spatial mixing. This, roughly, means that one uses worst-case analysis of the recursive procedure that creates the sub-instances. In this paper, we develop a new analysis method that is more refined than the worst-case analysis. We take the shape of instances in the computation tree into consideration and we amortise against certain "bad" instances that are created as the recursion proceeds. This enables us to show correlation decay and to obtain an FPTAS even when strong spatial mixing fails.
We apply our technique to the problem of approximately counting independent sets in hypergraphs with degree upper-bound Delta and with a lower bound k on the arity of hyperedges. Liu and Lin gave an FPTAS for k >= 2 and Delta <= 5 (lack of strong spatial mixing was the obstacle preventing this algorithm from being generalised to Delta = 6). Our technique gives a tight result for Delta = 6, showing that there is an FPTAS for k >= 3 and Delta <= 6. The best previously-known approximation scheme for Delta = 6 is the Markov-chain simulation based FPRAS of Bordewich, Dyer and Karpinski, which only works for k >= 8.
Our technique also applies for larger values of k, giving an FPTAS for k >= 1.66 Delta. This bound is not as strong as existing randomised results, for technical reasons that are discussed in the paper. Nevertheless, it gives the first deterministic approximation schemes in this regime. We further demonstrate that in the hypergraph independent set model, approximating the partition function is NP-hard even within the uniqueness regime.
approximate counting
independent sets in hypergraphs
correlation decay
45:1-45:13
Regular Paper
Ivona
Bezáková
Ivona Bezáková
Andreas
Galanis
Andreas Galanis
Leslie Ann
Goldberg
Leslie Ann Goldberg
Heng
Guo
Heng Guo
Daniel
Stefankovic
Daniel Stefankovic
10.4230/LIPIcs.ICALP.2016.45
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Complexity Trichotomy for Approximately Counting List H-Colourings
We examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph then counting list H-colourings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph then approximately counting list H-colourings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem which is believed to be of intermediate complexity - it is believed that it does not have an FPRAS, but that it is not as difficult as approximating the most difficult counting problems in #P. For every other graph H, approximately counting list H-colourings is complete for #P with respect to approximation-preserving reductions (so there is no FPRAS unless NP = RP). Two pleasing features of the trichotomy are (i) it has a natural formulation in terms of hereditary graph classes, and (ii) the proof is largely self-contained and does not require any universal algebra (unlike similar dichotomies in the weighted case). We are able to extend the hardness results to the bounded-degree setting, showing that all hardness results apply to input graphs with maximum degree at most 6.
approximate counting
graph homomorphisms
list colourings
46:1-46:13
Regular Paper
Andreas
Galanis
Andreas Galanis
Leslie Ann
Goldberg
Leslie Ann Goldberg
Mark
Jerrum
Mark Jerrum
10.4230/LIPIcs.ICALP.2016.46
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Parity Separation: A Scientifically Proven Method for Permanent Weight Loss
Given an edge-weighted graph G, let PerfMatch(G) denote the weighted sum over all perfect matchings M in G, weighting each matching M by the product of weights of edges in M. If G is unweighted, this plainly counts the perfect matchings of G.
In this paper, we introduce parity separation, a new method for reducing PerfMatch to unweighted instances: For graphs G with edge-weights 1 and -1, we construct two unweighted graphs G1 and G2 such that PerfMatch(G) = PerfMatch(G1) - PerfMatch(G2). This yields a novel weight removal technique for counting perfect matchings, in addition to those known from classical #P-hardness proofs. Our technique is based upon the Holant framework and matchgates. We derive the following applications:
Firstly, an alternative #P-completeness proof for counting unweighted perfect matchings.
Secondly, C=P-completeness for deciding whether two given unweighted graphs have the same number of perfect matchings. To the best of our knowledge, this is the first C=P-completeness result for the “equality-testing version” of any natural counting problem that is not already #P-hard under parsimonious reductions.
Thirdly, an alternative tight lower bound for counting unweighted perfect matchings under the counting exponential-time hypothesis #ETH.
perfect matchings
counting complexity
structural complexity
exponentialtime hypothesis
47:1-47:14
Regular Paper
Radu
Curticapean
Radu Curticapean
10.4230/LIPIcs.ICALP.2016.47
Amir Ben-Dor and Shai Halevi. Zero-one permanent is #P-complete, A simpler proof. In Second Israel Symposium on Theory of Computing Systems, ISTCS 1993, Natanya, Israel, June 7-9, 1993. Proceedings, pages 108-117, 1993.
Markus Bläser and Radu Curticapean. The complexity of the cover polynomials for planar graphs of bounded degree. In Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Warsaw, Poland, August 22-26, 2011. Proceedings, pages 96-107, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22993-0_12.
http://dx.doi.org/10.1007/978-3-642-22993-0_12
Markus Bläser and Radu Curticapean. Weighted counting of k-matchings is #W[1]-hard. In Parameterized and Exact Computation - 7th International Symposium, IPEC 2012, Ljubljana, Slovenia, September 12-14, 2012. Proceedings, pages 171-181, 2012. URL: http://dx.doi.org/10.1007/978-3-642-33293-7_17.
http://dx.doi.org/10.1007/978-3-642-33293-7_17
Markus Bläser and Holger Dell. Complexity of the cover polynomial. In Automata, Languages and Programming, 34th International Colloquium, ICALP 2007, Wroclaw, Poland, July 9-13, 2007, Proceedings, pages 801-812, 2007. URL: http://dx.doi.org/10.1007/978-3-540-73420-8_69.
http://dx.doi.org/10.1007/978-3-540-73420-8_69
Graham Brightwell and Peter Winkler. Counting linear extensions is #P-complete. In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 5-8, 1991, New Orleans, Louisiana, USA, pages 175-181, 1991. URL: http://dx.doi.org/10.1145/103418.103441.
http://dx.doi.org/10.1145/103418.103441
Jin-Yi Cai and Pinyan Lu. Holographic algorithms: From art to science. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 401-410, New York, NY, USA, 2007. ACM. URL: http://dx.doi.org/http://doi.acm.org/10.1145/1250790.1250850.
http://dx.doi.org/http://doi.acm.org/10.1145/1250790.1250850
Jin-Yi Cai, Pinyan Lu, and Mingji Xia. Holographic algorithms by Fibonacci gates and holographic reductions for hardness. In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 644-653. IEEE Computer Society, 2008. URL: http://dx.doi.org/10.1109/FOCS.2008.34.
http://dx.doi.org/10.1109/FOCS.2008.34
Mike Chen. The complexity of checking whether two DAG have the same number of topological sorts, November 2010. URL: http://cstheory.stackexchange.com/questions/3105.
http://cstheory.stackexchange.com/questions/3105
Radu Curticapean. Block interpolation: A framework for tight exponential-time counting complexity. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 380-392, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_31.
http://dx.doi.org/10.1007/978-3-662-47672-7_31
Radu Curticapean. Parity separation: A scientifically proven method for permanent weight loss. CoRR, abs/1511.07480, 2015. URL: http://arxiv.org/abs/1511.07480.
http://arxiv.org/abs/1511.07480
Radu Curticapean. The simple, little and slow things count: on parameterized counting complexity. PhD thesis, Saarland University, August 2015.
Radu Curticapean and Dániel Marx. Tight conditional lower bounds for counting perfect matchings on graphs of bounded treewidth, cliquewidth, and genus. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1650-1669, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch113.
http://dx.doi.org/10.1137/1.9781611974331.ch113
Holger Dell, Thore Husfeldt, Dániel Marx, Nina Taslaman, and Martin Wahlen. Exponential time complexity of the permanent and the Tutte polynomial. ACM Transactions on Algorithms, 10(4):21, 2014. URL: http://dx.doi.org/10.1145/2635812.
http://dx.doi.org/10.1145/2635812
Jack Edmonds. Paths, trees, and flowers. In Classic Papers in Combinatorics, Modern Birkhauser Classics, pages 361-379. Birkhauser Boston, 1987. URL: http://dx.doi.org/10.1007/978-0-8176-4842-8_26.
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Lane A. Hemaspaandra and Mitsunori Ogihara. The Complexity Theory Companion. Springer, 2002.
Lane A. Hemaspaandra and Heribert Vollmer. The satanic notations: counting classes beyond #P and other definitional adventures. SIGACT News, 26(1):2-13, 1995. URL: http://dx.doi.org/10.1145/203610.203611.
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Christian Hoffmann. Exponential time complexity of weighted counting of independent sets. In Parameterized and Exact Computation - 5th International Symposium, IPEC 2010, Chennai, India, December 13-15, 2010. Proceedings, pages 180-191, 2010. URL: http://dx.doi.org/10.1007/978-3-642-17493-3_18.
http://dx.doi.org/10.1007/978-3-642-17493-3_18
Thore Husfeldt and Nina Taslaman. The exponential time complexity of computing the probability that a graph is connected. In Parameterized and Exact Computation - 5th International Symposium, IPEC 2010, Chennai, India, December 13-15, 2010. Proceedings, pages 192-203, 2010. URL: http://dx.doi.org/10.1007/978-3-642-17493-3_19.
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Russel Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1727.
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Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
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Patrick Scharpfenecker and Jacobo Torán. Solution-graphs of boolean formulas and isomorphism. Electronic Colloquium on Computational Complexity (ECCC), 23:24, 2016. URL: http://eccc.hpi-web.de/report/2016/024.
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Leslie G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189-201, 1979.
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http://dx.doi.org/10.1142/S0129054191000066
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On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter
Conditional lower bounds for dynamic graph problems has received a great deal of attention in recent years. While many results are now known for the fully-dynamic case and such bounds often imply worst-case bounds for the partially dynamic setting, it seems much more difficult to prove amortized bounds for incremental and decremental algorithms. In this paper we consider partially dynamic versions of three classic problems in graph theory. Based on popular conjectures we show that:
- No algorithm with amortized update time O(n^{1-epsilon}) exists for incremental or decremental maximum cardinality bipartite matching. This significantly improves on the O(m^{1/2-epsilon}) bound for sparse graphs of Henzinger et al. [STOC'15] and O(n^{1/3-epsilon}) bound of Kopelowitz, Pettie and Porat. Our linear bound also appears more natural. In addition, the result we present separates the node-addition model from the edge insertion model, as an algorithm with total update time O(m*sqrt(n)) exists for the former by Bosek et al. [FOCS'14].
- No algorithm with amortized update time O(m^{1-epsilon}) exists for incremental or decremental maximum flow in directed and weighted sparse graphs. No such lower bound was known for partially dynamic maximum flow previously. Furthermore no algorithm with amortized update time O(n^{1-epsilon}) exists for directed and unweighted graphs or undirected and weighted graphs.
- No algorithm with amortized update time O(n^{1/2-epsilon}) exists for incremental or decremental (4/3 - epsilon')-approximating the diameter of an unweighted graph. We also show a slightly stronger bound if node additions are allowed. The result is then extended to the static case, where we show that no O((n*sqrt(m))^{1-epsilon}) algorithm exists. We also extend the result to the case when an additive error is allowed in the approximation. While our bounds are weaker than the already known bounds of Roditty and Vassilevska Williams [STOC'13], it is based on a weaker conjecture of Abboud et al. [STOC'15] and is the first known reduction from the 3SUM and APSP problems to diameter. Showing an equivalence between APSP and diameter is a major open problem in this area (Abboud et al. [SODA'15]), and thus showing even a weak connection in this direction is of interest.
Conditional lower bounds
Maximum cardinality matching
Diameter in graphs
Hardness in P
Partially dynamic problems
Maximum flow
48:1-48:14
Regular Paper
Søren
Dahlgaard
Søren Dahlgaard
10.4230/LIPIcs.ICALP.2016.48
Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. If the current clique algorithms are optimal, so is valiant’s parser. In Proc. 56th IEEE Symposium on Foundations of Computer Science (FOCS), pages 98-117, 2015.
Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight hardness results for LCS and other sequence similarity measures. In Proc. 56th IEEE Symposium on Foundations of Computer Science (FOCS), pages 59-78, 2015.
Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In Proc. 26th ACM/SIAM Symposium on Discrete Algorithms (SODA), pages 1681-1697, 2015.
Amir Abboud, Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Ryan Williams. Simulating branching programs with edit distance and friends or: A polylog shaved is a lower bound made. In Proc. 48th ACM Symposium on Theory of Computing (STOC), 2016. To appear.
Amir Abboud and Kevin Lewi. Exact weight subgraphs and the k-sum conjecture. In Proc. 40th International Colloquium on Automata, Languages and Programming (ICALP), pages 1-12, 2013.
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proc. 55th IEEE Symposium on Foundations of Computer Science (FOCS), pages 434-443, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In Proc. 41st International Colloquium on Automata, Languages and Programming (ICALP), pages 39-51, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Huacheng Yu. Matching triangles and basing hardness on an extremely popular conjecture. In Proc. 47th ACM Symposium on Theory of Computing (STOC), pages 41-50, 2015.
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Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In Proc. 25th ACM/SIAM Symposium on Discrete Algorithms (SODA), pages 217-226, 2014.
Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3sum conjecture. In Proc. 27th ACM/SIAM Symposium on Discrete Algorithms (SODA), pages 1272-1287, 2016.
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Aleksander Madry. From Graphs to Matrices, and Back: New Techniques for Graph Algorithms. PhD thesis, Massachusetts Institute of Technology, 6 2011.
Aleksander Madry. Navigating central path with electrical flows: From flows to matchings, and back. In Proc. 54th IEEE Symposium on Foundations of Computer Science (FOCS), pages 253-262, 2013.
Ofer Neiman and Shay Solomon. Simple deterministic algorithms for fully dynamic maximal matching. In Proc. 45th ACM Symposium on Theory of Computing (STOC), pages 745-754, 2013.
Mihai Patrascu. Towards polynomial lower bounds for dynamic problems. In Proc. 42nd ACM Symposium on Theory of Computing (STOC), pages 603-610, 2010.
Mihai Pǎtraşcu and Ryan Williams. On the possibility of faster SAT algorithms. In Proc. 21st ACM/SIAM Symposium on Discrete Algorithms (SODA), pages 1065-1075, 2010.
Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proc. 45th ACM Symposium on Theory of Computing (STOC), pages 515-524, 2013.
Liam Roditty and Uri Zwick. On dynamic shortest paths problems. Algorithmica, 61(2):389-401, 2011. See also ESA'04.
Piotr Sankowski. Faster dynamic matchings and vertex connectivity. In Proc. 18th ACM/SIAM Symposium on Discrete Algorithms (SODA), pages 118-126, 2007.
Jonah Sherman. Nearly maximum flows in nearly linear time. In Proc. 54th IEEE Symposium on Foundations of Computer Science (FOCS), pages 263-269, 2013.
Virginia Vassilevska and Ryan Williams. Finding, minimizing, and counting weighted subgraphs. In Proc. 41st ACM Symposium on Theory of Computing (STOC), pages 455-464, 2009.
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Incremental 2-Edge-Connectivity in Directed Graphs
We present an algorithm that can update the 2-edge-connected blocks of a directed graph with n vertices through a sequence of m edge insertions in a total of O(m*n) time. After each insertion, we can answer the following queries in asymptotically optimal time:
- Test in constant time if two query vertices v and w are 2-edge-connected. Moreover, if v and w are not 2-edge-connected, we can produce in constant time a “witness” of this property, by exhibiting an edge that is contained in all paths from v to w or in all paths from w to v.
- Report in O(n) time all the 2-edge-connected blocks of G.
This is the first dynamic algorithm for 2-connectivity problems on directed graphs, and it matches the best known bounds for simpler problems, such as incremental transitive closure.
2-edge connectivity on directed graphs; dynamic graph algorithms; incremental algorithms.
49:1-49:15
Regular Paper
Loukas
Georgiadis
Loukas Georgiadis
Giuseppe F.
Italiano
Giuseppe F. Italiano
Nikos
Parotsidis
Nikos Parotsidis
10.4230/LIPIcs.ICALP.2016.49
A. Abboud and V. Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proc. 55th IEEE Symp. on Foundations of Computer Science, FOCS, pages 434-443, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.53.
http://dx.doi.org/10.1109/FOCS.2014.53
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http://dx.doi.org/10.1145/2756553
A. L. Buchsbaum, L. Georgiadis, H. Kaplan, A. Rogers, R. E. Tarjan, and J. R. Westbrook. Linear-time algorithms for dominators and other path-evaluation problems. SIAM Journal on Computing, 38(4):1533-1573, 2008.
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http://dx.doi.org/10.1007/s00453-007-9051-4
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http://dx.doi.org/10.1145/265910.265914
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http://dx.doi.org/10.1145/2764909
L. Georgiadis, G. F. Italiano, L. Laura, and N. Parotsidis. 2-edge connectivity in directed graphs. In Proc. 26th ACM-SIAM Symp. on Discrete Algorithms, pages 1988-2005, 2015.
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L. Georgiadis, G. F. Italiano, and N. Parotsidis. A New Framework for Strong Connectivity and 2-Connectivity in Directed Graphs. ArXiv e-prints, abs/1511.02913, November 2015. URL: http://arxiv.org/abs/1511.02913.
http://arxiv.org/abs/1511.02913
B. Haeupler, T. Kavitha, R. Mathew, S. Sen, and R. E. Tarjan. Incremental cycle detection, topological ordering, and strong component maintenance. ACM Transactions on Algorithms, 8(1):3:1-3:33, January 2012. URL: http://dx.doi.org/10.1145/2071379.2071382.
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Unified Acceleration Method for Packing and Covering Problems via Diameter Reduction
In a series of recent breakthroughs, Allen-Zhu and Orecchia [Allen-Zhu/Orecchia, STOC 2015; Allen-Zhu/Orecchia, SODA 2015] leveraged insights from the linear coupling method [Allen-Zhu/Oreccia, arXiv 2014], which is a first-order optimization scheme, to provide improved algorithms for packing and covering linear programs. The result in [Allen-Zhu/Orecchia, STOC 2015] is particularly interesting, as the algorithm for packing LP achieves both width-independence and Nesterov-like acceleration, which was not known to be possible before. Somewhat surprisingly, however, while the dependence of the convergence rate on the error parameter epsilon for packing problems was improved to O(1/epsilon), which corresponds to what accelerated gradient methods are designed to achieve, the dependence for covering problems was only improved to O(1/epsilon^{1.5}), and even that required a different more complicated algorithm, rather than from Nesterov-like acceleration. Given the primal-dual connection between packing and covering problems and since previous algorithms for these very related problems have led to the same epsilon dependence, this discrepancy is surprising, and it leaves open the question of the exact role that the linear coupling is playing in coordinating the complementary gradient and mirror descent step of the algorithm. In this paper, we clarify these issues, illustrating that the linear coupling method can lead to improved O(1/epsilon) dependence for both packing and covering problems in a unified manner, i.e., with the same algorithm and almost identical analysis. Our main technical result is a novel dimension lifting method that reduces the coordinate-wise diameters of the feasible region for covering LPs, which is the key structural property to enable the same Nesterov-like acceleration as in the case of packing LPs. The technique is of independent interest and that may be useful in applying the accelerated linear coupling method to other combinatorial problems.
Convex optimization
Accelerated gradient descent
Linear program
Approximation algorithm
Packing and covering
50:1-50:13
Regular Paper
Di
Wang
Di Wang
Satish
Rao
Satish Rao
Michael W.
Mahoney
Michael W. Mahoney
10.4230/LIPIcs.ICALP.2016.50
Zeyuan Allen-Zhu and Lorenzo Orecchia. Nearly-linear time positive LP solver with faster convergence rate. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC'15, pages 229-236, 2015. Newer version available at URL: http://arxiv.org/abs/1411.1124.
http://arxiv.org/abs/1411.1124
Zeyuan Allen-Zhu and Lorenzo Orecchia. Using optimization to break the epsilon barrier: A faster and simpler width-independent algorithm for solving positive linear programs in parallel. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'15, pages 1439-1456, 2015.
Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. Theory of Computing, 8(6):121-164, 2012. URL: http://dx.doi.org/10.4086/toc.2012.v008a006.
http://dx.doi.org/10.4086/toc.2012.v008a006
Baruch Awerbuch and Rohit Khandekar. Stateless distributed gradient descent for positive linear programs. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 691-700, 2008. URL: http://dx.doi.org/10.1145/1374376.1374476.
http://dx.doi.org/10.1145/1374376.1374476
Lisa Fleischer. A fast approximation scheme for fractional covering problems with variable upper bounds. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, New Orleans, Louisiana, USA, January 11-14, 2004, pages 1001-1010, 2004. URL: http://dl.acm.org/citation.cfm?id=982792.982942.
http://dl.acm.org/citation.cfm?id=982792.982942
Christos Koufogiannakis and Neal E. Young. A nearly linear-time PTAS for explicit fractional packing and covering linear programs. Algorithmica, 70(4):648-674, 2014. URL: http://dx.doi.org/10.1007/s00453-013-9771-6.
http://dx.doi.org/10.1007/s00453-013-9771-6
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http://dx.doi.org/10.1137/S1052623403425629
Yurii Nesterov. Smooth minimization of non-smooth functions. Math. Program., 103(1):127-152, 2005. URL: http://dx.doi.org/10.1007/s10107-004-0552-5.
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Yurii Nesterov. Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM Journal on Optimization, 22(2):341-362, 2012. URL: http://dx.doi.org/10.1137/100802001.
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http://arxiv.org/abs/1409.5832
Neal E. Young. Sequential and parallel algorithms for mixed packing and covering. In 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14-17 October 2001, Las Vegas, Nevada, USA, pages 538-546, 2001. URL: http://dx.doi.org/10.1109/SFCS.2001.959930.
http://dx.doi.org/10.1109/SFCS.2001.959930
Neal E. Young. Nearly linear-time approximation schemes for mixed packing/covering and facility-location linear programs. CoRR, abs/1407.3015, 2014. URL: http://arxiv.org/abs/1407.3015.
http://arxiv.org/abs/1407.3015
Zeyuan Allen Zhu and Lorenzo Orecchia. Linear coupling: An ultimate unification of gradient and mirror descent. CoRR, abs/1407.1537, 2014. URL: http://arxiv.org/abs/1407.1537.
http://arxiv.org/abs/1407.1537
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Random-Edge Is Slower Than Random-Facet on Abstract Cubes
Random-Edge and Random-Facet are two very natural randomized pivoting rules for the simplex algorithm. The behavior of Random-Facet is fairly well understood. It performs an expected sub-exponential number of pivoting steps on any linear program, or more generally, on any Acyclic Unique Sink Orientation (AUSO) of an arbitrary polytope, making it the fastest known pivoting rule for the simplex algorithm. The behavior of Random-Edge is much less understood. We show that in the AUSO setting, Random-Edge is slower than Random-Facet. To do that, we construct AUSOs of the n-dimensional hypercube on which Random-Edge performs an expected number of 2^{Omega(sqrt(n*log(n)))} steps. This improves on a 2^{Omega(sqrt^3(n))} lower bound of Matoušek and Szabó. As Random-Facet performs an expected number of 2^{O(sqrt(n)} steps on any n-dimensional AUSO, this established our result. Improving our 2^{Omega(sqrt(n*log(n)))} lower bound seems to require radically new techniques.
Linear programming
the Simplex Algorithm
Pivoting rules
Acyclic Unique Sink Orientations
51:1-51:14
Regular Paper
Thomas Dueholm
Hansen
Thomas Dueholm Hansen
Uri
Zwick
Uri Zwick
10.4230/LIPIcs.ICALP.2016.51
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http://dx.doi.org/10.1145/1993636.1993675
O. Friedmann, T.D. Hansen, and U. Zwick. Errata for: A subexponential lower bound for the random facet algorithm for parity games. CoRR, abs/1410.7871, 2014. URL: http://arxiv.org/abs/1410.7871.
http://arxiv.org/abs/1410.7871
O. Friedmann, T.D. Hansen, and U. Zwick. Random-facet and random-bland require subexponential time even for shortest paths. CoRR, abs/1410.7530, 2014. URL: http://arxiv.org/abs/1410.7530.
http://arxiv.org/abs/1410.7530
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B. Gärtner and V. Kaibel. Two new bounds for the Random-Edge simplex-algorithm. SIAM J. Discrete Math., 21(1):178-190, 2007. URL: http://dx.doi.org/10.1137/05062370X.
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B. Gärtner and A. Thomas. The niceness of unique sink orientations. Unpublished manuscript, 2016. URL: https://sites.google.com/site/antonisthomas/research/niceness.pdf.
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http://dx.doi.org/10.1016/0166-218X(88)90042-X
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https://creativecommons.org/licenses/by/3.0/legalcode
Approximating the Solution to Mixed Packing and Covering LPs in Parallel O˜(epsilon^{-3}) Time
We study the problem of approximately solving positive linear programs (LPs). This class of LPs models a wide range of fundamental problems in combinatorial optimization and operations research, such as many resource allocation problems, solving non-negative linear systems, computing tomography, single/multi commodity flows on graphs, etc. For the special cases of pure packing or pure covering LPs, recent result by Allen-Zhu and Orecchia [Allen/Zhu/Orecchia, SODA'15] gives O˜(1/(epsilon^3))-time parallel algorithm, which breaks the longstanding O˜(1/(epsilon^4)) running time bound by the seminal work of Luby and Nisan [Luby/Nisan, STOC'93].
We present new parallel algorithm with running time O˜(1/(epsilon^3)) for the more general mixed packing and covering LPs, which improves upon the O˜(1/(epsilon^4))-time algorithm of Young [Young, FOCS'01; Young, arXiv 2014]. Our work leverages the ideas from both the optimization oriented approach [Allen/Zhu/Orecchia, SODA'15; Wang/Mahoney/Mohan/Rao, arXiv 2015], as well as the more combinatorial approach with phases [Young, FOCS'01; Young, arXiv 2014]. In addition, our algorithm, when directly applied to pure packing or pure covering LPs, gives a improved running time of O˜(1/(epsilon^2)).
Mixed packing and covering
Linear program
Approximation algorithm
Parallel algorithm
52:1-52:14
Regular Paper
Michael W.
Mahoney
Michael W. Mahoney
Satish
Rao
Satish Rao
Di
Wang
Di Wang
Peng
Zhang
Peng Zhang
10.4230/LIPIcs.ICALP.2016.52
Zeyuan Allen-Zhu and Lorenzo Orecchia. Nearly-linear time positive LP solver with faster convergence rate. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC'15, pages 229-236, 2015. Newer version available at URL: http://arxiv.org/abs/1411.1124.
http://arxiv.org/abs/1411.1124
Zeyuan Allen-Zhu and Lorenzo Orecchia. Using optimization to break the epsilon barrier: A faster and simpler width-independent algorithm for solving positive linear programs in parallel. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'15, pages 1439-1456, 2015.
Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. Theory of Computing, 8(6):121-164, 2012. URL: http://www.theoryofcomputing.org/articles/v008a006, URL: http://dx.doi.org/10.4086/toc.2012.v008a006.
http://dx.doi.org/10.4086/toc.2012.v008a006
Baruch Awerbuch and Rohit Khandekar. Stateless distributed gradient descent for positive linear programs. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 691-700, 2008.
S. Boyd and L. Vandenberghe. Convex Optimization. Camebridge University Press, 2004.
Lisa Fleischer. A fast approximation scheme for fractional covering problems with variable upper bounds. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, New Orleans, Louisiana, USA, January 11-14, 2004, pages 1001-1010, 2004.
Christos Koufogiannakis and Neal E. Young. A nearly linear-time PTAS for explicit fractional packing and covering linear programs. Algorithmica, 70(4):648-674, 2014. URL: http://dx.doi.org/10.1007/s00453-013-9771-6.
http://dx.doi.org/10.1007/s00453-013-9771-6
Yin Tat Lee and Aaron Sidford. Path finding methods for linear programming: Solving linear programs in õ(sqrt(rank)) iterations and faster algorithms for maximum flow. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 424-433, 2014.
Yin Tat Lee and Aaron Sidford. Efficient inverse maintenance and faster algorithms for linear programming. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 230-249, 2015.
Michael Luby and Noam Nisan. A parallel approximation algorithm for positive linear programming. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, May 16-18, 1993, San Diego, CA, USA, pages 448-457, 1993.
Faraz Makari Manshadi, Baruch Awerbuch, Rainer Gemula, Rohit Khandekar, Julián Mestre, and Mauro Sozio. A distributed algorithm for large-scale generalized matching. PVLDB, 6(9):613-624, 2013. Available at http://www.vldb.org/pvldb/vol6/p613-makarimanshadi.pdf.
Arkadi Nemirovski. Prox-method with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIAM Journal on Optimization, 15(1):229-251, 2004. URL: http://dx.doi.org/10.1137/S1052623403425629.
http://dx.doi.org/10.1137/S1052623403425629
Yurii Nesterov. Smooth minimization of non-smooth functions. Math. Program., 103(1):127-152, 2005. URL: http://dx.doi.org/10.1007/s10107-004-0552-5.
http://dx.doi.org/10.1007/s10107-004-0552-5
Yurii Nesterov. Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM Journal on Optimization, 22(2):341-362, 2012. URL: http://dx.doi.org/10.1137/100802001.
http://dx.doi.org/10.1137/100802001
Richard Peng and Kanat Tangwongsan. Faster and simpler width-independent parallel algorithms for positive semidefinite programming. In Proceedinbgs of the 24th ACM symposium on Parallelism in algorithms and architectures, SPAA '12, pages 101-108, 2012. Available at http://arxiv.org/abs/1201.5135.
James Renegar. Efficient first-order methods for linear programming and semidefinite programming. CoRR, abs/1409.5832, 2014. URL: http://arxiv.org/abs/1409.5832.
http://arxiv.org/abs/1409.5832
Di Wang, Michael W. Mahoney, Nishanth Mohan, and Satish Rao. Faster parallel solver for positive linear programs via dynamically-bucketed selective coordinate descent. CoRR, abs/1511.06468, 2015. URL: http://arxiv.org/abs/1511.06468.
http://arxiv.org/abs/1511.06468
Neal E. Young. Sequential and parallel algorithms for mixed packing and covering. In 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14-17 October 2001, Las Vegas, Nevada, USA, pages 538-546, 2001.
Neal E. Young. Nearly linear-time approximation schemes for mixed packing/covering and facility-location linear programs. CoRR, abs/1407.3015, 2014. URL: http://arxiv.org/abs/1407.3015.
http://arxiv.org/abs/1407.3015
Zeyuan Allen Zhu, Yin Tat Lee, and Lorenzo Orecchia. Using optimization to obtain a width-independent, parallel, simpler, and faster positive SDP solver. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1824-1831, 2016. Available at http://arxiv.org/abs/1507.02259.
Zeyuan Allen Zhu and Lorenzo Orecchia. Linear coupling: An ultimate unification of gradient and mirror descent. CoRR, abs/1407.1537, 2014. URL: http://arxiv.org/abs/1407.1537.
http://arxiv.org/abs/1407.1537
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Optimization Algorithms for Faster Computational Geometry
We study two fundamental problems in computational geometry: finding the maximum inscribed ball (MaxIB) inside a bounded polyhedron defined by m hyperplanes, and the minimum enclosing ball (MinEB) of a set of n points, both in d-dimensional space. We improve the running time of iterative algorithms on
MaxIB from ~O(m*d*alpha^3/epsilon^3) to ~O(m*d + m*sqrt(d)*alpha/epsilon), a speed-up up to ~O(sqrt(d)*alpha^2/epsilon^2), and
MinEB from ~O(n*d/sqrt(epsilon)) to ~O(n*d + n*sqrt(d)/sqrt(epsilon)), a speed-up up to ~O(sqrt(d)).
Our improvements are based on a novel saddle-point optimization framework. We propose a new algorithm L1L2SPSolver for solving a class of regularized saddle-point problems, and apply a randomized Hadamard space rotation which is a technique borrowed from compressive sensing. Interestingly, the motivation of using Hadamard rotation solely comes from our optimization view but not the original geometry problem: indeed, it is not immediately clear why MaxIB or MinEB, as a geometric problem, should be easier to solve if we rotate the space by a unitary matrix. We hope that our optimization perspective sheds lights on solving other geometric problems as well.
maximum inscribed balls
minimum enclosing balls
approximation algorithms
53:1-53:6
Regular Paper
Zeyuan
Allen-Zhu
Zeyuan Allen-Zhu
Zhenyu
Liao
Zhenyu Liao
Yang
Yuan
Yang Yuan
10.4230/LIPIcs.ICALP.2016.53
Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Geometric approximation via coresets. Combinatorial and computational geometry, 52:1-30, 2005.
Nir Ailon and Bernard Chazelle. Faster dimension reduction. Communications of the ACM, 53(May):97-104, 2010. URL: http://dx.doi.org/10.1145/1646353.1646379.
http://dx.doi.org/10.1145/1646353.1646379
Zeyuan Allen-Zhu, Zhenyu Liao, and Yang Yuan. Optimization Algorithms for Faster Computational Geometry. ArXiv e-prints, abs/1412.1001, December 2014.
Zeyuan Allen-Zhu and Lorenzo Orecchia. Using optimization to break the epsilon barrier: A faster and simpler width-independent algorithm for solving positive linear programs in parallel. In Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms, SODA'15, 2015.
Mihai Bădoiu and Kenneth L Clarkson. Optimal core-sets for balls. Computational Geometry, 40(1):14-22, 2008.
Mihai Bădoiu, Sariel Har-Peled, and Piotr Indyk. Approximate clustering via core-sets. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing - STOC'02, page 250, New York, New York, USA, 2002. ACM Press. URL: http://dx.doi.org/10.1145/509907.509947.
http://dx.doi.org/10.1145/509907.509947
Kenneth L Clarkson. Coresets, sparse greedy approximation, and the frank-wolfe algorithm. ACM Transactions on Algorithms (TALG), 6(4):63, 2010.
Kenneth L. Clarkson, Elad Hazan, and David P. Woodruff. Sublinear optimization for machine learning. Journal of the ACM, 59(5):23:1-23:49, October 2012. URL: http://dx.doi.org/10.1145/2371656.2371658.
http://dx.doi.org/10.1145/2371656.2371658
Jack Elzinga and Thomas G. Moore. A central cutting plane algorithm for the convex programming problem. Mathematical Programming, 8(1):134-145, 1975.
Bernd Gärtner and Martin Jaggi. Coresets for polytope distance. In Proceedings of the 25th annual symposium on computational geometry, pages 33-42. ACM, 2009.
Sariel Har-Peled, Dan Roth, and Dav Zimak. Maximum margin coresets for active and noise tolerant learning. In Proceedings of the 20th international joint conference on Artifical intelligence, pages 836-841. Morgan Kaufmann Publishers Inc., 2007.
Piyush Kumar, Joseph S. B. Mitchell, and E. Alper Yildirim. Approximate minimum enclosing balls in high dimensions using core-sets. Journal of Experimental Algorithmics, 8:1-29, January 2003. URL: http://dx.doi.org/10.1145/996546.996548.
http://dx.doi.org/10.1145/996546.996548
Chungmok Lee and Sungsoo Park. Chebyshev center based column generation. Discrete Applied Mathematics, 159(18):2251-2265, 2011.
Katta G. Murty. o(m) bound on number of iterations in sphere methods for lp. Algorithmic Operations Research, 7(1):30-40, 2012.
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Yurii Nesterov. Excessive Gap Technique in Nonsmooth Convex Minimization. SIAM Journal on Optimization, 16(1):235-249, January 2005. URL: http://dx.doi.org/10.1137/S1052623403422285.
http://dx.doi.org/10.1137/S1052623403422285
Ankan Saha, S. V. N. Vishwanathan, and Xinhua Zhang. New Approximation Algorithms for Minimum Enclosing Convex Shapes. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms - SODA'11, pages 1146-1160, September 2011. URL: http://arxiv.org/abs/0909.1062.
http://arxiv.org/abs/0909.1062
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http://dx.doi.org/10.1145/1137856.1137861
E. Alper Yildirim. Two algorithms for the minimum enclosing ball problem. SIAM Journal on Optimization, 19(3):1368-1391, 2008.
Yuchen Zhang and Lin Xiao. Stochastic Primal-Dual Coordinate Method for Regularized Empirical Risk Minimization. In Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, 2015. URL: http://arxiv.org/abs/1409.3257.
http://arxiv.org/abs/1409.3257
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A Fast Distributed Stateless Algorithm for alpha-Fair Packing Problems
We study weighted alpha-fair packing problems, that is, the problems of maximizing the objective functions (i) sum_j w_j*x_j^{1-alpha}/(1-alpha) when alpha > 0, alpha != 1 and (ii) sum_j w_j*ln(x_j) when alpha = 1, over linear constraints A*x <=b, x >= 0, where wj are positive weights and A and b are non-negative. We consider the distributed computation model that was used for packing linear programs and network utility maximization problems. Under this model, we provide a distributed algorithm for general alpha that converges to an epsilon-approximate solution in time (number of distributed iterations) that has an inverse polynomial dependence on the approximation parameter epsilon and poly-logarithmic dependence on the problem size. This is the first distributed algorithm for weighted alpha-fair packing with poly-logarithmic convergence in the input size. The algorithm uses simple local update rules and is stateless (namely, it allows asynchronous updates, is self-stabilizing, and allows incremental and local adjustments). We also obtain a number of structural results that characterize alpha-fair allocations as the value of alpha is varied. These results deepen our understanding of fairness guarantees in alpha-fair packing allocations, and also provide insight into the behavior of alpha-fair allocations in the asymptotic cases alpha -> 0, alpha -> 1, and alpha -> infinity.
Fairness
distributed and stateless algorithms
resource allocation
54:1-54:15
Regular Paper
Jelena
Marasevic
Jelena Marasevic
Clifford
Stein
Clifford Stein
Gil
Zussman
Gil Zussman
10.4230/LIPIcs.ICALP.2016.54
Zeyuan Allen-Zhu and Lorenzo Orecchia. Nearly-linear time positive LP solver with faster convergence rate. In Proc. ACM STOC'15, 2015.
Zeyuan Allen-Zhu and Lorenzo Orecchia. A novel, simple interpretation of Nesterov’s accelerated method as a combination of gradient and mirror descent, Jan. 2015. arXiv preprint, URL: http://arxiv.org/abs/1407.1537.
http://arxiv.org/abs/1407.1537
Zeyuan Allen-Zhu and Lorenzo Orecchia. Using optimization to break the epsilon barrier: A faster and simpler width-independent algorithm for solving positive linear programs in parallel. In Proc. ACM-SIAM SODA'15, 2015.
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Baruch Awerbuch, Yossi Azar, and Rohit Khandekar. Fast load balancing via bounded best response. In Proc. ACM-SIAM SODA'08, 2008.
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Baruch Awerbuch and Rohit Khandekar. Stateless distributed gradient descent for positive linear programs. SIAM J. Comput., 38(6):2468-2486, 2009.
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Ali Ghodsi, Matei Zaharia, Benjamin Hindman, Andy Konwinski, Scott Shenker, and Ion Stoica. Dominant resource fairness: Fair allocation of multiple resource types. In Proc. USENIX NSDI'11, 2011.
Sungjin Im, Janardhan Kulkarni, and Kamesh Munagala. Competitive algorithms from competitive equilibria: Non-clairvoyant scheduling under polyhedral constraints. In Proc. ACM STOC'14, 2014.
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Kamal Jain and Vijay Vazirani. Eisenberg-gale markets: Algorithms and structural properties. In Proc. ACM STOC'07, 2007.
Carlee Joe-Wong, Soumya Sen, Tian Lan, and Mung Chiang. Multiresource allocation: Fairness-efficiency tradeoffs in a unifying framework. IEEE/ACM Trans. Netw., 21(6):1785-1798, 2013.
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Jelena Marašević, Cliff Stein, and Gil Zussman. A fast distributed stateless algorithm for α-fair packing problems, Feb. 2016. arXiv preprint, URL: http://arxiv.org/abs/1502.03372v3.
http://arxiv.org/abs/1502.03372v3
Bill McCormick, Frank Kelly, Patrice Plante, Paul Gunning, and Peter Ashwood-Smith. Real time alpha-fairness based traffic engineering. In Proc. ACM HotSDN'14, 2014.
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Creative Commons Attribution 3.0 Unported license
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All-Pairs Approximate Shortest Paths and Distance Oracle Preprocessing
Given an undirected, unweighted graph G on n nodes, there is an O(n^2*poly log(n))-time algorithm that computes a data structure called distance oracle of size O(n^{5/3}*poly log(n)) answering approximate distance queries in constant time. For nodes at distance d the distance estimate is between d and 2d + 1.
This new distance oracle improves upon the oracles of Patrascu and Roditty (FOCS 2010), Abraham and Gavoille (DISC 2011), and Agarwal and Brighten Godfrey (PODC 2013) in terms of preprocessing time, and upon the oracle of Baswana and Sen (SODA 2004) in terms of stretch. The running time analysis is tight (up to logarithmic factors) due to a recent lower bound of Abboud and Bodwin (STOC 2016).
Techniques include dominating sets, sampling, balls, and spanners, and the main contribution lies in the way these techniques are combined. Perhaps the most interesting aspect from a technical point of view is the application of a spanner without incurring its constant additive stretch penalty.
graph algorithms
data structures
approximate shortest paths
distance oracles
distance labels
55:1-55:13
Regular Paper
Christian
Sommer
Christian Sommer
10.4230/LIPIcs.ICALP.2016.55
Amir Abboud and Greg Bodwin. The 4/3 additive spanner exponent is tight. In 48th ACM Symposium on Theory of Computing (STOC), 2016. To appear, available from: URL: http://arxiv.org/abs/1511.00700.
http://arxiv.org/abs/1511.00700
Ittai Abraham and Cyril Gavoille. On approximate distance labels and routing schemes with affine stretch. In 25th International Symposium on Distributed Computing (DISC), pages 404-415, 2011. URL: http://dx.doi.org/10.1007/978-3-642-24100-0_39.
http://dx.doi.org/10.1007/978-3-642-24100-0_39
Rachit Agarwal. The space-stretch-time tradeoff in distance oracles. In 22nd European Symposium on Algorithms (ESA), pages 49-60, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44777-2_5.
http://dx.doi.org/10.1007/978-3-662-44777-2_5
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Total Space in Resolution Is at Least Width Squared
Given an unsatisfiable k-CNF formula phi we consider two complexity measures in Resolution: width and total space. The width is the minimal W such that there exists a Resolution refutation of phi with clauses of at most W literals. The total space is the minimal size T of a memory used to write down a Resolution refutation of phi where the size of the memory is measured as the total number of literals it can contain. We prove that T = Omega((W - k)^2).
Resolution
width
total space
56:1-56:13
Regular Paper
Ilario
Bonacina
Ilario Bonacina
10.4230/LIPIcs.ICALP.2016.56
Michael Alekhnovich, Eli Ben-Sasson, Alexander A. Razborov, and Avi Wigderson. Space complexity in propositional calculus. SIAM J. Comput., 31(4):1184-1211, 2002. URL: http://dx.doi.org/10.1137/S0097539700366735.
http://dx.doi.org/10.1137/S0097539700366735
Albert Atserias and Víctor Dalmau. A combinatorial characterization of resolution width. J. Comput. Syst. Sci., 74(3):323-334, 2008. URL: http://dx.doi.org/10.1016/j.jcss.2007.06.025.
http://dx.doi.org/10.1016/j.jcss.2007.06.025
Albert Atserias, Massimo Lauria, and Jakob Nordström. Narrow proofs may be maximally long. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 286-297. IEEE, 2014. URL: http://dx.doi.org/10.1109/CCC.2014.36.
http://dx.doi.org/10.1109/CCC.2014.36
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http://www.aaai.org/Library/AAAI/1997/aaai97-032.php
Paul Beame, Christopher Beck, and Russell Impagliazzo. Time-space tradeoffs in resolution: superpolynomial lower bounds for superlinear space. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19-22, 2012, pages 213-232. ACM, 2012. URL: http://dx.doi.org/10.1145/2213977.2213999.
http://dx.doi.org/10.1145/2213977.2213999
Paul Beame, Richard M. Karp, Toniann Pitassi, and Michael E. Saks. On the complexity of unsatisfiability proofs for random k-CNF formulas. In Jeffrey Scott Vitter, editor, Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 561-571. ACM, 1998. URL: http://dx.doi.org/10.1145/276698.276870.
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Chris Beck, Jakob Nordström, and Bangsheng Tang. Some trade-off results for polynomial calculus: extended abstract. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 813-822. ACM, 2013. URL: http://dx.doi.org/10.1145/2488608.2488711.
http://dx.doi.org/10.1145/2488608.2488711
Eli Ben-Sasson and Nicola Galesi. Space complexity of random formulae in resolution. Random Struct. Algorithms, 23(1):92-109, 2003. URL: http://dx.doi.org/10.1002/rsa.10089.
http://dx.doi.org/10.1002/rsa.10089
Eli Ben-Sasson and Jakob Nordström. Understanding space in resolution: optimal lower bounds and exponential trade-offs. In Peter Bro Miltersen, Rüdiger Reischuk, Georg Schnitger, and Dieter van Melkebeek, editors, Computational Complexity of Discrete Problems, 14.09.-19.09.2008, volume 08381 of Dagstuhl Seminar Proceedings. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Germany, 2008. URL: http://drops.dagstuhl.de/opus/volltexte/2008/1781/.
http://drops.dagstuhl.de/opus/volltexte/2008/1781/
Eli Ben-Sasson and Jakob Nordström. A space hierarchy for k-DNF resolution. Electronic Colloquium on Computational Complexity (ECCC), 16:47, 2009. URL: http://eccc.hpi-web.de/report/2009/047.
http://eccc.hpi-web.de/report/2009/047
Eli Ben-Sasson and Jakob Nordström. Understanding space in proof complexity: Separations and trade-offs via substitutions. In Bernard Chazelle, editor, Innovations in Computer Science - ICS 2010, Tsinghua University, Beijing, China, January 7-9, 2011. Proceedings, pages 401-416. Tsinghua University Press, 2011. URL: http://conference.itcs.tsinghua.edu.cn/ICS2011/content/papers/3.html.
http://conference.itcs.tsinghua.edu.cn/ICS2011/content/papers/3.html
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http://dx.doi.org/10.1145/375827.375835
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http://arxiv.org/abs/1503.01613
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http://dx.doi.org/10.1137/060668250
Jakob Nordström. Pebble games, proof complexity, and time-space trade-offs. Logical Methods in Computer Science, 9(3), 2013. URL: http://dx.doi.org/10.2168/LMCS-9(3:15)2013.
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Jakob Nordström. On the interplay between proof complexity and SAT solving. ACM SIGLOG News, 2(3):19-44, August 2015.
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Alexander A. Razborov. Proof complexity of pigeonhole principles. In Werner Kuich, Grzegorz Rozenberg, and Arto Salomaa, editors, Developments in Language Theory, 5th International Conference, DLT 2001, Vienna, Austria, July 16-21, 2001, Revised Papers, volume 2295 of Lecture Notes in Computer Science, pages 100-116. Springer, 2001.
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http://dx.doi.org/10.1007/978-3-642-81955-1_28
Alasdair Urquhart. Hard examples for resolution. J. ACM, 34(1):209-219, 1987. URL: http://dx.doi.org/10.1145/7531.8928.
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http://dx.doi.org/10.1007/s11225-011-9356-9
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Supercritical Space-Width Trade-Offs for Resolution
We show that there are CNF formulas which can be refuted in resolution in both small space and small width, but for which any small-width resolution proof must have space exceeding by far the linear worst-case upper bound. This significantly strengthens the space-width trade-offs in [Ben-Sasson 2009], and provides one more example of trade-offs in the "supercritical" regime above worst case recently identified by [Razborov 2016]. We obtain our results by using Razborov’s new hardness condensation technique and combining it with the space lower bounds in [Ben-Sasson and Nordström 2008].
Proof complexity
resolution
space
width
trade-offs
supercritical
57:1-57:14
Regular Paper
Christoph
Berkholz
Christoph Berkholz
Jakob
Nordström
Jakob Nordström
10.4230/LIPIcs.ICALP.2016.57
Michael Alekhnovich, Eli Ben-Sasson, Alexander A. Razborov, and Avi Wigderson. Space complexity in propositional calculus. SIAM Journal on Computing, 31(4):1184-1211, 2002. Preliminary version in STOC'00.
Albert Atserias and Víctor Dalmau. A combinatorial characterization of resolution width. Journal of Computer and System Sciences, 74(3):323-334, May 2008. Preliminary version in CCC'03.
Albert Atserias, Massimo Lauria, and Jakob Nordström. Narrow proofs may be maximally long. In Proceedings of the 29th Annual IEEE Conference on Computational Complexity (CCC'14), pages 286-297, June 2014.
Roberto J. Bayardo Jr. and Robert Schrag. Using CSP look-back techniques to solve real-world SAT instances. In Proceedings of the 14th National Conference on Artificial Intelligence (AAAI'97), pages 203-208, July 1997.
Paul Beame, Chris Beck, and Russell Impagliazzo. Time-space tradeoffs in resolution: Superpolynomial lower bounds for superlinear space. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC'12), pages 213-232, May 2012.
Paul Beame, Richard Karp, Toniann Pitassi, and Michael Saks. The efficiency of resolution and Davis-Putnam procedures. SIAM Journal on Computing, 31(4):1048-1075, 2002. Preliminary versions of these results appeared in FOCS'96 and STOC'98.
Chris Beck, Jakob Nordström, and Bangsheng Tang. Some trade-off results for polynomial calculus. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC'13), pages 813-822, May 2013.
Eli Ben-Sasson. Size-space tradeoffs for resolution. SIAM Journal on Computing, 38(6):2511-2525, May 2009. Preliminary version in STOC'02.
Eli Ben-Sasson and Nicola Galesi. Space complexity of random formulae in resolution. Random Structures and Algorithms, 23(1):92-109, August 2003. Preliminary version in CCC'01.
Eli Ben-Sasson and Jakob Nordström. Short proofs may be spacious: An optimal separation of space and length in resolution. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS'08), pages 709-718, October 2008.
Eli Ben-Sasson and Jakob Nordström. Understanding space in proof complexity: Separations and trade-offs via substitutions. In Proceedings of the 2nd Symposium on Innovations in Computer Science (ICS'11), pages 401-416, January 2011.
Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow - resolution made simple. Journal of the ACM, 48(2):149-169, March 2001. Preliminary version in STOC'99.
Patrick Bennett, Ilario Bonacina, Nicola Galesi, Tony Huynh, Mike Molloy, and Paul Wollan. Space proof complexity for random 3-CNFs. Technical Report 1503.01613, arXiv.org, April 2015.
Christoph Berkholz. On the complexity of finding narrow proofs. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS'12), pages 351-360, October 2012.
Archie Blake. Canonical Expressions in Boolean Algebra. PhD thesis, University of Chicago, 1937.
Ilario Bonacina. Total space in resolution is at least width squared. In Proceedings of the 43rd International Colloquium on Automata, Languages and Programming (ICALP'16), July 2016. To appear.
Ilario Bonacina, Nicola Galesi, and Neil Thapen. Total space in resolution. In Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS'14), pages 641-650, October 2014.
Vašek Chvátal and Endre Szemerédi. Many hard examples for resolution. Journal of the ACM, 35(4):759-768, October 1988.
Matthew Clegg, Jeffery Edmonds, and Russell Impagliazzo. Using the Groebner basis algorithm to find proofs of unsatisfiability. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC'96), pages 174-183, May 1996.
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Juan Luis Esteban and Jacobo Torán. Space bounds for resolution. Information and Computation, 171(1):84-97, 2001. Preliminary versions of these results appeared in STACS'99 and CSL'99.
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Alexander Hertel. Applications of Games to Propositional Proof Complexity. PhD thesis, University of Toronto, May 2008. Available at URL: http://www.cs.utoronto.ca/~ahertel/.
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Jakob Nordström. Narrow proofs may be spacious: Separating space and width in resolution. SIAM Journal on Computing, 39(1):59-121, May 2009. Preliminary version in STOC'06.
Jakob Nordström. Pebble games, proof complexity and time-space trade-offs. Logical Methods in Computer Science, 9:15:1-15:63, September 2013.
Jakob Nordström and Johan Håstad. Towards an optimal separation of space and length in resolution. Theory of Computing, 9:471-557, May 2013. Preliminary version in STOC'08.
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Alasdair Urquhart. Hard examples for resolution. Journal of the ACM, 34(1):209-219, January 1987.
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Deterministic Time-Space Trade-Offs for k-SUM
Given a set of numbers, the k-SUM problem asks for a subset of k numbers that sums to zero. When the numbers are integers, the time and space complexity of k-SUM is generally studied in the word-RAM model; when the numbers are reals, the complexity is studied in the real-RAM model, and space is measured by the number of reals held in memory at any point. We present a time and space efficient deterministic self-reduction for the k-SUM problem which holds for both models, and has many interesting consequences. To illustrate:
- 3-SUM is in deterministic time O(n^2*lg(lg(n))/lg(n)) and space O(sqrt(n*lg(n)/lg(lg(n)))). In general, any polylogarithmic-time improvement over quadratic time for 3-SUM can be converted into an algorithm with an identical time improvement but low space complexity as well.
- 3-SUM is in deterministic time O(n^2) and space O(sqrt(n)), derandomizing an algorithm of Wang.
- A popular conjecture states that 3-SUM requires n^{2-o(1)} time on the word-RAM. We show that the 3-SUM Conjecture is in fact equivalent to the (seemingly weaker) conjecture that every O(n^{.51})-space algorithm for 3-SUM requires at least n^{2-o(1)} time on the word-RAM.
- For k >= 4, k-SUM is in deterministic O(n^{k-2+2/k}) time and O(sqrt(n)) space.
3SUM
kSUM
time-space tradeoff
algorithm
58:1-58:14
Regular Paper
Andrea
Lincoln
Andrea Lincoln
Virginia
Vassilevska Williams
Virginia Vassilevska Williams
Joshua R.
Wang
Joshua R. Wang
R. Ryan
Williams
R. Ryan Williams
10.4230/LIPIcs.ICALP.2016.58
Amir Abboud, Kevin Lewi, and Ryan Williams. Losing weight by gaining edges. In Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, pages 1-12, 2014.
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 39-51, 2014.
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 434-443, 2014.
Nir Ailon and Bernard Chazelle. Lower bounds for linear degeneracy testing. J. ACM, 52(2):157-171, 2005.
Amihood Amir, Timothy M. Chan, Moshe Lewenstein, and Noa Lewenstein. On hardness of jumbled indexing. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 114-125, 2014.
Per Austrin, Petteri Kaski, Mikko Koivisto, and Jussi Määttä. Space-time tradeoffs for subset sum: An improved worst case algorithm. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 45-56, 2013.
Ilya Baran, Erik D Demaine, and Mihai Patraşcu. Subquadratic algorithms for 3sum. In Algorithms and Data Structures, pages 409-421. Springer, 2005.
Paul Beame, Raphaël Clifford, and Widad Machmouchi. Element distinctness, frequency moments, and sliding windows. In FOCS, pages 290-299, 2013.
Paul Beame, Michael E. Saks, Xiaodong Sun, and Erik Vee. Time-space trade-off lower bounds for randomized computation of decision problems. J. ACM, 50(2):154-195, 2003.
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Semi-Streaming Algorithms for Annotated Graph Streams
Considerable effort has been devoted to the development of streaming algorithms for analyzing massive graphs. Unfortunately, many results have been negative, establishing that a wide variety of problems require Omega(n^2) space to solve. One of the few bright spots has been the development of semi-streaming algorithms for a handful of graph problems - these algorithms use space O(n*polylog(n)).
In the annotated data streaming model of Chakrabarti et al. [Chakrabarti/Cormode/Goyal/Thaler, ACM Trans. on Alg. 2014], a computationally limited client wants to compute some property of a massive input, but lacks the resources to store even a small fraction of the input, and hence cannot perform the desired computation locally. The client therefore accesses a powerful but untrusted service provider, who not only performs the requested computation, but also proves that the answer is correct.
We consider the notion of semi-streaming algorithms for annotated graph streams (semistreaming annotation schemes for short). These are protocols in which both the client's space usage and the length of the proof are O(n*polylog(n)). We give evidence that semi-streaming annotation schemes represent a more robust solution concept than does the standard semi-streaming model. On the positive side, we give semi-streaming annotation schemes for two dynamic graph problems that are intractable in the standard model: (exactly) counting triangles, and (exactly) computing maximum matchings. The former scheme answers a question of Cormode [Cormode, Problem 47]. On the negative side, we identify for the first time two natural graph problems (connectivity and bipartiteness in a certain edge update model) that can be solved in the standard semi-streaming model, but cannot be solved by annotation schemes of "sub-semi-streaming" cost. That is, these problems are as hard in the annotations model as they are in the standard model.
graph streams
stream verification
annotated data streams
probabilistic proof systems
59:1-59:14
Regular Paper
Justin
Thaler
Justin Thaler
10.4230/LIPIcs.ICALP.2016.59
Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In SODA, pages 459-467, 2012. URL: http://dl.acm.org/citation.cfm?id=2095156.
http://dl.acm.org/citation.cfm?id=2095156
Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Graph sketches: sparsification, spanners, and subgraphs. In PODS, pages 5-14, 2012. URL: http://dx.doi.org/10.1145/2213556.2213560.
http://dx.doi.org/10.1145/2213556.2213560
Noga Alon, Raphael Yuster, and Uri Zwick. Finding and counting given length cycles. Algorithmica, 17(3):209-223, 1997. URL: http://dx.doi.org/10.1007/BF02523189.
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http://dl.acm.org/citation.cfm?id=545381.545464
Amit Chakrabarti, Graham Cormode, Navin Goyal, and Justin Thaler. Annotations for sparse data streams. In SODA, pages 687-706, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.52.
http://dx.doi.org/10.1137/1.9781611973402.52
Amit Chakrabarti, Graham Cormode, Andrew McGregor, and Justin Thaler. Annotations in data streams. ACM Trans. Algorithms, 11(1):7:1-7:30, 2014. URL: http://dx.doi.org/10.1145/2636924.
http://dx.doi.org/10.1145/2636924
Amit Chakrabarti, Graham Cormode, Andrew McGregor, Justin Thaler, and Suresh Venkatasubramanian. Verifiable stream computation and arthur-merlin communication. In 30th Conference on Computational Complexity, CCC 2015, June 17-19, 2015, Portland, Oregon, USA, pages 217-243, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2015.217.
http://dx.doi.org/10.4230/LIPIcs.CCC.2015.217
Kai-Min Chung, Yael Tauman Kalai, Feng-Hao Liu, and Ran Raz. Memory delegation. In CRYPTO, pages 151-168, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22792-9_9.
http://dx.doi.org/10.1007/978-3-642-22792-9_9
Graham Cormode and Donatella Firmani. A unifying framework for l₀-sampling algorithms. Distributed and Parallel Databases, 32(3):315-335, 2014. URL: http://dx.doi.org/10.1007/s10619-013-7131-9.
http://dx.doi.org/10.1007/s10619-013-7131-9
Graham Cormode, Michael Mitzenmacher, and Justin Thaler. Practical verified computation with streaming interactive proofs. In ITCS, pages 90-112, 2012. URL: http://dx.doi.org/10.1145/2090236.2090245.
http://dx.doi.org/10.1145/2090236.2090245
Graham Cormode, Michael Mitzenmacher, and Justin Thaler. Streaming graph computations with a helpful advisor. Algorithmica, 65(2):409-442, 2013. URL: http://dx.doi.org/10.1007/s00453-011-9598-y.
http://dx.doi.org/10.1007/s00453-011-9598-y
Graham Cormode, Justin Thaler, and Ke Yi. Verifying computations with streaming interactive proofs. PVLDB, 5(1):25-36, 2011. URL: http://www.vldb.org/pvldb/vol5/p025_grahamcormode_vldb2012.pdf.
http://www.vldb.org/pvldb/vol5/p025_grahamcormode_vldb2012.pdf
Ashish Goel, Michael Kapralov, and Ian Post. Single pass sparsification in the streaming model with edge deletions. CoRR, abs/1203.4900, 2012. URL: http://arxiv.org/abs/1203.4900.
http://arxiv.org/abs/1203.4900
Shafi Goldwasser, Yael Tauman Kalai, and Guy N. Rothblum. Delegating computation: interactive proofs for muggles. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, STOC'08, pages 113-122, New York, NY, USA, 2008. ACM. URL: http://dx.doi.org/10.1145/1374376.1374396.
http://dx.doi.org/10.1145/1374376.1374396
Shafi Goldwasser, Silvio Micali, and Charles Rackoff. The knowledge complexity of interactive proof systems. SIAM J. Comput., 18(1):186-208, 1989. URL: http://dx.doi.org/10.1137/0218012.
http://dx.doi.org/10.1137/0218012
Michael T. Goodrich and Michael Mitzenmacher. Invertible bloom lookup tables. In Allerton, pages 792-799, 2011. URL: http://dx.doi.org/10.1109/Allerton.2011.6120248.
http://dx.doi.org/10.1109/Allerton.2011.6120248
Tom Gur and Ran Raz. Arthur-Merlin streaming complexity. In Proceedings of the 40th International Colloquium on Automata, Languages and Programming: Part I, ICALP'13, Berlin, Heidelberg, 2013. Springer-Verlag.
Hartmut Klauck and Ved Prakash. Streaming computations with a loquacious prover. In ITCS, pages 305-320, 2013. URL: http://dx.doi.org/10.1145/2422436.2422471.
http://dx.doi.org/10.1145/2422436.2422471
Hartmut Klauck and Ved Prakash. An improved interactive streaming algorithm for the distinct elements problem. In ICALP (1), pages 919-930, 2014. URL: http://dx.doi.org/10.1007/978-3-662-43948-7_76.
http://dx.doi.org/10.1007/978-3-662-43948-7_76
Troy Lee, Frédéric Magniez, and Miklos Santha. Improved quantum query algorithms for triangle finding and associativity testing. In SODA, pages 1486-1502, 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.107.
http://dx.doi.org/10.1137/1.9781611973105.107
List of open problems in sublinear algorithms: Problem 47. URL: http://sublinear.info/47.
http://sublinear.info/47
Carsten Lund, Lance Fortnow, Howard Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. J. ACM, 39:859-868, October 1992. URL: http://dx.doi.org/10.1145/146585.146605.
http://dx.doi.org/10.1145/146585.146605
Andrew McGregor. Graph mining on streams. In Ling Liu and M. Tamer Özsu, editors, Encyclopedia of Database Systems, pages 1271-1275. Springer US, 2009. URL: http://dx.doi.org/10.1007/978-0-387-39940-9_184.
http://dx.doi.org/10.1007/978-0-387-39940-9_184
Andrew McGregor. Graph stream algorithms: A survey. SIGMOD Rec., 43(1):9-20, May 2014. URL: http://dx.doi.org/10.1145/2627692.2627694.
http://dx.doi.org/10.1145/2627692.2627694
S. Muthukrishnan. Data Streams: Algorithms And Applications. Foundations and Trends in Theoretical Computer Science. Now Publishers Incorporated, 2005.
Charalampos Papamanthou, Elaine Shi, Roberto Tamassia, and Ke Yi. Streaming authenticated data structures. In EUROCRYPT, pages 353-370, 2013. URL: http://dx.doi.org/10.1007/978-3-642-38348-9_22.
http://dx.doi.org/10.1007/978-3-642-38348-9_22
A. Pavan, Kanat Tangwongsan, Srikanta Tirthapura, and Kun-Lung Wu. Counting and sampling triangles from a graph stream. Proc. VLDB Endow., 6(14):1870-1881, September 2013. URL: http://dx.doi.org/10.14778/2556549.2556569.
http://dx.doi.org/10.14778/2556549.2556569
Adi Shamir. IP = PSPACE. J. ACM, 39:869-877, October 1992. URL: http://dx.doi.org/10.1145/146585.146609.
http://dx.doi.org/10.1145/146585.146609
Siddharth Suri and Sergei Vassilvitskii. Counting triangles and the curse of the last reducer. In WWW, pages 607-614, 2011. URL: http://dx.doi.org/10.1145/1963405.1963491.
http://dx.doi.org/10.1145/1963405.1963491
Justin Thaler. Time-optimal interactive proofs for circuit evaluation. In CRYPTO (2), pages 71-89, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40084-1_5.
http://dx.doi.org/10.1007/978-3-642-40084-1_5
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Randomized Query Complexity of Sabotaged and Composed Functions
We study the composition question for bounded-error randomized query complexity: Is R(f circ g) = Omega(R(f)R(g))? We show that inserting a simple function h, whose query complexity is onlyTheta(log R(g)), in between f and g allows us to prove R(f circ h circ g) = Omega(R(f)R(h)R(g)).
We prove this using a new lower bound measure for randomized query complexity we call randomized sabotage complexity, RS(f). Randomized sabotage complexity has several desirable properties, such as a perfect composition theorem, RS(f circ g) >= RS(f) RS(g), and a composition theorem with randomized query complexity, R(f circ g) = Omega(R(f) RS(g)). It is also a quadratically tight lower bound for total functions and can be quadratically superior to the partition bound, the best known general lower bound for randomized query complexity.
Using this technique we also show implications for lifting theorems in communication complexity. We show that a general lifting theorem from zero-error randomized query to communication complexity implies a similar result for bounded-error algorithms for all total functions.
Randomized query complexity
decision tree complexity
composition theorem
partition bound
lifting theorem
60:1-60:14
Regular Paper
Ben-David
Shalev
Ben-David Shalev
Robin
Kothari
Robin Kothari
10.4230/LIPIcs.ICALP.2016.60
Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. To appear in Proceedings of STOC 2016. arXiv preprint http://arxiv.org/abs/arXiv:1511.01937, 2015.
Andris Ambainis, Martins Kokainis, and Robin Kothari. Nearly optimal separations between communication (or query) complexity and partitions. To appear in Proceedings of CCC 2016. arXiv preprints http://arxiv.org/abs/arXiv:1512.00661 and http://arxiv.org/abs/arXiv:1512.01210, 2015.
Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002. URL: http://dx.doi.org/10.1016/S0304-3975(01)00144-X.
http://dx.doi.org/10.1016/S0304-3975(01)00144-X
Andrew Drucker. Improved direct product theorems for randomized query complexity. Computational Complexity, 21(2):197-244, 2012. URL: http://dx.doi.org/10.1007/s00037-012-0043-7.
http://dx.doi.org/10.1007/s00037-012-0043-7
Mika Göös, T.S. Jayram, Toniann Pitassi, and Thomas Watson. Randomized communication vs. partition number. Electronic Colloquium on Computational Complexity (ECCC) http://eccc.hpi-web.de/report/2015/169/, 2015.
http://eccc.hpi-web.de/report/2015/169/
Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 1077-1088, Oct 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.70.
http://dx.doi.org/10.1109/FOCS.2015.70
Peter Høyer, Troy Lee, and Robert Špalek. Negative weights make adversaries stronger. In Proceedings of the 39th ACM Symposium on Theory of Computing (STOC 2007), pages 526-535, 2007. URL: http://dx.doi.org/10.1145/1250790.1250867.
http://dx.doi.org/10.1145/1250790.1250867
Rahul Jain and Hartmut Klauck. The partition bound for classical communication complexity and query complexity. In Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity, CCC'10, pages 247-258, 2010. URL: http://dx.doi.org/10.1109/CCC.2010.31.
http://dx.doi.org/10.1109/CCC.2010.31
Rahul Jain, Hartmut Klauck, and Miklos Santha. Optimal direct sum results for deterministic and randomized decision tree complexity. Information Processing Letters, 110(20):893-897, 2010. URL: http://dx.doi.org/10.1016/j.ipl.2010.07.020.
http://dx.doi.org/10.1016/j.ipl.2010.07.020
Rahul Jain, Troy Lee, and Nisheeth K. Vishnoi. A quadratically tight partition bound for classical communication complexity and query complexity. arXiv preprint http://arxiv.org/abs/arXiv:1401.4512, 2014.
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http://dx.doi.org/10.1007/BF01206317
Shelby Kimmel. Quantum adversary (upper) bound. In Automata, Languages, and Programming, volume 7391 of Lecture Notes in Computer Science, pages 557-568, 2012. URL: http://dx.doi.org/10.1007/978-3-642-31594-7_47.
http://dx.doi.org/10.1007/978-3-642-31594-7_47
Raghav Kulkarni and Avishay Tal. On fractional block sensitivity. Electronic Colloquium on Computational Complexity (ECCC) http://eccc.hpi-web.de/report/2013/168/, 2013.
http://eccc.hpi-web.de/report/2013/168/
Troy Lee, Rajat Mittal, Ben W. Reichardt, Robert Špalek, and Mario Szegedy. Quantum query complexity of state conversion. In Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science (FOCS 2011), pages 344-353, 2011. http://arxiv.org/abs/1011.3020, URL: http://dx.doi.org/10.1109/FOCS.2011.75.
http://dx.doi.org/10.1109/FOCS.2011.75
Ashley Montanaro. A composition theorem for decision tree complexity. Chicago Journal of Theoretical Computer Science, 2014(6), July 2014. URL: http://dx.doi.org/10.4086/cjtcs.2014.006.
http://dx.doi.org/10.4086/cjtcs.2014.006
Denis Pankratov. Direct sum questions in classical communication complexity. Master’s thesis, University of Chicago, 2012.
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http://dx.doi.org/10.1109/SFCS.1977.24
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Coding for Interactive Communication Correcting Insertions and Deletions
We consider the question of interactive communication, in which two remote parties perform a computation while their communication channel is (adversarially) noisy. We extend here the discussion into a more general and stronger class of noise, namely, we allow the channel to perform insertions and deletions of symbols. These types of errors may bring the parties "out of sync", so that there is no consensus regarding the current round of the protocol.
In this more general noise model, we obtain the first interactive coding scheme that has a constant rate and tolerates noise rates of up to 1/18 - epsilon. To this end we develop a novel primitive we name edit distance tree code. The edit distance tree code is designed to replace the Hamming distance constraints in Schulman's tree codes (STOC 93), with a stronger edit distance requirement. However, the straightforward generalization of tree codes to edit distance does not seem to yield a primitive that suffices for communication in the presence of synchronization problems. Giving the "right" definition of edit distance tree codes is a main conceptual contribution of this work.
Interactive communication
coding
edit distance
61:1-61:14
Regular Paper
Mark
Braverman
Mark Braverman
Ran
Gelles
Ran Gelles
Jieming
Mao
Jieming Mao
Rafail
Ostrovsky
Rafail Ostrovsky
10.4230/LIPIcs.ICALP.2016.61
Shweta Agrawal, Ran Gelles, and Amit Sahai. Adaptive protocols for interactive communication. arXiv preprint arXiv:1312.4182, 2013.
Zvika Brakerski and Yael Tauman Kalai. Efficient interactive coding against adversarial noise. In Foundations of Computer Science, IEEE Annual Symposium on, pages 160-166. IEEE Computer Society, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.22.
http://dx.doi.org/10.1109/FOCS.2012.22
Zvika Brakerski, Yael Tauman Kalai, and Moni Naor. Fast interactive coding against adversarial noise. J. ACM, 61(6):35:1-35:30, December 2014. URL: http://dx.doi.org/10.1145/2661628.
http://dx.doi.org/10.1145/2661628
Zvika Brakerski and Moni Naor. Fast algorithms for interactive coding. In SODA'13: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 443-456, 2013.
Gilles Brassard, Ashwin Nayak, Alain Tapp, Dave Touchette, and Falk Unger. Noisy interactive quantum communication. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 296-305, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.39.
http://dx.doi.org/10.1109/FOCS.2014.39
Mark Braverman and Klim Efremenko. List and unique coding for interactive communication in the presence of adversarial noise. In Foundations of Computer Science (FOCS), IEEE 55th Annual Symposium on, pages 236-245, 2014.
Mark Braverman and Anup Rao. Towards coding for maximum errors in interactive communication. In Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC'11, pages 159-166, New York, NY, USA, 2011. ACM. URL: http://dx.doi.org/10.1145/1993636.1993659.
http://dx.doi.org/10.1145/1993636.1993659
Mark Braverman and Anup Rao. Toward coding for maximum errors in interactive communication. Information Theory, IEEE Transactions on, 60(11):7248-7255, Nov 2014. URL: http://dx.doi.org/10.1109/TIT.2014.2353994.
http://dx.doi.org/10.1109/TIT.2014.2353994
Klim Efremenko, Ran Gelles, and Bernhard Haeupler. Maximal noise in interactive communication over erasure channels and channels with feedback. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS'15, pages 11-20, New York, NY, USA, 2015. ACM. URL: http://dx.doi.org/10.1145/2688073.2688077.
http://dx.doi.org/10.1145/2688073.2688077
Matthew Franklin, Ran Gelles, Rafail Ostrovsky, and Leonard J. Schulman. Optimal coding for streaming authentication and interactive communication. In Ran Canetti and Juan A. Garay, editors, CRYPTO'13, volume 8043 of LNCS, pages 258-276. Springer Berlin, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40084-1_15.
http://dx.doi.org/10.1007/978-3-642-40084-1_15
Matthew Franklin, Ran Gelles, Rafail Ostrovsky, and Leonard J. Schulman. Optimal coding for streaming authentication and interactive communication. Information Theory, IEEE Transactions on, 61(1):133-145, Jan 2015. URL: http://dx.doi.org/10.1109/TIT.2014.2367094.
http://dx.doi.org/10.1109/TIT.2014.2367094
Ran Gelles and Bernhard Haeupler. Capacity of interactive communication over erasure channels and channels with feedback. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'15, pages 1296-1311, 2015.
Ran Gelles, Ankur Moitra, and Amit Sahai. Efficient and explicit coding for interactive communication. In Foundations of Computer Science, 2011 IEEE 52nd Annual Symposium on, pages 768-777. IEEE Computer Society, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.51.
http://dx.doi.org/10.1109/FOCS.2011.51
Ran Gelles, Ankur Moitra, and Amit Sahai. Efficient coding for interactive communication. Information Theory, IEEE Transactions on, 60(3):1899-1913, March 2014. URL: http://dx.doi.org/10.1109/TIT.2013.2294186.
http://dx.doi.org/10.1109/TIT.2013.2294186
Mohsen Ghaffari and Bernhard Haeupler. Optimal Error Rates for Interactive Coding II: Efficiency and List Decoding. In Foundations of Computer Science (FOCS), IEEE 55th Annual Symposium on, pages 394-403, 2014.
Mohsen Ghaffari, Bernhard Haeupler, and Madhu Sudan. Optimal error rates for interactive coding I: Adaptivity and other settings. In STOC'14: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 794-803, 2014. URL: http://dx.doi.org/10.1145/2591796.2591872.
http://dx.doi.org/10.1145/2591796.2591872
Bernhard Haeupler. Interactive channel capacity revisited. In Foundations of Computer Science (FOCS), IEEE 55th Annual Symposium on, pages 226-235, 2014.
Jørn Justesen. Class of constructive asymptotically good algebraic codes. Information Theory, IEEE Transactions on, 18(5):652-656, Sep 1972. URL: http://dx.doi.org/10.1109/TIT.1972.1054893.
http://dx.doi.org/10.1109/TIT.1972.1054893
Gillat Kol and Ran Raz. Interactive channel capacity. In STOC'13: Proceedings of the 45th annual ACM Symposium on theory of computing, pages 715-724, 2013. URL: http://dx.doi.org/10.1145/2488608.2488699.
http://dx.doi.org/10.1145/2488608.2488699
Vladimir I. Levenshtein. Binary codes capable of correcting deletions, insertions and reversals. In Soviet physics doklady, volume 10, page 707, 1966.
Rafail Ostrovsky, Yuval Rabani, and Leonard J. Schulman. Error-correcting codes for automatic control. Information Theory, IEEE Transactions on, 55(7):2931-2941, July 2009. URL: http://dx.doi.org/10.1109/TIT.2009.2021303.
http://dx.doi.org/10.1109/TIT.2009.2021303
Leonard J. Schulman. Communication on noisy channels: a coding theorem for computation. Foundations of Computer Science, Annual IEEE Symposium on, pages 724-733, 1992. URL: http://dx.doi.org/10.1109/SFCS.1992.267778.
http://dx.doi.org/10.1109/SFCS.1992.267778
Leonard J. Schulman. Deterministic coding for interactive communication. In STOC'93: Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages 747-756, New York, NY, USA, 1993. ACM. URL: http://dx.doi.org/10.1145/167088.167279.
http://dx.doi.org/10.1145/167088.167279
Leonard J. Schulman. Coding for interactive communication. IEEE Trans. Inf. Theory, 42(6):1745-1756, 1996.
Leonard J. Schulman and David Zuckerman. Asymptotically good codes correcting insertions, deletions, and transpositions. Information Theory, IEEE Transactions on, 45(7):2552-2557, Nov 1999. URL: http://dx.doi.org/10.1109/18.796406.
http://dx.doi.org/10.1109/18.796406
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Amplifiers for the Moran Process
The Moran process, as studied by Lieberman, Hauert and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness r > 1. All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or non-mutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected then the algorithm will eventually reach fixation, in which all individuals are mutants, or extinction, in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every r > 1, the extinction probability tends to 0 as the number of vertices increases. Strong amplification is a rather surprising property - it means that in such graphs, the fixation probability of a uniformly-placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r > 1 (independently of n). The name "strongly amplifying" comes from the fact that this selective advantage is "amplified". Strong amplifiers have received quite a bit of attention, and Lieberman et al. proposed two potentially strongly-amplifying families - superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. We give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family "megastars". When the algorithm is run on an n-vertex graph in this family, starting with a uniformly-chosen mutant, the extinction probability is roughly n^{-1/2} (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of n). Finally, we prove that our analysis of megastars is fairly tight - there is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counterexample which clarifies the literature concerning the isothermal theorem of Lieberman et al. A full version [Galanis/Göbel/Goldberg/Lapinskas/Richerby, Preprint] containing detailed proofs is available at http://arxiv.org/abs/1512.05632. Theorem-numbering here matches the full version.
Moran process
randomised algorithm on graphs
evolutionary dynamics
62:1-62:13
Regular Paper
Andreas
Galanis
Andreas Galanis
Andreas
Göbel
Andreas Göbel
Leslie Ann
Goldberg
Leslie Ann Goldberg
John
Lapinskas
John Lapinskas
David
Richerby
David Richerby
10.4230/LIPIcs.ICALP.2016.62
B. Adlam, K. Chatterjee, and M. A. Nowak. Amplifiers of selection. Proceedings of the Royal Society A, 471(2181), 2015.
Chalee Asavathiratham, Sandip Roy, Bernard Lesieutre, and George Verghese. The influence model. IEEE Control Systems, 21(6):52-64, 2001.
Eli Berge. Dynamic monopolies of constant size. Journal of Combinatorial Theory, Series B, 83(2):191-200, 2001.
Carol Bezuidenhout and Geoffrey Grimmett. The critical contact process dies out. Ann. Probab., 18(4):1462-1482, 10 1990. URL: http://dx.doi.org/10.1214/aop/1176990627.
http://dx.doi.org/10.1214/aop/1176990627
M. Broom and J. Rychtár. An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proceedings of the Royal Society A, 464:2609-2627, 2008.
J. Díaz, L. A. Goldberg, G. B. Mertzios, D. Richerby, M. J. Serna, and P. G. Spirakis. On the fixation probability of superstars. Proceedings of the Royal Society A, 469(2156):20130193, 2013.
J. Díaz, L. A. Goldberg, G. B. Mertzios, D. Richerby, M. J. Serna, and P. G. Spirakis. Approximating fixation probabilities in the generalised Moran process. Algorithmica, 69(1):78-91, 2014.
Josep Díaz, Leslie Ann Goldberg, David Richerby, and Maria J. Serna. Absorption time of the Moran process. Random Structures and Algorithms, To Appear.
Richard Durrett and Jeffrey E. Steif. Fixation results for threshold voter systems. Ann. Probab., 21(1):232-247, 01 1993. URL: http://dx.doi.org/10.1214/aop/1176989403.
http://dx.doi.org/10.1214/aop/1176989403
Rick Durrett. Some features of the spread of epidemics and information on random graphs. Proceedings of the National Academy of Science, 107(10):4491-4498, 2010.
Andreas Galanis, Andreas Göbel, Leslie Ann Goldberg, John Lapinskas, and David Richerby. Amplifiers for the Moran process. Preprint.
Herbert Gintis. Game Theory Evolving: A Problem-Centered Introduction to Modeling Strategic Interaction. Princeton University Press, 2000.
C. Hauert. Evolutionary dynamics. In A. T. Skjeltorp and A. V. Belushkin, editors, Proceedings of the NATO Advanced Study Institute on Evolution from Cellular to Social Scales, pages 11-44. Springer, 2008.
A. Jamieson-Lane and C. Hauert. Fixation probabilities on superstars, revisited and revised. Journal of Theoretical Biology, 382:44-56, 2015.
David Kempe, Jon Kleinberg, and Éva Tardos. Maximizing the spread of influence through a social network. In Proc. 9th ACM International Conference on Knowledge Discovery and Data Mining, pages 137-146. ACM, 2003.
E. Lieberman, C. Hauert, and M. A. Nowak. Evolutionary dynamics on graphs. Nature, 433(7023):312-316, 2005. Supplementary material available at URL: http://www.nature.com/nature/journal/v433/n7023/full/nature03204.html.
http://www.nature.com/nature/journal/v433/n7023/full/nature03204.html
Thomas M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, 1999.
George B. Mertzios, Sotiris E. Nikoletseas, Christoforos Raptopoulos, and Paul G. Spirakis. Natural models for evolution on networks. Theor. Comput. Sci., 477:76-95, 2013. URL: http://dx.doi.org/10.1016/j.tcs.2012.11.032.
http://dx.doi.org/10.1016/j.tcs.2012.11.032
George B. Mertzios and Paul G. Spirakis. Strong bounds for evolution in networks. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part II, pages 669-680, 2013. URL: http://dx.doi.org/10.1007/978-3-642-39212-2_58.
http://dx.doi.org/10.1007/978-3-642-39212-2_58
P. A. P. Moran. Random processes in genetics. Proceedings of the Cambridge Philosophical Society, 54(1):60-71, 1958.
Devavrat Shah. Gossip algorithms. Found. Trends Netw., 3(1):1-125, January 2009. URL: http://dx.doi.org/10.1561/1300000014.
http://dx.doi.org/10.1561/1300000014
P. Shakarian, P. Roos, and A. Johnson. A review of evolutionary graph theory with applications to game theory. BioSystems, 107(2):66-80, 2012.
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Mixing Time of Markov Chains, Dynamical Systems and Evolution
In this paper we study the mixing time of evolutionary Markov chains over populations of a fixed size (N) in which each individual can be one of m types. These Markov chains have the property that they are guided by a dynamical system from the m-dimensional probability simplex to itself. Roughly, given the current state of the Markov chain, which can be viewed as a probability distribution over the m types, the next state is generated by applying this dynamical system to this distribution, and then sampling from it N times. Many processes in nature, from biology to sociology, are evolutionary and such chains can be used to model them. In this study, the mixing time is of particular interest as it determines the speed of evolution and whether the statistics of the steady state can be efficiently computed. In a recent result [Panageas, Srivastava, Vishnoi, Soda, 2016], it was suggested that the mixing time of such Markov chains is connected to the geometry of this guiding dynamical system. In particular, when the dynamical system has a fixed point which is a global attractor, then the mixing is fast. The limit sets of dynamical systems, however, can exhibit more complex behavior: they could have multiple fixed points that are not necessarily stable, periodic orbits, or even chaos. Such behavior arises in important evolutionary settings such as the dynamics of sexual evolution and that of grammar acquisition. In this paper we prove that the geometry of the dynamical system can also give tight mixing time bounds when the dynamical system has multiple fixed points and periodic orbits. We show that the mixing time continues to remain small in the presence of several unstable fixed points and is exponential in N when there are two or more stable fixed points. As a consequence of our results, we obtain a phase transition result for the mixing time of the sexual/grammar model mentioned above. We arrive at the conclusion that in the interesting parameter regime for these models, i.e., when there are multiple stable fixed points, the mixing is slow. Our techniques strengthen the connections between Markov chains and dynamical systems and we expect that the tools developed in this paper should have a wider applicability.
Markov chains
Mixing time
Dynamical Systems
Evolutionary dynamics
Language evolution
63:1-63:14
Regular Paper
Ioannis
Panageas
Ioannis Panageas
Nisheeth K.
Vishnoi
Nisheeth K. Vishnoi
10.4230/LIPIcs.ICALP.2016.63
L. Baum and J. Eagon. An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology. Bull. Amer. Math. Soc., 73:360-363, 1967.
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Information Cascades on Arbitrary Topologies
In this paper, we study information cascades on graphs. In this setting, each node in the graph represents a person. One after another, each person has to take a decision based on a private signal as well as the decisions made by earlier neighboring nodes. Such information cascades commonly occur in practice and have been studied in complete graphs where everyone can overhear the decisions of every other player. It is known that information cascades can be fragile and based on very little information, and that they have a high likelihood of being wrong.
Generalizing the problem to arbitrary graphs reveals interesting insights. In particular, we show that in a random graph G(n,q), for the right value of q, the number of nodes making a wrong decision is logarithmic in n. That is, in the limit for large n, the fraction of players that make a wrong decision tends to zero. This is intriguing because it contrasts to the two natural corner cases: empty graph (everyone decides independently based on his private signal) and complete graph (all decisions are heard by all nodes). In both of these cases a constant fraction of nodes make a wrong decision in expectation. Thus, our result shows that while both too little and too much information sharing causes nodes to take wrong decisions, for exactly the right amount of information sharing, asymptotically everyone can be right. We further show that this result in random graphs is asymptotically optimal for any topology, even if nodes follow a globally optimal algorithmic strategy. Based on the analysis of random graphs, we explore how topology impacts global performance and construct an optimal deterministic topology among layer graphs.
Information Cascades
Herding Effect
Random Graphs
64:1-64:14
Regular Paper
Jun
Wan
Jun Wan
Yu
Xia
Yu Xia
Liang
Li
Liang Li
Thomas
Moscibroda
Thomas Moscibroda
10.4230/LIPIcs.ICALP.2016.64
Daron Acemoglu, Munther A. Dahleh, Ilan Lobel, and Asuman Ozdaglar. Bayesian Learning in Social Networks. Review of Economic Studies, 78(4):1201-1236, 2011. URL: https://ideas.repec.org/a/oup/restud/v78y2011i4p1201-1236.html.
https://ideas.repec.org/a/oup/restud/v78y2011i4p1201-1236.html
Daron Acemoglu and Asuman Ozdaglar. Opinion dynamics and learning in social networks. Dynamic Games and Applications, 1(1):3-49, 2010.
Lisa R Anderson and Charles A Holt. Classroom games: Information cascades. The Journal of Economic Perspectives, 10(4):187-193, 1996.
Lisa R Anderson and Charles A Holt. Information cascades in the laboratory. The American economic review, pages 847-862, 1997.
Abhijit Banerjee and Drew Fudenberg. Word-of-mouth learning. Games and Economic Behavior, 46(1):1-22, January 2004.
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Sushil Bikhchandani, David Hirshleifer, and Ivo Welch. A theory of fads, fashion, custom, and cultural change as informational cascades. Journal of political Economy, pages 992-1026, 1992.
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MohammadTaghi Hajiaghayi, Hamid Mahini, and David Malec. The polarizing effect of network influences. In Proceedings of the Fifteenth ACM Conference on Economics and Computation, EC'14, pages 131-148, New York, NY, USA, 2014. ACM.
MohammadTaghi Hajiaghayi, Hamid Mahini, and Anshul Sawant. Scheduling a cascade with opposing influences. In Algorithmic Game Theory: 6th International Symposium, SAGT 2013, Aachen, Germany, October 21-23, 2013. Proceedings, pages 195-206, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg.
Elias Koutsoupias and Christos Papadimitriou. Worst-case equilibria. In STACS 99: 16th Annual Symposium on Theoretical Aspects of Computer Science Trier, Germany, March 4-6, 1999 Proceedings, pages 404-413, Berlin, Heidelberg, 1999. Springer Berlin Heidelberg.
David Krackhardt. A plunge into networks. Science, 326(5949):47-48, 2009. URL: http://dx.doi.org/10.1126/science.1167367.
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Arthur W Brian. Competing technologies, increasing returns, and lock-in by historical events. The Economic Journal, 99(394):116-131, 1989.
Jun Wan, Yu Xia, Liang Li, and Thomas Moscibroda. Information cascades on arbitrary topologies, 2016. URL: http://arxiv.org/abs/1604.07166.
http://arxiv.org/abs/1604.07166
Ivo Welch. Sequential sales, learning, and cascades. The Journal of finance, 47(2):695-732, 1992.
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Analysing Survey Propagation Guided Decimationon Random Formulas
Let vec(theta) be a uniformly distributed random k-SAT formula with n variables and m clauses. For clauses/variables ratio m/n <= r_{k-SAT} ~ 2^k*ln(2) the formula vec(theta) is satisfiable with high probability. However, no efficient algorithm is known to provably find a satisfying assignment beyond m/n ~ 2k*ln(k)/k with a non-vanishing probability. Non-rigorous statistical mechanics work on k-CNF led to the development of a new efficient "message passing algorithm" called Survey Propagation Guided Decimation [Mézard et al., Science 2002]. Experiments conducted for k=3,4,5 suggest that the algorithm finds satisfying assignments close to r_{k-SAT}. However, in the present paper we prove that the basic version of Survey Propagation Guided Decimation fails to solve random k-SAT formulas efficiently already for m/n = 2^{k}(1 + epsilon_k)*ln(k)/k with lim_{k -> infinity} epsilon_k = 0 almost a factor k below r_{k-SAT}.
Survey Propagation Guided Decimation
Message Passing Algorithm
Graph Theory
Random k-SAT
65:1-65:12
Regular Paper
Samuel
Hetterich
Samuel Hetterich
10.4230/LIPIcs.ICALP.2016.65
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Approximation Algorithms for Aversion k-Clustering via Local k-Median
In the aversion k-clustering problem, given a metric space, we want to cluster the points into k clusters. The cost incurred by each point is the distance to the furthest point in its cluster, and the cost of the clustering is the sum of all these per-point-costs. This problem is motivated by questions in generating automatic abstractions of extensive-form games.
We reduce this problem to a "local" k-median problem where each facility has a prescribed radius and can only connect to clients within that radius. Our main results is a constant-factor approximation algorithm for the aversion k-clustering problem via the local k-median problem.
We use a primal-dual approach; our technical contribution is a non-local rounding step which we feel is of broader interest.
Approximation algorithms
clustering
k-median
primal-dual
66:1-66:13
Regular Paper
Anupam
Gupta
Anupam Gupta
Guru
Guruganesh
Guru Guruganesh
Melanie
Schmidt
Melanie Schmidt
10.4230/LIPIcs.ICALP.2016.66
Hyung-Chan An, Aditya Bhaskara, Chandra Chekuri, Shalmoli Gupta, Vivek Madan, and Ola Svensson. Centrality of trees for capacitated k-center. Mathematical Programming, 154(1-2):29-53, 2015.
Vijay Arya, Naveen Garg, Rohit Khandekar, Adam Meyerson, Kamesh Munagala, and Vinayaka Pandit. Local search heuristics for k-median and facility location problems. SIAM Journal on Computing, 33(3):544-562, 2004.
Michel Louis Balinski. On finding integer solutions to linear programs. Technical report, DTIC Document, 1964.
Yair Bartal, Moses Charikar, and Danny Raz. Approximating min-sum k-clustering in metric spaces. In Proceedings of the 33rd STOC, pages 11-20, 2001.
Babak Behsaz, Zachary Friggstad, Mohammad R. Salavatipour, and Rohit Sivakumar. Approximation algorithms for min-sum k-clustering and balanced k-median. In Proceedings of the 42nd ICALP, pages 116-128, 2015.
Jaroslaw Byrka, Thomas Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An improved approximation for k-median, and positive correlation in budgeted optimization. In Proceedings of the 26th SODA, pages 737-756, 2015.
Moses Charikar and Sudipto Guha. Improved combinatorial algorithms for facility location problems. SIAM Journal on Computing, 34(4):803-824, 2005.
Moses Charikar, Sudipto Guha, Éva Tardos, and David B. Shmoys. A constant-factor approximation algorithm for the k-median problem. Journal of Computer and System Sciences, 65(1):129-149, 2002.
Moses Charikar and Rina Panigrahy. Clustering to minimize the sum of cluster diameters. Journal of Computer and System Sciences, 68(2):417-441, 2004.
Julia Chuzhoy and Yuval Rabani. Approximating k-median with non-uniform capacities. In Proceedings of the 16th SODA, pages 952-958, 2005.
Marek Cygan, MohammadTaghi Hajiaghayi, and Samir Khuller. LP rounding for k-centers with non-uniform hard capacities. In Proceedings of the 53rd FOCS, pages 273-282, 2012.
Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-306, 1985.
Sudipto Guha and Samir Khuller. Greedy strikes back: Improved facility location algorithms. Journal of Algorithms, 31(1):228-248, 1999.
Dorit S. Hochbaum and David B. Shmoys. A best possible heuristic for the k-center problem. Mathematics of Operations Research, 10:180-184, 1985.
Wen-Lian Hsu and George L. Nemhauser. Easy and hard bottleneck location problems. Discrete Applied Mathematics, 1:209-215, 1979.
Kamal Jain, Mohammad Mahdian, and Amin Saberi. A new greedy approach for facility location problems. In Proceedings of the 34th STOC, pages 731-740, 2002.
Kamal Jain and Vijay V. Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. Journal of the ACM, 48(2):274-296, 2001.
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Christian Kroer and Tuomas Sandholm. Extensive-Form Game Imperfect-Recall Abstractions With Bounds. CoRR, abs/1409.3302, 2014. also published at the Algorithmic Game Theory workshop at IJCAI, 2015.
Shi Li. A 1.488 approximation algorithm for the uncapacitated facility location problem. Information and Computation, 222:45-58, 2013.
Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. In Proceedings of the 45th STOC, pages 901-910, 2013.
David P. Williamson and David B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, 2011. Available at URL: http://www.designofapproxalgs.com.
http://www.designofapproxalgs.com
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The Non-Uniform k-Center Problem
In this paper, we introduce and study the Non-Uniform k-Center (NUkC) problem. Given a finite metric space (X, d) and a collection of balls of radii {r_1 >= ... >= r_k}, the NUkC problem is to find a placement of their centers on the metric space and find the minimum dilation alpha, such that the union of balls of radius alpha*r_i around the i-th center covers all the points in X. This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds, or as a wireless router placement problem with routers of different powers/ranges.
The NUkC problem generalizes the classic k-center problem when all the k radii are the same (which can be assumed to be 1 after scaling). It also generalizes the k-center with outliers (kCwO for short) problem when there are k balls of radius 1 and l balls of radius 0. There are 2-approximation and 3-approximation algorithms known for these problems respectively; the former is best possible unless P=NP and the latter remains unimproved for 15 years.
We first observe that no O(1)-approximation is to the optimal dilation is possible unless P=NP, implying that the NUkC problem is more non-trivial than the above two problems. Our main algorithmic result is an (O(1), O(1))-bi-criteria approximation result: we give an O(1)-approximation to the optimal dilation, however, we may open Theta(1) centers of each radii. Our techniques also allow us to prove a simple (uni-criteria), optimal 2-approximation to the kCwO problem improving upon the long-standing 3-factor. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees which have been studied recently in the TCS community. We show NUkC is as hard as the firefighter problem.
While we don't know if the converse is true, we are able to adapt ideas from recent works [Chalermsook/Chuzhoy, SODA 2010; Asjiashvili/Baggio/Zenklusen, arXiv 2016] in non-trivial ways to obtain our constant factor bi-criteria approximation.
Clustering
k-Center
Approximation Algorithms
Firefighter Problem
67:1-67:15
Regular Paper
Deeparnab
Chakrabarty
Deeparnab Chakrabarty
Prachi
Goyal
Prachi Goyal
Ravishankar
Krishnaswamy
Ravishankar Krishnaswamy
10.4230/LIPIcs.ICALP.2016.67
D. Adjiashvili, A. Baggio, and R. Zenklusen. Firefighting on trees beyond integrality gaps. CoRR, abs/1601.00271, 2016. URL: http://arxiv.org/abs/1601.00271.
http://arxiv.org/abs/1601.00271
J. Byrka, T. Pensyl, B. Rybicki, A. Srinivasan, and K. Trinh. An improved approximation for k-median, and positive correlation in budgeted optimization. Proceedings, ACM-SIAM Symposium on Discrete Algorithms (SODA), 2015.
D. Chakrabarty, P. Goyal, and R. Krishnaswamy. The non-uniform k-center problem. Available on arXiv, and authors webpage (May, 2016), 2016.
P. Chalermsook and J. Chuzhoy. Resource minimization for fire containment. Proceedings, ACM-SIAM Symposium on Discrete Algorithms (SODA), 2010.
M. Charikar, L. O' Callaghan, and R. Panigrahy. Better streaming algorithms for clustering problems. ACM Symp. on Theory of Computing (STOC), 2003.
M. Charikar, C. Chekuri, T. Feder, and R. Motwani. Incremental clustering and dynamic infomation retrieval. ACM Symp. on Theory of Computing (STOC), 1997.
M. Charikar, S. Guha, D. Shmoys, and E. Tardos. A constant-factor approximation algorithm for the k-median problem. ACM Symp. on Theory of Computing (STOC), 1999.
M. Charikar, S. Khuller, D. M. Mount, and G. Narasimhan. Algorithms for facility location problems with outliers. Proceedings, ACM-SIAM Symposium on Discrete Algorithms (SODA), 2001.
S. Finbow, A. King, G. MacGillivray, and R. Rizzi. The firefighter problem for graphs of maximum degree three. Discrete Mathematics, 307(16):2094-2105, 2007.
I. L. Goertz and V. Nagarajan. Locating depots for capacitated vehicle routing. Proceedings, International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, 2011.
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S. Guha, R. Rastogi, and K. Shim. CURE: An efficient clustering algorithm for large databases. Proceedings of SIGMOD, 1998.
S. Har-Peled and S. Mazumdar. Coresets for k-means and k-median clustering and their applications. ACM Symp. on Theory of Computing (STOC), 2004.
D. S. Hochbaum and D. B. Shmoys. A best possible heuristic for the k-center problem. Mathematics of operations research, 10(2):180-184, 1985.
S. Im and B. Moseley. Fast and better distributed mapreduce algorithms for k-center clustering. Proceedings, ACM Symposium on Parallelism in Algorithms and Architectures, 2015.
K. Jain and V. V. Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. J. ACM, 48(2):274-296, 2001.
T. Kanungo, D. M. Mount, N. S. Netanyahu, C. D. Piatko, R. Silverman, and A. Y. Wu. A local search approximation algorithm for k-means clustering. In Proceedings of the 18th Annual Symposium on Computational Geometry (SoCG'02), 2002.
A. King and G. MacGillivray. The firefighter problem for cubic graphs. Discrete Mathematics, 310(3):614-621, 2010.
A. Kumar, Y. Sabharwal, and S. Sen. A simple linear time (1 + )-approximation algorithm for k-means clustering in any dimensions. Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), 2004.
G. Laporte. Location routing problems. In B. L. Golden and A. A. Assad, editors, Vehicle Routing: Methods and Studies, pages 163-198. 1998.
S. Li and O. Svensson. Approximating k-median via pseudo-approximation. ACM Symp. on Theory of Computing (STOC), 2013.
G. Malkomes, M. J. Kusner, W. Chen, K. Q. Weinberger, and B. Moseley. Fast distributed k-center clustering with outliers on massive data. Advances in Neur. Inf. Proc. Sys. (NIPS), 2015.
R. McCutchen and S. Khuller. Streaming algorithms for k-center clustering with outliers and with anonymity. Proceedings, International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, 2008.
H. Min, V. Jayaraman, and R. Srivastava. Combined location-routing problems: A synthesis and future research directions. European Journal of Operational Research, 108:1-15, 1998.
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k-Center Clustering Under Perturbation Resilience
The k-center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case instances: a 2-approximation for symmetric kcenter and an O(log*(k))-approximation for the asymmetric version. Therefore to improve on these ratios, one must go beyond the worst case.
In this work, we take this approach and provide strong positive results both for the asymmetric and symmetric k-center problems under a very natural input stability (promise) condition called alpha-perturbation resilience [Bilu Linial, 2012], which states that the optimal solution does not change under any alpha-factor perturbation to the input distances. We show that by assuming 2-perturbation resilience, the exact solution for the asymmetric k-center problem can be found in polynomial time. To our knowledge, this is the first problem that is hard to approximate to any constant factor in the worst case, yet can be optimally solved in polynomial time under perturbation resilience for a constant value of alpha. Furthermore, we prove our result is tight by showing symmetric k-center under (2-epsilon)-perturbation resilience is hard unless NP=RP.
This is the first tight result for any problem under perturbation resilience, i.e., this is the first time the exact value of alpha for which the problem switches from being NP-hard to efficiently computable has been found.
Our results illustrate a surprising relationship between symmetric and asymmetric k-center instances under perturbation resilience. Unlike approximation ratio, for which symmetric k-center is easily solved to a factor of 2 but asymmetric k-center cannot be approximated to any constant factor, both symmetric and asymmetric k-center can be solved optimally under resilience
to 2-perturbations.
k-center
clustering
perturbation resilience
68:1-68:14
Regular Paper
Maria-Florina
Balcan
Maria-Florina Balcan
Nika
Haghtalab
Nika Haghtalab
Colin
White
Colin White
10.4230/LIPIcs.ICALP.2016.68
Aaron Archer. Two o (log* k)-approximation algorithms for the asymmetric k-center problem. In Integer Programming and Combinatorial Optimization, pages 1-14. Springer, 2001.
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http://dx.doi.org/10.1145/2688073.2688116
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Maria-Florina Balcan and Mark Braverman. Approximate nash equilibria under stability conditions. Technical report, 2010.
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http://dl.acm.org/citation.cfm?id=313852.313861
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Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers
We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {L_i}_{i in F} and a set D of clients located in a common metric space {c(i,j)}_{i,j in F union D}, and bounds k, m. A feasible solution is a pair (S subseteq F, sigma: D -> S union {out}), where sigma specifies the client assignments, such that |S| <=k, |sigma^{-1}(i)| >= L_i for all i in S, and |sigma^{-1}(out)| <= m. In the lower-bounded min-sum-of-radii with outliers P (LBkSRO) problem, the objective is to minimize sum_{i in S} max_{j in sigma^{-1})i)}, and in the lower-bounded k-supplier with outliers (LBkSupO) problem, the objective is to minimize max_{i in S} max_{j in sigma^{-1})i)} c(i,j).
We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the non-outlier version (i.e., m = 0). These also constitute the first approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers separately. We apply the primal-dual method to the relaxation where we Lagrangify the |S| <= k constraint. The chief technical contribution and novelty of our algorithm is that, departing from the standard paradigm used for such constrained problems, we obtain an O(1)-approximation despite the fact that we do not obtain a Lagrangian-multiplier-preserving algorithm for the Lagrangian relaxation. We believe that our ideas have broader applicability to other clustering problems with outliers as well.
We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds.
Approximation algorithms
facililty-location problems
primal-dual method
Lagrangian relaxation
k-center problems
minimizing sum of radii
69:1-69:15
Regular Paper
Sara
Ahmadian
Sara Ahmadian
Chaitanya
Swamy
Chaitanya Swamy
10.4230/LIPIcs.ICALP.2016.69
Gagan Aggarwal, Tomás Feder, Krishnaram Kenthapadi, Samir Khuller, Rina Panigrahy, Dilys Thomas, and An Zhu. Achieving anonymity via clustering. ACM Transactions on Algorithms (TALG), 6(3):49, 2010.
Gagan Aggarwal, Tomás Feder, Krishnaram Kenthapadi, Rajeev Motwani, Rina Panigrahy, Dilys Thomas, and An Zhu. Approximation algorithms for k-anonymity. Journal of Privacy Technology (JOPT), 2005.
Sara Ahmadian and Chaitanya Swamy. Improved approximation guarantees for lower-bounded facility location. In Proceedings of the 10th International Workshop on Approximation and Online Algorithms, pages 257-271. Springer, 2012.
Hyung-Chan An, Aditya Bhaskara, Chandra Chekuri, Shalmoli Gupta, Vivek Madan, and Ola Svensson. Centrality of trees for capacitated k-center. Mathematical Programming, 154(1-2):29-53, 2015.
Babak Behsaz and Mohammad R Salavatipour. On minimum sum of radii and diameters clustering. Algorithmica, 73(1):143-165, 2015.
Jarosław Byrka, Thomas Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An improved approximation for k-median, and positive correlation in budgeted optimization. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 737-756. SIAM, 2015.
Vasilis Capoyleas, Günter Rote, and Gerhard Woeginger. Geometric clusterings. Journal of Algorithms, 12(2):341-356, 1991.
Moses Charikar and Sudipto Guha. Improved combinatorial algorithms for facility location problems. SIAM Journal on Computing, 34(4):803-824, 2005.
Moses Charikar, Sudipto Guha, Éva Tardos, and David B Shmoys. A constant-factor approximation algorithm for the k-median problem. In Proceedings of the 31st annual ACM Symposium on Theory of Computing, pages 1-10. ACM, 1999.
Moses Charikar, Samir Khuller, David Mount, and Giri Narasimhan. Algorithms for facility location problems with outliers. In Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 642-651. SIAM, 2001.
Moses Charikar and Rina Panigrahy. Clustering to minimize the sum of cluster diameters. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pages 1-10. ACM, 2001.
Ke Chen. A constant factor approximation algorithm for k-median clustering with outliers. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 826-835. SIAM, 2008.
Marek Cygan, MohammadTaghi Hajiaghayi, and Samir Khuller. Lp rounding for k-centers with non-uniform hard capacities. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, pages 273-282. IEEE, 2012.
Marek Cygan and Tomasz Kociumaka. Constant factor approximation for capacitated k-center with outliers. In Proc. 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), volume 25 of LIPIcs - Leibniz International Proceedings in Informatics, pages 251-262. Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik, 2014. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2014.251.
http://dx.doi.org/10.4230/LIPIcs.STACS.2014.251
Srinivas R Doddi, Madhav V Marathe, SS Ravi, David Scot Taylor, and Peter Widmayer. Approximation algorithms for clustering to minimize the sum of diameters. In Proceedings of the 11th Scandinavian Workshop on Algorithm Theory, pages 237-250, 2000.
Alina Ene, Sariel Har-Peled, and Benjamin Raichel. Fast clustering with lower bounds: No customer too far, no shop too small. arXiv preprint arXiv:1304.7318, 2013.
Matt Gibson, Gaurav Kanade, Erik Krohn, Imran A Pirwani, and Kasturi Varadarajan. On metric clustering to minimize the sum of radii. Algorithmica, 57(3):484-498, 2010.
Matt Gibson, Gaurav Kanade, Erik Krohn, Imran A Pirwani, and Kasturi Varadarajan. On clustering to minimize the sum of radii. SIAM Journal on Computing, 41(1):47-60, 2012.
Sudipto Guha, Adam Meyerson, and Kamesh Munagala. Hierarchical placement and network design problems. In Proceedings of 41st Annual IEEE Symposium on Foundations of Computer Science, pages 603-612. IEEE, 2000.
Dorit S Hochbaum and David B Shmoys. A best possible heuristic for the k-center problem. Mathematics of Operations Research, 10(2):180-184, 1985.
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Kamal Jain and Vijay V Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and lagrangian relaxation. Journal of the ACM, 48(2):274-296, 2001.
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A Duality Based 2-Approximation Algorithm for Maximum Agreement Forest
We give a 2-approximation algorithm for the Maximum Agreement Forest problem on two rooted binary trees. This NP-hard problem has been studied extensively in the past two decades, since it can be used to compute the Subtree Prune-and-Regraft (SPR) distance between two phylogenetic trees. Our result improves on the very recent 2.5-approximation algorithm due to Shi, Feng, You and Wang (2015). Our algorithm is the first approximation algorithm for this problem that uses LP duality in its analysis.
Maximum agreement forest
phylogenetic tree
SPR distance
subtree prune-and-regraft distance
computational biology
70:1-70:14
Regular Paper
Frans
Schalekamp
Frans Schalekamp
Anke
van Zuylen
Anke van Zuylen
Suzanne
van der Ster
Suzanne van der Ster
10.4230/LIPIcs.ICALP.2016.70
Benjamin L. Allen and Mike Steel. Subtree transfer operations and their induced metrics on evolutionary trees. Annals of Combinatorics, 5(1):1-15, 2001.
Maria Luisa Bonet, Katherine St John, Ruchi Mahindru, and Nina Amenta. Approximating subtree distances between phylogenies. Journal of Computational Biology, 13(8):1419-1434, 2006.
Magnus Bordewich, Catherine McCartin, and Charles Semple. A 3-approximation algorithm for the subtree distance between phylogenies. Journal of Discrete Algorithms, 6(3):458-471, 2008. URL: http://dx.doi.org/10.1016/j.jda.2007.10.002.
http://dx.doi.org/10.1016/j.jda.2007.10.002
Magnus Bordewich and Charles Semple. On the computational complexity of the rooted subtree prune and regraft distance. Ann. Comb., 8(4):409-423, 2004. URL: http://dx.doi.org/10.1007/s00026-004-0229-z.
http://dx.doi.org/10.1007/s00026-004-0229-z
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Estela M. Rodrigues. Algoritmos para Comparação de Árvores Filogenéticas e o Problema dos Pontos de Recombinação. PhD thesis, University of São Paulo, Brazil, 2003. Chapter 7, available at URL: http://www.ime.usp.br/~estela/studies/tese-traducao-cp7.ps.gz.
http://www.ime.usp.br/~estela/studies/tese-traducao-cp7.ps.gz
Estela M. Rodrigues, Marie-France Sagot, and Yoshiko Wakabayashi. The maximum agreement forest problem: approximation algorithms and computational experiments. Theoretical Computer Science, 374(1-3):91-110, 2007. URL: http://dx.doi.org/10.1016/j.tcs.2006.12.011.
http://dx.doi.org/10.1016/j.tcs.2006.12.011
Frans Schalekamp, Anke van Zuylen, and Suzanne van der Ster. A duality based 2-approximation algorithm for maximum agreement forest. CoRR, abs/1511.06000, 2015. URL: http://arxiv.org/abs/1511.06000.
http://arxiv.org/abs/1511.06000
Feng Shi, Qilong Feng, Jie You, and Jianxin Wang. Improved approximation algorithm for maximum agreement forest of two rooted binary phylogenetic trees. Journal of Combinatorial Optimization, 2015. URL: http://dx.doi.org/10.1007/s10878-015-9921-7.
http://dx.doi.org/10.1007/s10878-015-9921-7
Mike Steel and Tandy Warnow. Kaikoura tree theorems: Computing the maximum agreement subtree. Information Processing Letters, 48(2):77-82, November 1993. URL: http://dx.doi.org/10.1016/0020-0190(93)90181-8.
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Chris Whidden, Robert G. Beiko, and Norbert Zeh. Fixed-parameter algorithms for maximum agreement forests. SIAM Journal on Computing, 42(4):1431-1466, 2013. URL: http://dx.doi.org/10.1137/110845045.
http://dx.doi.org/10.1137/110845045
Chris Whidden and Norbert Zeh. A unifying view on approximation and FPT of agreement forests. In Algorithms in Bioinformatics, volume 5724 of Lecture Notes in Computer Science, pages 390-402. Springer Berlin Heidelberg, 2009. URL: http://dx.doi.org/10.1007/978-3-642-04241-6_32.
http://dx.doi.org/10.1007/978-3-642-04241-6_32
Yufeng Wu. A practical method for exact computation of subtree prune and regraft distance. Bioinformatics, 25(2):190-196, 2009. URL: http://dx.doi.org/10.1093/bioinformatics/btn606.
http://dx.doi.org/10.1093/bioinformatics/btn606
Yufeng Wu and Jiayin Wang. Fast computation of the exact hybridization number of two phylogenetic trees. In Bioinformatics Research and Applications, volume 6053 of Lecture Notes in Computer Science, pages 203-214. Springer Berlin Heidelberg, 2010. URL: http://dx.doi.org/10.1007/978-3-642-13078-6_23.
http://dx.doi.org/10.1007/978-3-642-13078-6_23
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Robust Assignments via Ear Decompositions and Randomized Rounding
Many real-life planning problems require making a priori decisions before all parameters of the problem have been revealed. An important special case of such problem arises in scheduling and transshipment problems, where a set of jobs needs to be assigned to the available set of machines or personnel (resources), in a way that all jobs have assigned resources, and no two jobs share the same resource. In its nominal form, the resulting computational problem becomes the assignment problem.
This paper deals with the Robust Assignment Problem (RAP) which models situations in which certain assignments are vulnerable and may become unavailable after the solution has been chosen. The goal is to choose a minimum-cost collection of assignments (edges in the corresponding bipartite graph) so that if any vulnerable edge becomes unavailable, the remaining part of the solution contains an assignment of all jobs.
We develop algorithms and hardness results for RAP and establish several connections to well-known concepts from matching theory, robust optimization, LP-based techniques and combinations
thereof.
robust optimization
matching theory
ear decomposition
randomized rounding
approximation algorithm
71:1-71:14
Regular Paper
David
Adjiashvili
David Adjiashvili
Viktor
Bindewald
Viktor Bindewald
Dennis
Michaels
Dennis Michaels
10.4230/LIPIcs.ICALP.2016.71
D. Adjiashvili. Non-uniform robust network design in planar graphs. In Proc. of APPROX, pages 61-77, 2015.
D. Adjiashvili, S. Stiller, and R. Zenklusen. Bulk-robust combinatorial optimization. Math Program, 149(1-2):361-390, 2014.
H. Aissi, C. Bazgan, and D. Vanderpooten. Complexity of the min-max and min-max regret assignment problems. Oper Res Lett, 33(6):634-640, 2005.
H. Aissi, C. Bazgan, and D. Vanderpooten. Min–max and min–max regret versions of combinatorial optimization problems: A survey. Eur J Oper Res, 197(2):427-438, 2009.
D. Bertsimas, D.B. Brown, and C. Caramanis. Theory and applications of robust optimization. SIAM review, 53:464-501, 2011.
R.C. Brigham, F. Harary, E.C. Violin, and J. Yellen. Perfect-matching preclusion. Congressus Numerantium, 174:185-192, 2005. 00042.
G. Carello and E. Lanzarone. A cardinality-constrained robust model for the assignment problem in Home Care services. Eur J Oper Res, 236(2):748-762, 2014.
S. Chechik, M. Langberg, D. Peleg, and L. Roditty. Fault-tolerant spanners for general graphs. In Proc. of STOC, pages 435-444, 2009.
S. Chechik and D. Peleg. Robust fault tolerant uncapacitated facility location. In Proc. of STACS, pages 191-202, 2010.
C. Chekuri. Routing and network design with robustness to changing or uncertain traffic demands. ACM SIGACT News, 38(3):106-129, 2007.
E. Cheng, L. Lesniak, M.J. Lipman, and L. Liptak. Conditional matching preclusion sets. Information Sciences, 179(8):1092-1101, 2009. 00039.
J. Cheriyan, A. Sebo, and Z. Szigeti. Improving on the 1.5-Approximation of a Smallest 2-Edge Connected Spanning Subgraph. SIAM J Discrete Math, 14(2):170-180, 2001.
M.H. de Carvalho and J. Cheriyan. An O(VE) algorithm for ear decompositions of matching-covered graphs. In Proc. of SODA, pages 415-423, 2005.
V.G. Deineko and G.J. Woeginger. On the robust assignment problem under a fixed number of cost scenarios. Oper Res Lett, 34(2):175-179, 2006.
M. Dinitz and R. Krauthgamer. Fault-tolerant spanners: better and simpler. In Proc. of PODC, pages 169-178. ACM, 2011.
M.C. Dourado, D. Meierling, L.D. Penso, D. Rautenbach, F. Protti, and A.R. de Almeida. Robust recoverable perfect matchings. Networks, 66(3):210-213, 2015.
R. Fujita, Y. Kobayashi, and K. Makino. Robust matchings and matroid intersections. In Proc. of ESA, pages 123-134. Springer, 2010.
H. N. Gabow, M. X. Goemans, É. Tardos, and D. P. Williamson. Approximating the smallest k-edge connected spanning subgraph by LP-rounding. In Proc. of SODA, pages 562-571, 2005.
F. Grandoni, R. Ravi, M. Singh, and R. Zenklusen. New approaches to multi-objective optimization. Math Program, 146(1-2):525-554, 2014.
M. Grötschel, L. Lovasz, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Algorithms and Combinatorics. Springer Berlin Heidelberg, 1993.
M. Hajiaghayi, N. Immorlica, and V.S. Mirrokni. Power optimization in fault-tolerant topology control algorithms for wireless multi-hop networks. In Proc. of MobiCom, pages 300-312. ACM, 2003.
R. Hassin and S. Rubinstein. Robust matchings. SIAM J Discrete Math, 15(4):530-537, 2002.
W. Herroelen and R. Leus. Project scheduling under uncertainty: Survey and research potentials. Eur J Oper Res, 165(2):289-306, 2005.
K. Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica, 21:39-60, 2001.
K. Jain and V.V. Vazirani. An approximation algorithm for the fault tolerant metric facility location problem. In Proc. of APPROX, pages 177-183, 2000.
P. Kouvelis and G. Yu. Robust discrete optimization and its applications, volume 14. Springer Science &Business Media, 1997.
P. Laroche, F. Marchetti, S. Martin, and Z. Roka. Bipartite complete matching vertex interdiction problem: Application to robust nurse assignment. In Proc. of CoDIT, pages 182-187, 2014.
L. Lovász and M.D. Plummer. Matching theory. North-Holland, Amsterdam, 1986.
J. Plesník. Connectivity of regular graphs and the existence of 1-factors. Matematickỳ časopis, 22(4):310-318, 1972.
A. Sebő and J. Vygen. Shorter tours by nicer ears: 7/5-approximation for the graph-tsp, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. Combinatorica, 5(34):597-629, 2014.
C. Swamy and D. B. Shmoys. Fault-tolerant facility location. In Proc. of SODA, pages 735-736, 2003.
C.S. Tang. Robust strategies for mitigating supply chain disruptions. International Journal of Logistics: Research and Applications, 9(1):33-45, 2006.
G. Yu and J. Yang. On the robust shortest path problem. Computers & Operations Research, 25(6):457-468, 1998.
R. Zenklusen. Matching interdiction. Discrete Appl Math, 158(15):1676-1690, 2010.
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Closing the Gap for Makespan Scheduling via Sparsification Techniques
Makespan scheduling on identical machines is one of the most basic and fundamental packing problem studied in the discrete optimization literature. It asks for an assignment of n jobs to a set of m identical machines that minimizes the makespan. The problem is strongly NPhard, and thus we do not expect a (1 + epsilon)-approximation algorithm with a running time that depends polynomially on 1/epsilon. Furthermore, Chen et al. [Chen/JansenZhang, SODA'13] recently showed that a running time of 2^{1/epsilon}^{1-delta} + poly(n) for any delta > 0 would imply that the Exponential Time Hypothesis (ETH) fails. A long sequence of algorithms have been developed that try to obtain low dependencies on 1/epsilon, the better of which achieves a running time of 2^{~O(1/epsilon^{2})} + O(n*log(n)) [Jansen, SIAM J. Disc. Math. 2010]. In this paper we obtain an algorithm with a running time of 2^{~O(1/epsilon)} + O(n*log(n)), which is tight under ETH up to logarithmic factors on the exponent.
Our main technical contribution is a new structural result on the configuration-IP. More precisely, we show the existence of a highly symmetric and sparse optimal solution, in which all but a constant number of machines are assigned a configuration with small support. This structure can then be exploited by integer programming techniques and enumeration. We believe that our structural result is of independent interest and should find applications to other settings.
In particular, we show how the structure can be applied to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines. For all these cases we obtain an efficient PTAS with running time 2^{~O(1/epsilon)} + poly(n).
scheduling
approximation
PTAS
makespan
ETH
72:1-72:13
Regular Paper
Klaus
Jansen
Klaus Jansen
Kim-Manuel
Klein
Kim-Manuel Klein
José
Verschae
José Verschae
10.4230/LIPIcs.ICALP.2016.72
N. Alon, Y. Azar, G. Woeginger, and T. Yadid. Approximation schemes for scheduling. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '97), pages 493-500. ACM/SIAM, 1997.
N. Alon, Y. Azar, G. J. Woeginger, and T. Yadid. Approximation schemes for scheduling on parallel machines. Journal of Scheduling, 1:55-66, 1998.
L. Chen, K. Jansen, and G. Zhang. On the optimality of approximation schemes for the classical scheduling problem. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'13), pages 657-668. ACM/SIAM, 2013.
F. Eisenbrand and G. Shmonin. Carathéodory bounds for integer cones. Operations Research Letters, 34:564-568, 2006.
M. X. Goemans and T. Rothvoß. Polynomiality for bin packing with a constant number of item types. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'14), pages 830-839. ACM/SIAM, 2014.
R. L. Graham. Bounds for certain multiprocessing anomalies. Bell System Technical Journal, 45:1563-1581, 1966.
R. L. Graham. Bounds on multiprocessing timing anomalies. SIAM Journal on Applied Mathematics, 17:416-429, 1969.
D. Hochbaum, editor. Approximation algorithms for NP-hard problems. PWS Publishing Company, 1997.
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K. Jansen. An EPTAS for scheduling jobs on uniform processors: Using an MILP relaxation with a constant number of integral variables. SIAM Journal on Discrete Mathematics, 24:457-485, 2010.
K. Jansen and C. Robenek. Scheduling jobs on identical and uniform processors revisited. In Approximation and Online Algorithms (WAOA'11), number 7164 in Lecture Notes in Computer Science, pages 109-122. Springer, 2011.
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Constant Approximation for Capacitated k-Median with (1+epsilon)-Capacity Violation
We study the Capacitated k-Median problem for which existing constant-factor approximation algorithms are all pseudo-approximations that violate either the capacities or the upper bound k on the number of open facilities. Using the natural LP relaxation for the problem, one can only hope to get the violation factor down to 2. Li [SODA'16] introduced a novel LP to go beyond the limit of 2 and gave a constant-factor approximation algorithm that opens (1 + epsilon)*k facilities.
We use the configuration LP of Li [SODA'16] to give a constant-factor approximation for the Capacitated k-Median problem in a seemingly harder configuration: we violate only the capacities by 1 + epsilon. This result settles the problem as far as pseudo-approximation algorithms are concerned.
Approximation Algorithms
Capacitated k-Median
Pseudo Approximation
Capacity Violation
73:1-73:14
Regular Paper
Gökalp
Demirci
Gökalp Demirci
Shi
Li
Shi Li
10.4230/LIPIcs.ICALP.2016.73
Karen Aardal, Pieter L. van den Berg, Dion Gijswijt, and Shanfei Li. Approximation algorithms for hard capacitated k-facility location problems. European Journal of Operational Research, 242(2):358-368, 2015. URL: http://dx.doi.org/10.1016/j.ejor.2014.10.011.
http://dx.doi.org/10.1016/j.ejor.2014.10.011
Hyung-Chan An, Mohit Singh, and Ola Svensson. LP-based algorithms for capacitated facility location. In Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014, 2014.
V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala, and V. Pandit. Local search heuristic for k-median and facility location problems. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, STOC'01, pages 21-29, New York, NY, USA, 2001. ACM. URL: http://dx.doi.org/10.1145/380752.380755.
http://dx.doi.org/10.1145/380752.380755
Jarosław Byrka, Krzysztof Fleszar, Bartosz Rybicki, and Joachim Spoerhase. Bi-factor approximation algorithms for hard capacitated k-median problems. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015), 2015.
Jarosław Byrka, Thomas Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An improved approximation for k-median, and positive correlation in budgeted optimization. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015), 2015.
Jarosław Byrka, Bartosz Rybicki, and Sumedha Uniyal. An approximation algorithm for uniform capacitated k-median problem with 1+ε capacity violation, 2015. arXiv:1511.07494. URL: http://arxiv.org/abs/arXiv:1511.07494.
http://arxiv.org/abs/arXiv:1511.07494
Robert D. Carr, Lisa K. Fleischer, Vitus J. Leung, and Cynthia A. Phillips. Strengthening integrality gaps for capacitated network design and covering problems. In Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'00, pages 106-115, Philadelphia, PA, USA, 2000. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=338219.338241.
http://dl.acm.org/citation.cfm?id=338219.338241
M. Charikar and S. Guha. Improved combinatorial algorithms for the facility location and k-median problems. In In Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, pages 378-388, 1999.
M. Charikar, S. Guha, É. Tardos, and D. B. Shmoys. A constant-factor approximation algorithm for the k-median problem (extended abstract). In Proceedings of the thirty-first annual ACM symposium on Theory of computing, STOC'99, pages 1-10, New York, NY, USA, 1999. ACM. URL: http://dx.doi.org/10.1145/301250.301257.
http://dx.doi.org/10.1145/301250.301257
Julia Chuzhoy and Yuval Rabani. Approximating k-median with non-uniform capacities. In SODA ’05, pages 952-958, 2005.
Sudipto Guha. Approximation Algorithms for Facility Location Problems. PhD thesis, Stanford University, Stanford, CA, USA, 2000.
K. Jain, M. Mahdian, and A. Saberi. A new greedy approach for facility location problems. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, STOC'02, pages 731-740, New York, NY, USA, 2002. ACM. URL: http://dx.doi.org/10.1145/509907.510012.
http://dx.doi.org/10.1145/509907.510012
K Jain and V. V. Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. J. ACM, 48(2):274-296, 2001. URL: http://dx.doi.org/10.1145/375827.375845.
http://dx.doi.org/10.1145/375827.375845
Shanfei Li. An improved approximation algorithm for the hard uniform capacitated k-median problem. In APPROX'14/RANDOM'14: Proceedings of the 17th International Workshop on Combinatorial Optimization Problems and the 18th International Workshop on Randomization and Computation, APPROX'14/RANDOM'14, 2014.
Shi Li. On uniform capacitated k-median beyond the natural LP relaxation. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015), 2015.
Shi Li. Approximating capacitated k-median with (1 + ε)k open facilities. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2016), pages 786-796, 2016.
Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. In Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC'13, pages 901-910, New York, NY, USA, 2013. ACM. URL: http://dx.doi.org/10.1145/2488608.2488723.
http://dx.doi.org/10.1145/2488608.2488723
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D. B. Shmoys, É. Tardos, and K. Aardal. Approximation algorithms for facility location problems (extended abstract). In STOC'97: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pages 265-274, New York, NY, USA, 1997. ACM. URL: http://dx.doi.org/10.1145/258533.258600.
http://dx.doi.org/10.1145/258533.258600
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Approximating Directed Steiner Problems via Tree Embedding
Directed Steiner problems are fundamental problems in Combinatorial Optimization and Theoretical Computer Science. An important problem in this genre is the k-edge connected directed Steiner tree (k-DST) problem. In this problem, we are given a directed graph G on n vertices with edge-costs, a root vertex r, a set of h terminals T and an integer k. The goal is to find a min-cost subgraph H subseteq G that connects r to each terminal t in T by k edge-disjoint r, t-paths. This problem includes as special cases the well-known directed Steiner tree (DST) problem (the case k=1) and the group Steiner tree (GST) problem. Despite having been studied and mentioned many times in literature, e.g., by Feldman et al. [SODA'09, JCSS'12], by Cheriyan et al. [SODA'12, TALG'14], by Laekhanukit [SODA'14] and in a survey by Kortsarz and Nutov [Handbook of Approximation Algorithms and Metaheuristics], there was no known non-trivial approximation algorithm for k-DST for k >= 2 even in a special case that an input graph is directed acyclic and has a constant number of layers. If an input graph is not acyclic, the complexity status of k-DST is not known even for a very strict special case that k=2 and h=2.
In this paper, we make a progress toward developing a non-trivial approximation algorithm for k-DST. We present an O(D*k^{D-1}*log(n))-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D layers, which can be extended to a special case of k-DST on "general graphs" when an instance has a D-shallow optimal solution, i.e., there exist k edge-disjoint r, t-paths, each of length at most D, for every terminal t in T. For the case k=1 (DST), our algorithm yields an approximation ratio of O(D*log(h)), thus implying an O(log^3(h))-approximation algorithm for DST that runs in quasi-polynomial-time (due to the height-reduction of Zelikovsky [Algorithmica'97]). Our algorithm is based on an LP-formulation that allows us to embed a solution to a tree-instance of GST, which does not preserve connectivity. We show, however, that one can randomly extract a solution of k-DST from the tree-instance of GST.
Our algorithm is almost tight when k and D are constants since the case that k=1 and D=3 is NP-hard to approximate to within a factor of O(log(h)), and our algorithm archives the same approximation ratio for this special case. We also remark that the k^{1/4-epsilon}-hardness instance of k-DST is a DAG with 6 layers, and our algorithm gives O(k^5*log(n))-approximation for this special case. Consequently, as our algorithm works for general graphs, we obtain an O(D*k^{D-1}*log(n))-approximation algorithm for a D-shallow instance of the k edge-connected directed Steiner subgraph problem, where we wish to connect every pair of terminals by k edgedisjoint paths.
Approximation Algorithms
Network Design
Graph Connectivity
Directed Graph
74:1-74:13
Regular Paper
Bundit
Laekhanukit
Bundit Laekhanukit
10.4230/LIPIcs.ICALP.2016.74
Parinya Chalermsook, Fabrizio Grandoni, and Bundit Laekhanukit. On survivable set connectivity. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 25-36, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.3.
http://dx.doi.org/10.1137/1.9781611973730.3
Moses Charikar, Chandra Chekuri, To-Yat Cheung, Zuo Dai, Ashish Goel, Sudipto Guha, and Ming Li. Approximation algorithms for directed steiner problems. J. Algorithms, 33(1):73-91, 1999. URL: http://dx.doi.org/10.1006/jagm.1999.1042.
http://dx.doi.org/10.1006/jagm.1999.1042
Joseph Cheriyan, Bundit Laekhanukit, Guyslain Naves, and Adrian Vetta. Approximating rooted steiner networks. ACM Transactions on Algorithms, 11(2):8:1-8:22, 2014. URL: http://dx.doi.org/10.1145/2650183.
http://dx.doi.org/10.1145/2650183
Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998. URL: http://dx.doi.org/10.1145/285055.285059.
http://dx.doi.org/10.1145/285055.285059
Moran Feldman, Guy Kortsarz, and Zeev Nutov. Improved approximation algorithms for directed steiner forest. J. Comput. Syst. Sci., 78(1):279-292, 2012. URL: http://dx.doi.org/10.1016/j.jcss.2011.05.009.
http://dx.doi.org/10.1016/j.jcss.2011.05.009
Zachary Friggstad, Jochen Könemann, Young Kun-Ko, Anand Louis, Mohammad Shadravan, and Madhur Tulsiani. Linear programming hierarchies suffice for directed steiner tree. In Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Bonn, Germany, June 23-25, 2014. Proceedings, pages 285-296, 2014. URL: http://dx.doi.org/10.1007/978-3-319-07557-0_24.
http://dx.doi.org/10.1007/978-3-319-07557-0_24
Naveen Garg, Goran Konjevod, and R. Ravi. A polylogarithmic approximation algorithm for the group steiner tree problem. J. Algorithms, 37(1):66-84, 2000. URL: http://dx.doi.org/10.1006/jagm.2000.1096.
http://dx.doi.org/10.1006/jagm.2000.1096
Christopher S. Helvig, Gabriel Robins, and Alexander Zelikovsky. An improved approximation scheme for the group steiner problem. Networks, 37(1):8-20, 2001. URL: http://dx.doi.org/10.1002/1097-0037(200101)37:1<8::AID-NET2>3.0.CO;2-R.
http://dx.doi.org/10.1002/1097-0037(200101)37:1<8::AID-NET2>3.0.CO;2-R
Guy Kortsarz and Zeev Nutov. Approximating minimum-cost connectivity problems. In Teofilo F. Gonzalez, editor, Handbook of Approximation Algorithms and Metaheuristics. Chapman and Hall/CRC, 2007. URL: http://dx.doi.org/10.1201/9781420010749.ch58.
http://dx.doi.org/10.1201/9781420010749.ch58
Bundit Laekhanukit. Parameters of two-prover-one-round game and the hardness of connectivity problems. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1626-1643, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.118.
http://dx.doi.org/10.1137/1.9781611973402.118
Carsten Lund and Mihalis Yannakakis. On the hardness of approximating minimization problems. J. ACM, 41(5):960-981, 1994. URL: http://dx.doi.org/10.1145/185675.306789.
http://dx.doi.org/10.1145/185675.306789
Zeev Nutov. Approximability status of survivable network problems. Preprint available at URL: http://www.openu.ac.il/home/nutov/Survivable-Network.pdf.
http://www.openu.ac.il/home/nutov/Survivable-Network.pdf
Thomas Rothvoß. Directed steiner tree and the lasserre hierarchy. CoRR, abs/1111.5473, 2011. URL: http://arxiv.org/abs/1111.5473.
http://arxiv.org/abs/1111.5473
Alexander Zelikovsky. A series of approximation algorithms for the acyclic directed steiner tree problem. Algorithmica, 18(1):99-110, 1997. URL: http://dx.doi.org/10.1007/BF02523690.
http://dx.doi.org/10.1007/BF02523690
Creative Commons Attribution 3.0 Unported license
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Tight Analysis of a Multiple-Swap Heurstic for Budgeted Red-Blue Median
Budgeted Red-Blue Median is a generalization of classic k-Median in that there are two sets of facilities, say R and B, that can be used to serve clients located in some metric space. The goal is to open kr facilities in R and kb facilities in B for some given bounds kr, kb and connect each client to their nearest open facility in a way that minimizes the total connection cost.
We extend work by Hajiaghayi, Khandekar, and Kortsarz [2012] and show that a multipleswap local search heuristic can be used to obtain a (5 + epsilon)-approximation for Budgeted RedBlue Median for any constant epsilon > 0. This is an improvement over their single swap analysis and beats the previous best approximation guarantee of 8 by Swamy [2014].
We also present a matching lower bound showing that for every p >= 1, there are instances of Budgeted Red-Blue Median with local optimum solutions for the p-swap heuristic whose cost is 5 + Omega(1/p) times the optimum solution cost. Thus, our analysis is tight up to the lower order terms. In particular, for any epsilon > 0 we show the single-swap heuristic admits local optima whose cost can be as bad as 7 - epsilon times the optimum solution cost.
Approximation Algorithms
Local search
Red-Blue Meidan
75:1-75:13
Regular Paper
Zachary
Friggstad
Zachary Friggstad
Yifeng
Zhang
Yifeng Zhang
10.4230/LIPIcs.ICALP.2016.75
Ankit Aggarwal, Anand Louis, Manisha Bansal, Naveen Garg, Neelima Gupta, Shubham Gupta, and Surabhi Jain. A 3-approximation for facility location with uniform capacities. In Proc. of IPCO, pages 149-162, 2010.
Sara Ahmadian, Zachary Friggstad, and Chaitanya Swamy. Local-search based approximation algorithms for mobile facility location problems. In Proc. of SODA, pages 1607-1621, 2013.
Vijay Arya, Naveen Garg, Rohit Khandekar, Adam Meyerson, Kamesh Munagala, and Vinayaka Pandit. Local search heuristics for k-median and facility location problems. SIAM J. Comput., 33(3):544-562, 2004.
Manisha Bansal, Naveen Garg, and Neelima Gupta. A 5-approximation for capacitated facility location. In Proc. of ESA, pages 133-144, 2012.
Jaroslaw Byrka, Thomas Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An improved approximation for k-median, and positive correlation in budgeted optimization. In Proc. of SODA, pages 737-756, 2015.
Moses Charikar and Sudipto Guha. Improved combinatorial algorithms for facility location problems. SIAM J. Comput., 34(4):803-824, 2005.
Inge Li Gørtz and Viswanath Nagarajan. Locating depots for capacitated vehicle routing. In Proc. of APPROX, pages 230-241, 2011.
Anupam Gupta and Kanat Tangwongsan. Simpler analyses of local search algorithms for facility location. CoRR, abs/0809.2554, 2008.
MohammadTaghi Hajiaghayi, Rohit Khandekar, and Guy Kortsarz. Local search algorithms for the red-blue median problem. Algorithmica, 63(4):795-814, 2012.
Ravishankar Krishnaswamy, Amit Kumar, Viswanath Nagarajan, Yogish Sabharwal, and Barna Saha. The matroid median problem. In Proc. of SODA, pages 1117-1130, 2011.
Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. In Proc. of STOC, pages 901-910, 2013.
Mohammad Mahdian and Martin Pál. Universal facility location. In Proc. of ESA, pages 409-421, 2003.
Martin Pál, Éva Tardos, and Tom Wexler. Facility location with nonuniform hard capacities. In Proc. of FOCS, pages 329-338, 2001.
Zoya Svitkina and Éva Tardos. Facility location with hierarchical facility costs. ACM Trans. Algorithms, 6(2), 2010.
Chaitanya Swamy. Improved approximation algorithms for matroid and knapsack median problems and applications. In Proc. of APPROX, pages 403-418, 2014.
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Improved Reduction from the Bounded Distance Decoding Problem to the Unique Shortest Vector Problem in Lattices
We present a probabilistic polynomial-time reduction from the lattice Bounded Distance Decoding (BDD) problem with parameter 1/( sqrt(2) * gamma) to the unique Shortest Vector Problem (uSVP) with parameter gamma for any gamma > 1 that is polynomial in the lattice dimension n. It improves the BDD to uSVP reductions of [Lyubashevsky and Micciancio, CRYPTO, 2009] and [Liu, Wang, Xu and Zheng, Inf. Process. Lett., 2014], which rely on Kannan's embedding technique. The main ingredient to the improvement is the use of Khot's lattice sparsification [Khot, FOCS, 2003] before resorting to Kannan's embedding, in order to boost the uSVP parameter.
Lattices
Bounded Distance Decoding Problem
Unique Shortest Vector Problem
Sparsification
76:1-76:12
Regular Paper
Shi
Bai
Shi Bai
Damien
Stehlé
Damien Stehlé
Weiqiang
Wen
Weiqiang Wen
10.4230/LIPIcs.ICALP.2016.76
M. Ajtai. The shortest vector problem in l₂ is NP-hard for randomized reductions (extended abstract). In Proc. of STOC, pages 284-293. ACM, 1998.
L. Babai. On Lovász lattice reduction and the nearest lattice point problem. Combinatorica, 6:1-13, 1986.
S. Bai and S. Galbraith. Private communication, 2015.
D. Dadush and G. Kun. Lattice sparsification and the approximate closest vector problem. In Proc. of SODA, pages 1088-1102. SIAM, 2013.
D. Dadush, O. Regev, and N. Stephens-Davidowitz. On the closest vector problem with a distance guarantee. In Proc. of CCC, pages 98-109. IEEE Computer Society Press, 2014.
R. de Buda. The upper error bound of a new near-optimal code. IEEE Trans. on Information Theory, 21(4):441-445, 1975.
U. Fincke and M. Pohst. A procedure for determining algebraic integers of given norm. In Proc. of EUROCAL, volume 162 of LNCS, pages 194-202, 1983.
R. Kannan. Improved algorithms for integer programming and related lattice problems. In Proc. of STOC, pages 99-108. ACM, 1983.
R. Kannan. Minkowski’s convex body theorem and integer programming. Math. Oper. Res., 12(3):415-440, 1987.
S. Khot. Hardness of approximating the shortest vector problem in high L_p norms. In Proc. of FOCS, pages 290-297. IEEE Computer Society Press, 2003.
S. Khot. Hardness of approximating the shortest vector problem in lattices. J. ACM, 52(5):789-808, 2005.
A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Math. Ann, 261:515-534, 1982.
M. Liu, X. Wang, G. Xu, and X. Zheng. A note on BDD problems with λ₂-gap. Inf. Process. Lett., 114(1-2):9-12, January 2014.
Y. K. Liu, V. Lyubashevsky, and D. Micciancio. On bounded distance decoding for general lattices. In Proc. of RANDOM, volume 4110 of LNCS, pages 450-461. Springer, 2006.
V. Lyubashevsky and D. Micciancio. On bounded distance decoding, unique shortest vectors, and the minimum distance problem. In Proc. of CRYPTO, pages 577-594, 2009.
D. Micciancio. The shortest vector problem is NP-hard to approximate to within some constant. SIAM J. Comput, 30(6):2008-2035, 2001.
D. Micciancio. Private communication, 2015.
D. Micciancio and S. Goldwasser. Complexity of Lattice problem: A Cryptography Perspective. Kluwer, 2009.
O. Regev. On lattices, learning with errors, random linear codes, and cryptography. J. ACM, 56(6), 2009.
C. P. Schnorr. A hierarchy of polynomial lattice basis reduction algorithms. Theor. Comput. Science, 53:201-224, 1987.
N. Stephens-Davidowitz. Discrete Gaussian sampling reduces to CVP and SVP. In Proc. of SODA, pages 1748-1764. SIAM, 2016.
A. Vardy. Algorithmic complexity in coding theory and the minimum distance problem. In Proc. of STOC, pages 92-109. ACM, 1997.
Creative Commons Attribution 3.0 Unported license
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A Parallel Repetition Theorem for All Entangled Games
The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Raz's classical parallel repetition theorem have been proved for many special classes of games. However, for general entangled games no parallel repetition theorem was known.
We prove that the entangled value of a two-player game G repeated n times in parallel is at most c_G*n^{-1/4}*log(n) for a constant c_G depending on G, provided that the entangled value of G is less than 1. In particular, this gives the first proof that the entangled value of a parallel repeated game must converge to 0 for all games whose entangled value is less than 1. Central to our proof is a combination of both classical and quantum correlated sampling.
parallel repetition
direct product theorems
entangled games
quantum games
77:1-77:13
Regular Paper
Henry
Yuen
Henry Yuen
10.4230/LIPIcs.ICALP.2016.77
Mohammad Bavarian, Thomas Vidick, and Henry Yuen. Anchoring games for parallel repetition. arXiv preprint arXiv:1509.07466, 2015.
Mohammad Bavarian, Thomas Vidick, and Henry Yuen. Parallel repetition via fortification: analytic view and the quantum case. arXiv preprint arXiv:1603.05349, 2016.
John S Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1(3), 1964.
Mark Braverman and Ankit Garg. Small value parallel repetition for general games. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing (STOC), 2015.
Mark Braverman, Anup Rao, Omri Weinstein, and Amir Yehudayoff. Direct products in communication complexity. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pages 746-755. IEEE, 2013.
André Chailloux and Giannicola Scarpa. Parallel repetition of entangled games with exponential decay via the superposed information cost. In Automata, Languages, and Programming, pages 296-307. Springer, 2014.
Kai-Min Chung, Xiaodi Wu, and Henry Yuen. Parallel repetition for entangled k-player games via fast quantum search. In the 30th Conference on Computational Complexity (CCC), pages 512-536, 2015.
Richard Cleve, William Slofstra, Falk Unger, and Sarvagya Upadhyay. Perfect parallel repetition theorem for quantum xor proof systems. Computational Complexity, 17(2):282-299, 2008.
Irit Dinur, David Steurer, and Thomas Vidick. A parallel repetition theorem for entangled projection games. In the 29th Conference on Computational Complexity (CCC), pages 197-208, 2014.
Uriel Feige and Joe Kilian. Two-prover protocols - low error at affordable rates. SIAM Journal on Computing, 30(1):324-346, 2000.
Johan Håstad. Some optimal inapproximability results. Journal of the ACM (JACM), 48(4), 2001.
Thomas Holenstein. Parallel repetition: Simplification and the no-signaling case. Theory of Computing, 5(8):141-172, 2009. URL: http://dx.doi.org/10.4086/toc.2009.v005a008.
http://dx.doi.org/10.4086/toc.2009.v005a008
Rahul Jain. New strong direct product results in communication complexity. Electronic Colloquium on Computational Complexity (ECCC), 18(24):2, 2011.
Rahul Jain, Attila Pereszlényi, and Penghui Yao. A parallel repetition theorem for entangled two-player one-round games under product distributions. In Proceedings of 29th Conference on Computational Complexity (CCC), pages 209-216, 2014.
Julia Kempe, Oded Regev, and Ben Toner. Unique games with entangled provers are easy. In Proceedings of Foundations of Computer Science (FOCS), 2008.
Julia Kempe and Thomas Vidick. Parallel repetition of entangled games. In Proceedings of the forty-third annual ACM symposium on Theory of computing (STOC), pages 353-362, 2011.
Dana Moshkovitz. Parallel repetition from fortification. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), pages 414-423. IEEE, 2014.
Michael A Nielsen and Isaac L Chuang. Quantum computation and quantum information. Cambridge university press, 2010.
Ran Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763-803, 1998.
Wim van Dam and Patrick Hayden. Universal entanglement transformations without communication. Physical Review A, 67(6):060302, 2003.
Mark M Wilde. Quantum information theory. Cambridge University Press, 2013.
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Tight Sum-Of-Squares Lower Bounds for Binary Polynomial Optimization Problems
We give two results concerning the power of the Sum-Of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree 2d and an odd number of variables n, we prove that (n+2d-1)/2 levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito.
Additionally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is n-1. We disprove this conjecture and derive lower and upper bounds for the rank.
SoS/Lasserre hierarchy
lift and project methods
binary polynomial optimization
78:1-78:14
Regular Paper
Adam
Kurpisz
Adam Kurpisz
Samuli
Leppänen
Samuli Leppänen
Monaldo
Mastrolilli
Monaldo Mastrolilli
10.4230/LIPIcs.ICALP.2016.78
Yu Hin Au. A Comprehensive Analysis of Lift-and-Project Methods for Combinatorial Optimization. PhD thesis, University of Waterloo, 2014.
Boaz Barak, Fernando G. S. L. Brandão, Aram Wettroth Harrow, Jonathan A. Kelner, David Steurer, and Yuan Zhou. Hypercontractivity, sum-of-squares proofs, and their applications. In STOC, pages 307-326, 2012. URL: http://dx.doi.org/10.1145/2213977.2214006.
http://dx.doi.org/10.1145/2213977.2214006
Grigoriy Blekherman. Symmetric sums of squares on the hypercube. Manuscript in preparation, 2015.
William Cook and Sanjeeb Dash. On the matrix-cut rank of polyhedra. Mathematics of Operations Research, 26(1):19-30, 2001.
Gérard Cornuéjols and Yanjun Li. On the rank of mixed 0, 1 polyhedra. In IPCO, pages 71-77. Springer, 2001.
AC Dixon. Summation of a certain series. Proceedings of the London Mathematical Society, 1(1):284-291, 1902.
Hamza Fawzi, James Saunderson, and Pablo A. Parrilo. Sparse sums of squares on finite abelian groups and improved semidefinite lifts. Mathematical Programming, pages 1-43, 2016. URL: http://dx.doi.org/10.1007/s10107-015-0977-z.
http://dx.doi.org/10.1007/s10107-015-0977-z
Michel X. Goemans and Levent Tunçel. When does the positive semidefiniteness constraint help in lifting procedures? Mathematics of Operations Research, 26(4):796-815, 2001. URL: http://dx.doi.org/10.1287/moor.26.4.796.10012.
http://dx.doi.org/10.1287/moor.26.4.796.10012
Dima Grigoriev. Complexity of positivstellensatz proofs for the knapsack. Computational Complexity, 10(2):139-154, 2001. URL: http://dx.doi.org/10.1007/s00037-001-8192-0.
http://dx.doi.org/10.1007/s00037-001-8192-0
Dima Grigoriev. Linear lower bound on degrees of positivstellensatz calculus proofs for the parity. Theoretical Computer Science, 259(1-2):613-622, 2001. URL: http://dx.doi.org/10.1016/S0304-3975(00)00157-2.
http://dx.doi.org/10.1016/S0304-3975(00)00157-2
Dima Grigoriev and Nicolai Vorobjov. Complexity of null-and positivstellensatz proofs. Annals of Pure and Applied Logic, 113(1-3):153-160, 2001. URL: http://dx.doi.org/10.1016/S0168-0072(01)00055-0.
http://dx.doi.org/10.1016/S0168-0072(01)00055-0
Adam Kurpisz, Samuli Leppänen, and Monaldo Mastrolilli. On the hardest problem formulations for the 0/1 lasserre hierarchy. In ICALP, pages 872-885, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_71.
http://dx.doi.org/10.1007/978-3-662-47672-7_71
Adam Kurpisz, Samuli Leppänen, and Monaldo Mastrolilli. Sum-of-squares lower bounds for maximally symmetric formulations. In To appear in IPCO, 2016. URL: http://arxiv.org/abs/1407.1746.
http://arxiv.org/abs/1407.1746
Jean B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11(3):796-817, 2001. URL: http://dx.doi.org/10.1137/S1052623400366802.
http://dx.doi.org/10.1137/S1052623400366802
Monique Laurent. A comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre relaxations for 0-1 programming. Mathematics of Operations Research, 28(3):470-496, 2003. URL: http://dx.doi.org/10.1287/moor.28.3.470.16391.
http://dx.doi.org/10.1287/moor.28.3.470.16391
Monique Laurent. Lower bound for the number of iterations in semidefinite hierarchies for the cut polytope. Mathematics of Operations Research, 28(4):871-883, 2003. URL: http://dx.doi.org/10.1287/moor.28.4.871.20508.
http://dx.doi.org/10.1287/moor.28.4.871.20508
James R. Lee, Prasad Raghavendra, and David Steurer. Lower bounds on the size of semidefinite programming relaxations. In STOC, pages 567-576, 2015. URL: http://dx.doi.org/10.1145/2746539.2746599.
http://dx.doi.org/10.1145/2746539.2746599
Troy Lee, Anupam Prakash, Ronald de Wolf, and Henry Yuen. On the sum-of-squares degree of symmetric quadratic functions. In To appear in CCC, 2016. URL: http://arxiv.org/abs/1601.02311.
http://arxiv.org/abs/1601.02311
Claire Mathieu and Alistair Sinclair. Sherali-adams relaxations of the matching polytope. In STOC, pages 293-302, 2009. URL: http://dx.doi.org/10.1145/1536414.1536456.
http://dx.doi.org/10.1145/1536414.1536456
Raghu Meka, Aaron Potechin, and Avi Wigderson. Sum-of-squares lower bounds for planted clique. In STOC, pages 87-96, 2015. URL: http://dx.doi.org/10.1145/2746539.2746600.
http://dx.doi.org/10.1145/2746539.2746600
Pablo Parrilo. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology, 2000.
Shinsaku Sakaue, Akiko Takeda, Sunyoung Kim, and Naoki Ito. Exact sdp relaxations with truncated moment matrix for binary polynomial optimization problems. Technical report, University of Tokyo, 2016. URL: http://www.keisu.t.u-tokyo.ac.jp/research/techrep/data/2016/METR16-01.pdf.
http://www.keisu.t.u-tokyo.ac.jp/research/techrep/data/2016/METR16-01.pdf
Tamon Stephen and Levent Tunçel. On a representation of the matching polytope via semidefinite liftings. Mathematics of Operations Research, 24(1):1-7, 1999.
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Correlation Decay and Tractability of CSPs
The algebraic dichotomy conjecture of Bulatov, Krokhin and Jeavons yields an elegant characterization of the complexity of constraint satisfaction problems. Roughly speaking, the characterization asserts that a CSP L is tractable if and only if there exist certain non-trivial operations known as polymorphisms to combine solutions to L to create new ones.
In this work, we study the dynamical system associated with repeated applications of a polymorphism to a distribution over assignments. Specifically, we exhibit a correlation decay phenomenon that makes two variables or groups of variables that are not perfectly correlated become independent after repeated applications of a polymorphism.
We show that this correlation decay phenomenon can be utilized in designing algorithms for CSPs by exhibiting two applications:
1. A simple randomized algorithm to solve linear equations over a prime field, whose analysis crucially relies on correlation decay.
2. A sufficient condition for the simple linear programming relaxation for a 2-CSP to be sound (have no integrality gap) on a given instance.
Constraint Satisfaction
Polymorphisms
Linear Equations
Correlation Decay
79:1-79:13
Regular Paper
Jonah
Brown-Cohen
Jonah Brown-Cohen
Prasad
Raghavendra
Prasad Raghavendra
10.4230/LIPIcs.ICALP.2016.79
Libor Barto and Marcin Kozik. Constraint satisfaction problems of bounded width. In FOCS, pages 595-603, 2009. URL: http://dx.doi.org/10.1109/FOCS.2009.32.
http://dx.doi.org/10.1109/FOCS.2009.32
Libor Barto and Marcin Kozik. Absorbing subalgebras, cyclic terms, and the constraint satisfaction problem. Logical Methods in Computer Science, 8(1), 2012. URL: http://dx.doi.org/10.2168/LMCS-8(1:7)2012.
http://dx.doi.org/10.2168/LMCS-8(1:7)2012
Andrei A. Bulatov. Tractable conservative constraint satisfaction problems. In LICS, pages 321-330, 2003. URL: http://dx.doi.org/10.1109/LICS.2003.1210072.
http://dx.doi.org/10.1109/LICS.2003.1210072
Andrei A. Bulatov. A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM, 53(1):66-120, 2006. URL: http://dx.doi.org/10.1145/1120582.1120584.
http://dx.doi.org/10.1145/1120582.1120584
Andrei A. Bulatov and Víctor Dalmau. A simple algorithm for mal'tsev constraints. SIAM J. Comput., 36(1):16-27, 2006. URL: http://dx.doi.org/10.1137/050628957.
http://dx.doi.org/10.1137/050628957
Andrei A. Bulatov, Andrei A. Krokhin, and Peter Jeavons. Constraint satisfaction problems and finite algebras. In ICALP, pages 272-282, 2000. URL: http://dx.doi.org/10.1007/3-540-45022-X_24.
http://dx.doi.org/10.1007/3-540-45022-X_24
Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic snp and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput., 28(1):57-104, 1998. URL: http://dx.doi.org/10.1137/S0097539794266766.
http://dx.doi.org/10.1137/S0097539794266766
Gábor Kun and Mario Szegedy. A new line of attack on the dichotomy conjecture. In STOC, pages 725-734, 2009. URL: http://dx.doi.org/10.1145/1536414.1536512.
http://dx.doi.org/10.1145/1536414.1536512
Miklós Maróti and Ralph McKenzie. Existence theorems for weakly symmetric operations. Algebra Universalis, 59:463-489, 2008. URL: http://dx.doi.org/10.1007/s00012-008-2122-9.
http://dx.doi.org/10.1007/s00012-008-2122-9
E. Mossel. Gaussian bounds for noise correlation of functions. In FOCS'08: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, 2008.
Thomas J. Schaefer. The complexity of satisfiability problems. In STOC, pages 216-226, 1978. URL: http://dx.doi.org/10.1145/800133.804350.
http://dx.doi.org/10.1145/800133.804350
Creative Commons Attribution 3.0 Unported license
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On Percolation and NP-Hardness
The edge-percolation and vertex-percolation random graph models start with an arbitrary graph G, and randomly delete edges or vertices of G with some fixed probability. We study the computational hardness of problems whose inputs are obtained by applying percolation to worst-case instances. Specifically, we show that a number of classical N P-hard graph problems remain essentially as hard on percolated instances as they are in the worst-case (assuming NP !subseteq BPP). We also prove hardness results for other NP-hard problems such as Constraint Satisfaction Problems, where random deletions are applied to clauses or variables.
We focus on proving the hardness of the Maximum Independent Set problem and the Graph Coloring problem on percolated instances. To show this we establish the robustness of the corresponding parameters alpha(.) and Chi(.) to percolation, which may be of independent interest. Given a graph G, let G' be the graph obtained by randomly deleting edges of G. We show that if alpha(G) is small, then alpha(G') remains small with probability at least 0.99. Similarly, we show that if Chi(G) is large, then Chi(G') remains large with probability at least 0.99.
percolation
NP-hardness
random subgraphs
chromatic number
80:1-80:14
Regular Paper
Huck
Bennett
Huck Bennett
Daniel
Reichman
Daniel Reichman
Igor
Shinkar
Igor Shinkar
10.4230/LIPIcs.ICALP.2016.80
N. Alon and J. H. Spencer. The Probabilistic Method. Wiley-Interscience series in discrete mathematics and optimization. J. Wiley &Sons, New York, 2000.
B. Barak, M. Hardt, T. Holenstein, and Steurer D. Subsampling mathematical relaxations and average-case complexity. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, San Francisco, California, USA, pages 512-531, 2011. URL: http://dx.doi.org/10.1137/1.9781611973082.41.
http://dx.doi.org/10.1137/1.9781611973082.41
B. Bollobás. The chromatic number of random graphs. Combinatorica, 8(1):49-55, 1988. URL: http://dx.doi.org/10.1007/BF02122551.
http://dx.doi.org/10.1007/BF02122551
B. Bollobás. Random graphs. Springer, 1998.
B. Bollobás, B. P. Narayanan, and A. M. Raigorodskii. On the stability of the erdős-ko-rado theorem. J. Comb. Theory, Ser. A, 137:64-78, 2016. URL: http://dx.doi.org/10.1016/j.jcta.2015.08.002.
http://dx.doi.org/10.1016/j.jcta.2015.08.002
R. L. Brooks. On colouring the nodes of a network. Mathematical Proceedings of the Cambridge Philosophical Society, 37:194-197, 4 1941. URL: http://dx.doi.org/10.1017/S030500410002168X.
http://dx.doi.org/10.1017/S030500410002168X
B. Bukh. Interesting problems that I cannot solve. Problem 2. URL: http://www.borisbukh.org/problems.html.
http://www.borisbukh.org/problems.html
I. Dinur, E. Mossel, and O. Regev. Conditional hardness for approximate coloring. SIAM J. Comput., 39(3):843-873, 2009. URL: http://dx.doi.org/10.1137/07068062X.
http://dx.doi.org/10.1137/07068062X
I. Dinur and I. Shinkar. On the conditional hardness of coloring a 4-colorable graph with super-constant number of colors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 13th International Workshop, APPROX 2010, and 14th International Workshop, RANDOM 2010, Barcelona, Spain, September 1-3, 2010. Proceedings, pages 138-151, 2010. URL: http://dx.doi.org/10.1007/978-3-642-15369-3_11.
http://dx.doi.org/10.1007/978-3-642-15369-3_11
M. Etscheid and H. Röglin. Smoothed analysis of local search for the maximum-cut problem. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, Portland, Oregon, USA, pages 882-889, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.66.
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Tight Hardness Results for Maximum Weight Rectangles
Given n weighted points (positive or negative) in d dimensions, what is the axis-aligned box which maximizes the total weight of the points it contains?
The best known algorithm for this problem is based on a reduction to a related problem, the Weighted Depth problem [Chan, FOCS, 2013], and runs in time O(n^d). It was conjectured [Barbay et al., CCCG, 2013] that this runtime is tight up to subpolynomial factors. We answer this conjecture affirmatively by providing a matching conditional lower bound. We also provide conditional lower bounds for the special case when points are arranged in a grid (a well studied problem known as Maximum Subarray problem) as well as for other related problems.
All our lower bounds are based on assumptions that the best known algorithms for the All-Pairs Shortest Paths problem (APSP) and for the Max-Weight k-Clique problem in edge-weighted graphs are essentially optimal.
Maximum Rectangles
Hardness in P
81:1-81:13
Regular Paper
Arturs
Backurs
Arturs Backurs
Nishanth
Dikkala
Nishanth Dikkala
Christos
Tzamos
Christos Tzamos
10.4230/LIPIcs.ICALP.2016.81
Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. If the current clique algorithms are optimal, so is valiant’s parser. In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 98-117. IEEE, 2015.
Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic equivalences between graph centrality problems, apsp and diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1681-1697. SIAM, 2015.
Amir Abboud, Virginia Vassilevska Williams, and Huacheng Yu. Matching triangles and basing hardness on an extremely popular conjecture. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 41-50. ACM, 2015.
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The Johnson-Lindenstrauss Lemma Is Optimal for Linear Dimensionality Reduction
For any n > 1, 0 < epsilon < 1/2, and N > n^C for some constant C > 0, we show the existence of an N-point subset X of l_2^n such that any linear map from X to l_2^m with distortion at most 1 + epsilon must have m = Omega(min{n, epsilon^{-2}*lg(N)). This improves a lower bound of Alon [Alon, Discre. Mathem., 1999], in the linear setting, by a lg(1/epsilon) factor. Our lower bound matches the upper bounds provided by the identity matrix and the Johnson-Lindenstrauss lemma [Johnson and Lindenstrauss, Contem. Mathem., 1984].
dimensionality reduction
lower bounds
Johnson-Lindenstrauss
82:1-82:11
Regular Paper
Kasper Green
Larsen
Kasper Green Larsen
Jelani
Nelson
Jelani Nelson
10.4230/LIPIcs.ICALP.2016.82
Radosław Adamczak and Paweł Wolff. Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order. CoRR, abs/1304.1826, 2013.
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Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost
Transportation cost metrics, also known as the Wasserstein distances W_p, are a natural choice for defining distances between two pointsets, or distributions, and have been applied in numerous fields. From the computational perspective, there has been an intensive research effort for understanding the W_p metrics over R^k, with work on the W_1 metric (a.k.a earth mover distance) being most successful in terms of theoretical guarantees. However, the W_2 metric, also known as the root-mean square (RMS) bipartite matching distance, is often a more suitable choice in many application areas, e.g. in graphics. Yet, the geometry of this metric space is currently poorly understood, and efficient algorithms have been elusive. For example, there are no known non-trivial algorithms for nearest-neighbor search or sketching for this metric.
In this paper we take the first step towards explaining the lack of efficient algorithms for the W_2 metric, even over the three-dimensional Euclidean space R^3. We prove that there are no meaningful embeddings of W_2 over R^3 into a wide class of normed spaces, as well as that there are no efficient sketching algorithms for W_2 over R^3 achieving constant approximation. For example, our results imply that: 1) any embedding into L1 must incur a distortion of Omega(sqrt(log(n))) for pointsets of size n equipped with the W_2 metric; and 2) any sketching algorithm of size s must incur Omega(sqrt(log(n))/sqrt(s)) approximation. Our results follow from a more general statement, asserting that W_2 over R^3 contains the 1/2-snowflake of all finite metric spaces with a uniformly bounded distortion. These are the first non-embeddability/non-sketchability results for W_2.
Transportation metric
embedding
snowflake
sketching
83:1-83:14
Regular Paper
Alexandr
Andoni
Alexandr Andoni
Assaf
Naor
Assaf Naor
Ofer
Neiman
Ofer Neiman
10.4230/LIPIcs.ICALP.2016.83
Pankaj Agarwal and Kasturi Varadarajan. A near-linear constant-factor approximation for euclidean bipartite matching? In Proceedings of the Twentieth Annual Symposium on Computational Geometry, SCG'04, pages 247-252, New York, NY, USA, 2004. ACM. URL: http://dx.doi.org/10.1145/997817.997856.
http://dx.doi.org/10.1145/997817.997856
Pankaj K. Agarwal, Alon Efrat, and Micha Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM Journal on Computing, 29(3):912-953, 2000.
Pankaj K. Agarwal and R. Sharathkumar. Approximation algorithms for bipartite matching with metric and geometric costs. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC'14, pages 555-564, New York, NY, USA, 2014. ACM. URL: http://dx.doi.org/10.1145/2591796.2591844.
http://dx.doi.org/10.1145/2591796.2591844
Alexandr Andoni, Khanh Do Ba, Piotr Indyk, and David Woodruff. Efficient sketches for Earth-Mover Distance, with applications. In Proceedings of the Symposium on Foundations of Computer Science (FOCS), 2009.
Alexandr Andoni, Piotr Indyk, and Robert Krauthgamer. Earth mover distance over high-dimensional spaces. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 343-352, 2008. Previously ECCC Report TR07-048.
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http://arxiv.org/abs/1411.2577
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http://dx.doi.org/10.1145/2591796.2591805
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Patrick Mullen, Pooran Memari, Fernando de Goes, and Mathieu Desbrun. Hot: Hodge-optimized triangulations. ACM Transactions on Graphics (TOG), 30(4):103, 2011.
Assaf Naor. Comparison of metric spectral gaps. Anal. Geom. Metr. Spaces, 2:1-52, 2014. URL: http://dx.doi.org/10.2478/agms-2014-0001.
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Assaf Naor, Yuval Peres, Oded Schramm, and Scott Sheffield. Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J., 134(1):165-197, 2006. URL: http://dx.doi.org/10.1215/S0012-7094-06-13415-4.
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Assaf Naor and Gideon Schechtman. Planar earthmover is not in L₁. SIAM J. Comput. (SICOMP), 37(3):804-826, 2007. An extended abstract appeared in FOCS'06.
Kangyu Ni, Xavier Bresson, Tony F. Chan, and Selim Esedoglu. Local histogram based segmentation using the wasserstein distance. International Journal of Computer Vision, 84(1):97-111, 2009. URL: http://dx.doi.org/10.1007/s11263-009-0234-0.
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Yann Ollivier, Herve Pajot, and Cedric Villani. Optimal Transport, Theory and Applications. Cambridge University Press, New York, NY, USA, 2014.
List of open problems in sublinear algorithms: Problem 7. URL: http://sublinear.info/7.
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List of open problems in sublinear algorithms: Problem 49. URL: http://sublinear.info/49.
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Yossi Rubner, Carlo Tomasi, and Leonidas J. Guibas. The earth mover’s distance as a metric for image retrieval. International Journal of Computer Vision, 40(2):99-121, 2000.
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Simple Average-Case Lower Bounds for Approximate Near-Neighbor from Isoperimetric Inequalities
We prove an Omega(d/log(sw/nd)) lower bound for the average-case cell-probe complexity of deterministic or Las Vegas randomized algorithms solving approximate near-neighbor (ANN) problem in ddimensional Hamming space in the cell-probe model with w-bit cells, using a table of size s. This lower bound matches the highest known worst-case cell-probe lower bounds for any static data structure problems.
This average-case cell-probe lower bound is proved in a general framework which relates the cell-probe complexity of ANN to isoperimetric inequalities in the underlying metric space. A tighter connection between ANN lower bounds and isoperimetric inequalities is established by a stronger richness lemma proved by cell-sampling techniques.
nearest neighbor search
approximate near-neighbor
cell-probe model
isoperimetric inequality
84:1-84:13
Regular Paper
Yitong
Yin
Yitong Yin
10.4230/LIPIcs.ICALP.2016.84
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Quasimetric Embeddings and Their Applications
We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces.
Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any n-point quasimetric space supported on a graph of treewidth t admits a random embedding into quasiultrametric spaces with distortion O(t*log^2(n)), where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain t*log^{O(1)}(n)-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut and the Directed Multicut problems on n-vertex graphs of treewidth t, with running time polynomial in both n and t.
The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of [Chuzhoy and Khanna 2009] we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph G into graphs that exclude G as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting.
metric embeddings
quasimetrics
outliers
random embeddings
treewidth
Directed Sparsest-Cut
Directed Multicut
85:1-85:14
Regular Paper
Facundo
Mémoli
Facundo Mémoli
Anastasios
Sidiropoulos
Anastasios Sidiropoulos
Vijay
Sridhar
Vijay Sridhar
10.4230/LIPIcs.ICALP.2016.85
Ittai Abraham, Yair Bartal, and Ofer Neiman. Nearly tight low stretch spanning trees. arXiv preprint arXiv:0808.2017, 2008.
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The Landscape of Communication Complexity Classes
We prove several results which, together with prior work, provide a nearly-complete picture of the relationships among classical communication complexity classes between P and PSPACE, short of proving lower bounds against classes for which no explicit lower bounds were already known. Our article also serves as an up-to-date survey on the state of structural communication complexity.
Among our new results we show that MA !subseteq ZPP^{NP[1]}, that is, Merlin–Arthur proof systems cannot be simulated by zero-sided error randomized protocols with one NP query. Here the class ZPP^{NP[1]} has the property that generalizing it in the slightest ways would make it contain AM intersect coAM, for which it is notoriously open to prove any explicit lower bounds. We also prove that US !subseteq ZPP^{NP[1]}, where US is the class whose canonically complete problem is the variant of set-disjointness where yes-instances are uniquely intersecting. We also prove that US !subseteq coDP, where DP is the class of differences of two NP sets. Finally, we explore an intriguing open issue: are rank-1 matrices inherently more powerful than rectangles in communication complexity? We prove a new separation concerning PP that sheds light on this issue and strengthens some previously known separations.
Landscape
communication
complexity
classes
86:1-86:15
Regular Paper
Mika
Göös
Mika Göös
Toniann
Pitassi
Toniann Pitassi
Thomas
Watson
Thomas Watson
10.4230/LIPIcs.ICALP.2016.86
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http://dx.doi.org/10.1145/2636924
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http://dx.doi.org/10.1109/CCC.2008.32
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http://dx.doi.org/10.1007/s00037-008-0242-4
Alexander Sherstov. The unbounded-error communication complexity of symmetric functions. Combinatorica, 31(5):583-614, 2011. URL: http://dx.doi.org/10.1007/s00493-011-2580-0.
http://dx.doi.org/10.1007/s00493-011-2580-0
Rahul Tripathi. The 1-versus-2 queries problem revisited. Theory of Computing Systems, 46(2):193-221, 2010. URL: http://dx.doi.org/10.1007/s00224-008-9126-x.
http://dx.doi.org/10.1007/s00224-008-9126-x
Leslie Valiant. Graph-theoretic arguments in low-level complexity. In Proceedings of the 6th Symposium on Mathematical Foundations of Computer Science (MFCS), pages 162-176. Springer, 1977. URL: http://dx.doi.org/10.1007/3-540-08353-7_135.
http://dx.doi.org/10.1007/3-540-08353-7_135
Leslie Valiant and Vijay Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47(3):85-93, 1986. URL: http://dx.doi.org/10.1016/0304-3975(86)90135-0.
http://dx.doi.org/10.1016/0304-3975(86)90135-0
Nikolai Vereshchagin. Relativizability in complexity theory. In Provability, Complexity, Grammars, volume 192 of AMS Translations, Series 2, pages 87-172. American Mathematical Society, 1999.
Henning Wunderlich. On a theorem of Razborov. Computational Complexity, 21(3):431-477, 2012. URL: http://dx.doi.org/10.1007/s00037-011-0021-5.
http://dx.doi.org/10.1007/s00037-011-0021-5
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Information Complexity Is Computable
The information complexity of a function f is the minimum amount of information Alice and Bob need to exchange to compute the function f. In this paper we provide an algorithm for approximating the information complexity of an arbitrary function f to within any additive error epsilon > 0, thus resolving an open question as to whether information complexity is computable.
In the process, we give the first explicit upper bound on the rate of convergence of the information complexity of f when restricted to b-bit protocols to the (unrestricted) information complexity of f.
Communication complexity
convergence rate
information complexity
87:1-87:10
Regular Paper
Mark
Braverman
Mark Braverman
Jon
Schneider
Jon Schneider
10.4230/LIPIcs.ICALP.2016.87
Noga Alon and Eyal Lubetzky. The shannon capacity of a graph and the independence numbers of its powers. Information Theory, IEEE Transactions on, 52(5):2172-2176, 2006.
Ziv Bar-Yossef, Thathachar S Jayram, Ravindra Kumar, and D Sivakumar. An information statistics approach to data stream and communication complexity. In Foundations of Computer Science, 2002. Proceedings. The 43rd Annual IEEE Symposium on, pages 209-218. IEEE, 2002.
Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. SIAM Journal on Computing, 42(3):1327-1363, 2013.
Richard Beigel and Jun Tarui. On acc [circuit complexity]. In Foundations of Computer Science, 1991. Proceedings., 32nd Annual Symposium on, pages 783-792. IEEE, 1991.
Mark Braverman. Interactive information complexity. SIAM Journal on Computing, 44(6):1698-1739, 2015.
Mark Braverman, Ankit Garg, Denis Pankratov, and Omri Weinstein. From information to exact communication. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 151-160. ACM, 2013.
Mark Braverman and Akhila Rao. Information equals amortized communication. Information Theory, IEEE Transactions on, 60(10):6058-6069, 2014.
Amit Chakrabart, Yaoyun Shi, Anthony Wirth, and Andrew Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In Foundations of Computer Science, 2001. Proceedings. 42nd IEEE Symposium on, pages 270-278. IEEE, 2001.
Richard Cleve, Peter Høyer, Benjamin Toner, and John Watrous. Consequences and limits of nonlocal strategies. In Computational Complexity, 2004. Proceedings. 19th IEEE Annual Conference on, pages 236-249. IEEE, 2004.
Thomas M. Cover and Joy A. Thomas. Elements of Information Theory. Wiley series in telecommunications. J. Wiley and Sons, New York, 1991.
Rahul Jain. New strong direct product results in communication complexity. In Electronic Colloquium on Computational Complexity (ECCC), volume 18, page 2, 2011.
Mauricio Karchmer and Avi Wigderson. Monotone circuits for connectivity require super-logarithmic depth. SIAM Journal on Discrete Mathematics, 3(2):255-265, 1990.
Nan Ma and Prakash Ishwar. Two-terminal distributed source coding with alternating messages for function computation. In Information Theory, 2008. ISIT 2008. IEEE International Symposium on, pages 51-55. IEEE, 2008.
Nan Ma and Prakash Ishwar. Some results on distributed source coding for interactive function computation. Information Theory, IEEE Transactions on, 57(9):6180-6195, 2011.
Claude Elwood Shannon. A mathematical theory of communication. ACM SIGMOBILE Mobile Computing and Communications Review, 5(1):3-55, 2001.
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Rényi Information Complexity and an Information Theoretic Characterization of the Partition Bound
In this work we introduce a new information-theoretic complexity measure for 2-party functions, called Rényi information complexity. It is a lower-bound on communication complexity, and has the two leading lower-bounds on communication complexity as its natural relaxations: (external) information complexity and logarithm of partition complexity. These two lower-bounds had so far appeared conceptually quite different from each other, but we show that they are both obtained from Rényi information complexity using two different, but natural relaxations:
1. The relaxation of Rényi information complexity that yields information complexity is to change the order of Rényi mutual information used in its definition from infinity to 1.
2. The relaxation that connects Rényi information complexity with partition complexity is to replace protocol transcripts used in the definition of Rényi information complexity with what we term "pseudotranscripts", which omits the interactive nature of a protocol, but only requires that the probability of any transcript given inputs x and y to the two parties, factorizes into two terms which depend on x and y separately. While this relaxation yields an apparently different definition than (log of) partition function, we show that the two are in fact identical. This gives us a surprising characterization of the partition bound in terms of an information-theoretic quantity.
We also show that if both the above relaxations are simultaneously applied to Rényi information complexity, we obtain a complexity measure that is lower-bounded by the (log of) relaxed partition complexity, a complexity measure introduced by Kerenidis et al. (FOCS 2012). We obtain a sharper connection between (external) information complexity and relaxed partition complexity than Kerenidis et al., using an arguably more direct proof.
Further understanding Rényi information complexity (of various orders) might have consequences for important direct-sum problems in communication complexity, as it lies between communication complexity and information complexity.
Information Complexity
Communication Complexity
Rényi Mutual Information
88:1-88:14
Regular Paper
Manoj M.
Prabhakaran
Manoj M. Prabhakaran
Vinod M.
Prabhakaran
Vinod M. Prabhakaran
10.4230/LIPIcs.ICALP.2016.88
Farid M. Ablayev. Lower bounds for one-way probabilistic communication complexity and their application to space complexity. Theor. Comput. Sci., 157(2):139-159, 1996. URL: http://dx.doi.org/10.1016/0304-3975(95)00157-3.
http://dx.doi.org/10.1016/0304-3975(95)00157-3
Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci., 68(4):702-732, 2004. URL: http://dx.doi.org/10.1016/j.jcss.2003.11.006.
http://dx.doi.org/10.1016/j.jcss.2003.11.006
Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. SIAM J. Comput., 42(3):1327-1363, 2013. URL: http://dx.doi.org/10.1137/100811969.
http://dx.doi.org/10.1137/100811969
Mark Braverman. Interactive information complexity. In STOC, pages 505-524, 2012. URL: http://dx.doi.org/10.1145/2213977.2214025.
http://dx.doi.org/10.1145/2213977.2214025
Mark Braverman and Anup Rao. Information equals amortized communication. In FOCS, pages 748-757, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.86.
http://dx.doi.org/10.1109/FOCS.2011.86
Mark Braverman and Omri Weinstein. A discrepancy lower bound for information complexity. In APPROX-RANDOM, pages 459-470, 2012. URL: http://dx.doi.org/10.1007/978-3-642-32512-0_39.
http://dx.doi.org/10.1007/978-3-642-32512-0_39
Amit Chakrabarti, Ranganath Kondapally, and Zhenghui Wang. Information complexity versus corruption and applications to orthogonality and gap-hamming. In APPROX-RANDOM, pages 483-494, 2012. URL: http://dx.doi.org/10.1007/978-3-642-32512-0_41.
http://dx.doi.org/10.1007/978-3-642-32512-0_41
Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, and Andrew Chi-Chih Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In FOCS, pages 270-278, 2001. URL: http://dx.doi.org/10.1109/SFCS.2001.959901.
http://dx.doi.org/10.1109/SFCS.2001.959901
Lila Fontes, Rahul Jain, Iordanis Kerenidis, Sophie Laplante, Mathieu Laurière, and Jérémie Roland. Relative discrepancy does not separate information and communication complexity. In ICALP, pages 506-516, 2015.
Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of information and communication. In FOCS, pages 176-185, 2014.
Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of communication and external information. In Electronic Colloquium on Computational Complexity (ECCC), 2015.
Prahladh Harsha, Rahul Jain, David McAllester, and Jaikumar Radhakrishnan. The communication complexity of correlation. IEEE Transactions on Information Theory, 56(1):438-449, 2010. URL: http://dx.doi.org/10.1109/TIT.2009.2034824.
http://dx.doi.org/10.1109/TIT.2009.2034824
Siu-Wai Ho and Sergio Verdú. Convexity/concavity of Rényi entropy and α-mutual information. In Information Theory (ISIT), 2015 IEEE International Symposium on, pages 745-749, 2015.
Rahul Jain and Hartmut Klauck. The partition bound for classical communication complexity and query complexity. In IEEE Conference on Computational Complexity, pages 247-258, 2010.
Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. A direct sum theorem in communication complexity via message compression. In ICALP, pages 300-315, 2003. URL: http://dx.doi.org/10.1007/3-540-45061-0_26.
http://dx.doi.org/10.1007/3-540-45061-0_26
Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. Prior entanglement, message compression and privacy in quantum communication. In IEEE Conference on Computational Complexity, pages 285-296, 2005. URL: http://dx.doi.org/10.1109/CCC.2005.24.
http://dx.doi.org/10.1109/CCC.2005.24
T. S. Jayram, Ravi Kumar, and D. Sivakumar. Two applications of information complexity. In STOC, pages 673-682, 2003. URL: http://dx.doi.org/10.1145/780542.780640.
http://dx.doi.org/10.1145/780542.780640
Iordanis Kerenidis, Sophie Laplante, Virginie Lerays, Jérémie Roland, and David Xiao. Lower bounds on information complexity via zero-communication protocols and applications. In FOCS, pages 500-509, 2012.
Eyal Kushilevitz and Noam Nisan. Communication complexity. Cambridge University Press, New York, 1997.
Stephen J Ponzio, Jaikumar Radhakrishnan, and Srinivasan Venkatesh. The communication complexity of pointer chasing. Journal of Computer and System Sciences, 62(2):323-355, 2001.
Manoj Prabhakaran and Vinod M. Prabhakaran. Tension bounds for information complexity. CoRR, abs/1408.6285, 2014. URL: http://arxiv.org/abs/1408.6285.
http://arxiv.org/abs/1408.6285
Alfred Rényi. On measures of information and entropy. In Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability, pages 547-561, 1960.
Michael E. Saks and Xiaodong Sun. Space lower bounds for distance approximation in the data stream model. In STOC, pages 360-369, 2002. URL: http://dx.doi.org/10.1145/509907.509963.
http://dx.doi.org/10.1145/509907.509963
R. Sibson. Information radius. Z. Wahrscheinlichkeitstheorie und Verw. Geb., 14:149-161, 1969.
Sergio Verdú. α-mutual information. In Information Theory and Applications Workshop (ITA), 2015.
Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (preliminary report). In STOC, pages 209-213, 1979. URL: http://dx.doi.org/10.1145/800135.804414.
http://dx.doi.org/10.1145/800135.804414
Moshe Zakai and Jacob Ziv. A generalization of the rate-distortion theory and applications. In Information Theory New Trends and Open Problems, pages 87-123. Springer, 1975.
Jacob Ziv and Moshe Zakai. On functionals satisfying a data-processing theorem. Information Theory, IEEE Transactions on, 19(3):275-283, 1973.
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On Isoperimetric Profiles and Computational Complexity
The isoperimetric profile of a graph is a function that measures, for an integer k, the size of the smallest edge boundary over all sets of vertices of size k. We observe a connection between isoperimetric profiles and computational complexity. We illustrate this connection by an example from communication complexity, but our main result is in algebraic complexity.
We prove a sharp super-polynomial separation between monotone arithmetic circuits and monotone arithmetic branching programs. This shows that the classical simulation of arithmetic circuits by arithmetic branching programs by Valiant, Skyum, Berkowitz, and Rackoff (1983) cannot be improved, as long as it preserves monotonicity.
A key ingredient in the proof is an accurate analysis of the isoperimetric profile of finite full binary trees. We show that the isoperimetric profile of a full binary tree constantly fluctuates between one and almost the depth of the tree.
Monotone computation
separations
communication complexity
isoperimetry
89:1-89:12
Regular Paper
Pavel
Hrubes
Pavel Hrubes
Amir
Yehudayoff
Amir Yehudayoff
10.4230/LIPIcs.ICALP.2016.89
M. Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In FOCS, pages 67-75, 2008.
S. Arora and B. Barak. Computational complexity: a modern approach. Cambridge University Press, 2009.
B. S. Bharadwaj, L. S. Chandran, and A. Das. Isoperimetric problem and meta-fibonacci sequences. In Computing and Combinatorics, pages 22-30. Springer, 2008.
A. Borodin, A. A. Razborov, and R. Smolensky. On lower bounds for read-k-times branching programs. Computational Complexity, 3(1):1-18, 1993.
P. Bürgisser, M. Clausen, and M. A. Shokrollahi. Algebraic complexity theory, volume 315. Springer Science and Business Media, 1997.
H. Fournier, N. Limaye, G. Malod, and S. Srinivasan. Lower bounds for depth 4 formulas computing iterated matrix multiplication. In STOC, pages 128-135, 2014.
J. von zur Gathen. Algebraic complexity theory. Annual review of computer science, 3:317-347, 1988.
A. Gupta, P. Kamath, N. Kayal, and R. Saptharishi. Arithmetic circuits: A chasm at depth three. In FOCS, pages 578-587, 2013.
A. Gupta, P. Kamath, N. Kayal, and R. Saptharishi. Approaching the chasm at depth four. Journal of the ACM, 61(6):33, 2014.
P. Hrubes and A. Yehudayoff. Monotone separations for constant degree polynomials. Information Processing Letters, 110(1):1-3, 2009.
L. Hyafil. On the parallel evaluation of multivariate polynomials. SIAM J. Comput., 8(2):120-123, 1979.
S. Jukna. Boolean function complexity: advances and frontiers, volume 27. Springer Science and Business Media, 2012.
P. Koiran. Arithmetic circuits: The chasm at depth four gets wider. Theoretical Computer Science, 448:56-65, 2012.
M. Kumar and S. Saraf. The limits of depth reduction for arithmetic formulas: It’s all about the top fan-in. In STOC, pages 136-145, 2014.
E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, New York, NY, USA, 1997.
F. Morgan. Manifolds with density. Notices of the AMS, pages 853-858, 2005.
N. Nisan. Lower bounds for non-commutative computation. In STOC, pages 410-418, 1991.
C. H. Papadimitriou and M. Sipser. Communication complexity. In STOC, pages 196-200, 1982.
R. Raz and A. Yehudayoff. Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors. J. Comput. Syst. Sci., 77(1):167-190, 2011.
E. Shamir and M. Snir. Lower bounds on the number of multiplications and the number of additions in monotone computations. Technical Report RC-6757, IBM, 1977.
A. Shpilka and A. Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Found. Trends Theor. Comput. Sci., 5:207-388, 2010. URL: 10.1561/0400000039, URL: http://dx.doi.org/10.1561/0400000039.
http://dx.doi.org/10.1561/0400000039
S. Tavenas. Improved bounds for reduction to depth 4 and depth 3. Information and Computation, 240:2-11, 2015.
L. G. Valiant, S. Skyum, S. Berkowitz, and C. Rackoff. Fast parallel computation of polynomials using few processors. SIAM J. on Computing, 12(4):641-644, 1983.
A. C. Yao. Some complexity questions related to distributive computing (preliminary report). In STOC, pages 209-213, 1979.
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Tolerant Testers of Image Properties
We initiate a systematic study of tolerant testers of image properties or, equivalently, algorithms that approximate the distance from a given image to the desired property (that is, the smallest fraction of pixels that need to change in the image to ensure that the image satisfies the desired property). Image processing is a particularly compelling area of applications for sublinear-time algorithms and, specifically, property testing. However, for testing algorithms to reach their full potential in image processing, they have to be tolerant, which allows them to be resilient to noise. Prior to this work, only one tolerant testing algorithm for an image property (image partitioning) has been published.
We design efficient approximation algorithms for the following fundamental questions: What fraction of pixels have to be changed in an image so that it becomes a half-plane? a representation of a convex object? a representation of a connected object? More precisely, our algorithms approximate the distance to three basic properties (being a half-plane, convexity, and connectedness) within a small additive error epsilon, after reading a number of pixels polynomial in 1/epsilon and independent of the size of the image. The running time of the testers for half-plane and convexity is also polynomial in 1/epsilon. Tolerant testers for these three properties were not investigated previously. For convexity and connectedness, even the existence of distance approximation algorithms with query complexity independent of the input size is not implied by previous work. (It does not follow from the VC-dimension bounds, since VC dimension of convexity and connectedness, even in two dimensions, depends on the input size. It also does not follow from the existence of non-tolerant testers.)
Our algorithms require very simple access to the input: uniform random samples for the half-plane property and convexity, and samples from uniformly random blocks for connectedness. However, the analysis of the algorithms, especially for convexity, requires many geometric and combinatorial insights. For example, in the analysis of the algorithm for convexity, we define a set of reference polygons P_{epsilon} such that (1) every convex image has a nearby polygon in P_{epsilon} and (2) one can use dynamic programming to quickly compute the smallest empirical distance to a polygon in P_{epsilon}. This construction might be of independent interest.
Computational geometry
convexity
half-plane
connectedness
propertytesting
tolerant property testing
90:1-90:14
Regular Paper
Piotr
Berman
Piotr Berman
Meiram
Murzabulatov
Meiram Murzabulatov
Sofya
Raskhodnikova
Sofya Raskhodnikova
10.4230/LIPIcs.ICALP.2016.90
Imre Barany. Extremal problems for convex lattice polytopes: a survey. Contemporary Mathematics, 2000.
Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. Testing convexity of figures under the uniform distribution. In SoCG, 2016.
Piotr Berman, Sofya Raskhodnikova, and Grigory Yaroslavtsev. L_p-testing. In STOC, pages 164-173, 2014. URL: http://dx.doi.org/10.1145/2591796.2591887.
http://dx.doi.org/10.1145/2591796.2591887
Avrim Blum. Machine learning theory. Lecture notes. URL: http://www.cs.cmu.edu/~avrim/ML12/lect0201.pdf.
http://www.cs.cmu.edu/~avrim/ML12/lect0201.pdf
Andrea Campagna, Alan Guo, and Ronitt Rubinfeld. Local reconstructors and tolerant testers for connectivity and diameter. In APPROX-RANDOM, pages 411-424, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40328-6_29.
http://dx.doi.org/10.1007/978-3-642-40328-6_29
Artur Czumaj and Christian Sohler. Property testing with geometric queries. In Algorithms - ESA 2001, 9th Annual European Symposium, Aarhus, Denmark, August 28-31, 2001, Proceedings, pages 266-277, 2001. URL: http://dx.doi.org/10.1007/3-540-44676-1_22.
http://dx.doi.org/10.1007/3-540-44676-1_22
Artur Czumaj, Christian Sohler, and Martin Ziegler. Property testing in computational geometry. In Algorithms - ESA 2000, 8th Annual European Symposium, Saarbrücken, Germany, September 5-8, 2000, Proceedings, pages 155-166, 2000. URL: http://dx.doi.org/10.1007/3-540-45253-2_15.
http://dx.doi.org/10.1007/3-540-45253-2_15
Andrzej Ehrenfeucht, David Haussler, Michael J. Kearns, and Leslie G. Valiant. A general lower bound on the number of examples needed for learning. Inf. Comput., 82(3):247-261, 1989.
Eldar Fischer and Lance Fortnow. Tolerant versus intolerant testing for boolean properties. Theory of Computing, 2(1):173-183, 2006. URL: http://dx.doi.org/10.4086/toc.2006.v002a009.
http://dx.doi.org/10.4086/toc.2006.v002a009
Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, 1998. URL: http://dx.doi.org/10.1145/285055.285060.
http://dx.doi.org/10.1145/285055.285060
Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. Algorithmica, 32(2):302-343, 2002. URL: http://dx.doi.org/10.1007/s00453-001-0078-7.
http://dx.doi.org/10.1007/s00453-001-0078-7
David Haussler. Decision theoretic generalizations of the PAC model for neural net and other learning applications. Inf. Comput., 100(1):78-150, 1992. URL: http://dx.doi.org/10.1016/0890-5401(92)90010-D.
http://dx.doi.org/10.1016/0890-5401(92)90010-D
Adam Tauman Kalai, Adam R. Klivans, Yishay Mansour, and Rocco A. Servedio. Agnostically learning halfspaces. SIAM J. Comput., 37(6):1777-1805, 2008.
Michael J. Kearns, Robert E. Schapire, and Linda Sellie. Toward efficient agnostic learning. Machine Learning, 17(2-3):115-141, 1994. URL: http://dx.doi.org/10.1007/BF00993468.
http://dx.doi.org/10.1007/BF00993468
Igor Kleiner, Daniel Keren, Ilan Newman, and Oren Ben-Zwi. Applying property testing to an image partitioning problem. IEEE Trans. Pattern Anal. Mach. Intell., 33(2):256-265, 2011. URL: http://dx.doi.org/10.1109/TPAMI.2010.165.
http://dx.doi.org/10.1109/TPAMI.2010.165
Simon Korman, Daniel Reichman, and Gilad Tsur. Tight approximation of image matching. CoRR, abs/1111.1713, 2011. URL: http://arxiv.org/abs/1111.1713.
http://arxiv.org/abs/1111.1713
Simon Korman, Daniel Reichman, Gilad Tsur, and Shai Avidan. Fast-match: Fast affine template matching. In CVPR, pages 2331-2338, 2013. URL: http://dx.doi.org/10.1109/CVPR.2013.302.
http://dx.doi.org/10.1109/CVPR.2013.302
Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. J. Comput. Syst. Sci., 72(6):1012-1042, 2006. URL: http://dx.doi.org/10.1016/j.jcss.2006.03.002.
http://dx.doi.org/10.1016/j.jcss.2006.03.002
Sofya Raskhodnikova. Approximate testing of visual properties. In RANDOM-APPROX, pages 370-381, 2003. URL: http://dx.doi.org/10.1007/978-3-540-45198-3_31.
http://dx.doi.org/10.1007/978-3-540-45198-3_31
Dana Ron and Gilad Tsur. Testing properties of sparse images. ACM Trans. Algorithms, 10(4):17:1-17:52, 2014. URL: http://dx.doi.org/10.1145/2635806.
http://dx.doi.org/10.1145/2635806
Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252-271, 1996.
Bernd Schmeltz. Learning convex sets under uniform distribution. In Data Structures and Efficient Algorithms, Final Report on the DFG Special Joint Initiative, pages 204-213, 1992.
Leslie G. Valiant. A theory of the learnable. Commun. ACM, 27(11):1134-1142, 1984. URL: http://dx.doi.org/10.1145/1968.1972.
http://dx.doi.org/10.1145/1968.1972
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Erasure-Resilient Property Testing
Property testers form an important class of sublinear algorithms. In the standard property testing model, an algorithm accesses the input function f:D -> R via an oracle. With very few exceptions, all property testers studied in this model rely on the oracle to provide function values at all queried domain points. However, in many realistic situations, the oracle may be unable to reveal the function values at some domain points due to privacy concerns, or when some of the values get erased by mistake or by an adversary. The testers do not learn anything useful about the property by querying those erased points. Moreover, the knowledge of a tester may enable an adversary to erase some of the values so as to increase the query complexity of the tester arbitrarily or, in some cases, make the tester entirely useless.
In this work, we initiate a study of property testers that are resilient to the presence of adversarially erased function values. An alpha-erasure-resilient epsilon-tester is given parameters alpha, epsilon in (0,1), along with oracle access to a function f such that at most an alpha fraction of function values have been erased. The tester does not know whether a value is erased until it queries the corresponding domain point. The tester has to accept with high probability if there is a way to assign values to the erased points such that the resulting function satisfies the desired property P. It has to reject with high probability if, for every assignment of values to the erased points, the resulting function has to be changed in at least an epsilon-fraction of the non-erased domain points to satisfy P.
We design erasure-resilient property testers for a large class of properties. For some properties, it is possible to obtain erasure-resilient testers by simply using standard testers as a black box. However, there are more challenging properties for which all known testers rely on querying a specific point. If this point is erased, all these testers break. We give efficient erasure-resilient testers for several important classes of such properties of functions including monotonicity, the Lipschitz property, and convexity. Finally, we show a separation between the standard testing and erasure-resilient testing. Specifically, we describe a property that can be epsilon-tested with O(1/epsilon) queries in the standard model, whereas testing it in the erasure-resilient model requires number of queries polynomial in the input size.
Randomized algorithms
property testing
error correction
monotoneand Lipschitz functions
91:1-91:15
Regular Paper
Kashyap
Dixit
Kashyap Dixit
Sofya
Raskhodnikova
Sofya Raskhodnikova
Abhradeep
Thakurta
Abhradeep Thakurta
Nithin
Varma
Nithin Varma
10.4230/LIPIcs.ICALP.2016.91
N. Ailon and B. Chazelle. Information theory in property testing and monotonicity testing in higher dimension. Inform. and Comput., 204(11):1704-1717, 2006.
N. Ailon, B. Chazelle, S. Comandur, and D. Liu. Estimating the distance to a monotone function. Random Structures Algorithms, 31(3):371-383, 2007.
Pranjal Awasthi, Madhav Jha, Marco Molinaro, and Sofya Raskhodnikova. Testing Lipschitz functions on hypergrid domains. Algorithmica, 74(3):1055-1081, 2016. URL: http://dx.doi.org/10.1007/s00453-015-9984-y.
http://dx.doi.org/10.1007/s00453-015-9984-y
Maria-Florina Balcan, Eric Blais, Avrim Blum, and Liu Yang. Active property testing. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 21-30, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.64.
http://dx.doi.org/10.1109/FOCS.2012.64
T. Batu, R. Rubinfeld, and P. White. Fast approximate PCPs for multidimensional bin-packing problems. Inform. and Comput., 196(1):42-56, 2005.
Tugkan Batu, Lance Fortnow, Ronitt Rubinfeld, Warren D. Smith, and Patrick White. Testing closeness of discrete distributions. J. ACM, 60(1):4, 2013. URL: http://dx.doi.org/10.1145/2432622.2432626.
http://dx.doi.org/10.1145/2432622.2432626
Piotr Berman, Meiram Murzabulatov, and Sofya Raskhodnikova. Testing Convexity of Figures Under the Uniform Distribution, 2015. To appear in SoCG 2016.
Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, and David P. Woodruff. Transitive-closure spanners. SIAM J. Comput., 41(6):1380-1425, 2012.
E. Blais, J. Brody, and K. Matulef. Property testing lower bounds via communication complexity. Comp. Complexity, 21(2):311-358, 2012.
Eric Blais, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Lower bounds for testing properties of functions over hypergrid domains. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 309-320, 2014.
J. Briët, S. Chakraborty, D. García-Soriano, and A. Matsliah. Monotonicity testing and shortest-path routing on the cube. Combinatorica, 32(1):35-53, 2012.
Deeparnab Chakrabarty, Kashyap Dixit, Madhav Jha, and C. Seshadhri. Property testing on product distributions: Optimal testers for bounded derivative properties. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1809-1828, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.121.
http://dx.doi.org/10.1137/1.9781611973730.121
Deeparnab Chakrabarty and C. Seshadhri. Optimal bounds for monotonicity and lipschitz testing over hypercubes and hypergrids. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 419-428, 2013. URL: http://dx.doi.org/10.1145/2488608.2488661.
http://dx.doi.org/10.1145/2488608.2488661
Deeparnab Chakrabarty and C. Seshadhri. An optimal lower bound for monotonicity testing over hypergrids. Theory of Computing, 10:453-464, 2014. URL: http://dx.doi.org/10.4086/toc.2014.v010a017.
http://dx.doi.org/10.4086/toc.2014.v010a017
Kashyap Dixit, Madhav Jha, Sofya Raskhodnikova, and Abhradeep Thakurta. Testing the lipschitz property over product distributions with applications to data privacy. In TCC, pages 418-436, 2013. URL: http://dx.doi.org/10.1007/978-3-642-36594-2_24.
http://dx.doi.org/10.1007/978-3-642-36594-2_24
Yevgeniy Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, and Alex Samorodnitsky. Improved testing algorithms for monotonicity. In Randomization, Approximation, and Combinatorial Algorithms and Techniques, Third International Workshop on Randomization and Approximation Techniques in Computer Science, and Second International Workshop on Approximation Algorithms for Combinatorial Optimization Problems RANDOM-APPROX'99, Berkeley, CA, USA, August 8-11, 1999, Proceedings, pages 97-108, 1999.
Funda Ergün, Sampath Kannan, Ravi Kumar, Ronitt Rubinfeld, and Mahesh Viswanathan. Spot-checkers. J. Comput. Syst. Sci., 60(3):717-751, 2000. URL: http://dx.doi.org/10.1006/jcss.1999.1692.
http://dx.doi.org/10.1006/jcss.1999.1692
Shahar Fattal and Dana Ron. Approximating the distance to convexity. Unpublished manuscript. Uploaded at URL: http://www.eng.tau.ac.il/~danar/Public-pdf/app-conv.pdf.
http://www.eng.tau.ac.il/~danar/Public-pdf/app-conv.pdf
Shahar Fattal and Dana Ron. Approximating the distance to monotonicity in high dimensions. ACM Transactions on Algorithms, 6(3), 2010. URL: http://dx.doi.org/10.1145/1798596.1798605.
http://dx.doi.org/10.1145/1798596.1798605
E. Fischer. On the strength of comparisons in property testing. Inform. and Comput., 189(1):107-116, 2004.
Eldar Fischer and Lance Fortnow. Tolerant versus intolerant testing for boolean properties. Theory of Computing, 2(9):173-183, 2006. URL: http://dx.doi.org/10.4086/toc.2006.v002a009.
http://dx.doi.org/10.4086/toc.2006.v002a009
Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. Monotonicity testing over general poset domains. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, STOC'02, pages 474-483, New York, NY, USA, 2002. ACM. URL: http://dx.doi.org/10.1145/509907.509977.
http://dx.doi.org/10.1145/509907.509977
O. Goldreich, S. Goldwasser, E. Lehman, D. Ron, and A. Samorodnitsky. Testing monotonicity. Combinatorica, 20:301-337, 2000.
O. Goldreich, S. Goldwasser, and D. Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, 1998.
Oded Goldreich and Dana Ron. On proximity-oblivious testing. SIAM J. Comput., 40(2):534-566, 2011. URL: http://dx.doi.org/10.1137/100789646.
http://dx.doi.org/10.1137/100789646
Oded Goldreich and Dana Ron. On sample-based testers. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 337-345, 2015. URL: http://dx.doi.org/10.1145/2688073.2688080.
http://dx.doi.org/10.1145/2688073.2688080
Oded Goldreich and Igor Shinkar. Two-sided error proximity oblivious testing. Random Struct. Algorithms, 48(2):341-383, 2016. URL: http://dx.doi.org/10.1002/rsa.20582.
http://dx.doi.org/10.1002/rsa.20582
S. Halevy and E. Kushilevitz. Testing monotonicity over graph products. Random Structures Algorithms, 33(1):44-67, 2008.
Madhav Jha and Sofya Raskhodnikova. Testing and reconstruction of Lipschitz functions with applications to data privacy. SIAM J. Comput., 42(2):700-731, 2013.
Michael J. Kearns and Dana Ron. Testing problems with sublearning sample complexity. J. Comput. Syst. Sci., 61(3):428-456, 2000. URL: http://dx.doi.org/10.1006/jcss.1999.1656.
http://dx.doi.org/10.1006/jcss.1999.1656
E. Lehman and D. Ron. On disjoint chains of subsets. J. Combin. Theory Ser. A, 94(2):399-404, 2001.
M. Parnas, D. Ron, and R. Rubinfeld. Tolerant property testing and distance approximation. J. Comput. System Sci., 6(72):1012-1042, 2006.
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http://dx.doi.org/10.1137/S0097539702414026
Bruce A. Reed. The height of a random binary search tree. J. ACM, 50(3):306-332, 2003. URL: http://dx.doi.org/10.1145/765568.765571.
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http://dx.doi.org/10.1109/FOCS.2010.51
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Towards Tight Lower Bounds for Range Reporting on the RAM
In the orthogonal range reporting problem, we are to preprocess a set of n points with integer coordinates on a UxU grid. The goal is to support reporting all k points inside an axis-aligned query rectangle. This is one of the most fundamental data structure problems in databases and computational geometry. Despite the importance of the problem its complexity remains unresolved in the word-RAM.
On the upper bound side, three best tradeoffs exist, all derived by reducing range reporting to a ball-inheritance problem. Ball-inheritance is a problem that essentially encapsulates all previous attempts at solving range reporting in the word-RAM. In this paper we make progress towards closing the gap between the upper and lower bounds for range reporting by proving cell probe lower bounds for ball-inheritance. Our lower bounds are tight for a large range of parameters, excluding any further progress for range reporting using the ball-inheritance reduction.
Data Structures
Lower Bounds
Cell Probe Model
Range Reporting
92:1-92:12
Regular Paper
Allan
Grønlund
Allan Grønlund
Kasper Green
Larsen
Kasper Green Larsen
10.4230/LIPIcs.ICALP.2016.92
Stephen Alstrup, Gerth Stølting Brodal, and Theis Rauhe. New data structures for orthogonal range searching. In Proc. 41st IEEE Symposium on Foundations of Computer Science, pages 198-207, 2000.
Lars Arge, Vasilis Samoladas, and Jeffrey Scott Vitter. On two-dimensional indexability and optimal range search indexing. In Proc. 18th ACM Symposium on Principles of Database Systems, pages 346-357, 1999.
Jon Louis Bentley. Multidimensional binary search trees used for associative searching. Communications of the ACM, 18(9):509-517, 1975.
Gerth Stølting Brodal and Kasper Green Larsen. Optimal planar orthogonal skyline counting queries. In Proc. 14th Scandinavian Workshop on Algorithms Theory, pages 110-121, 2014.
Timothy M. Chan, Kasper Larsen, and Mihai Pǎtraşcu. Orthogonal range searching on the ram, revisited. In Proc. 27th ACM Symposium on Computational Geometry, pages 354-363, 2011. See also arXiv:1011.5200.
Timothy M. Chan and Bryan T. Wilkinson. Adaptive and approximate orthogonal range counting. In Proc. 24th ACM/SIAM Symposium on Discrete Algorithms, pages 241-251, 2013.
Bernard Chazelle. Filtering search: a new approach to query answering. SIAM Journal on Computing, 15(3):703-724, 1986. URL: http://dx.doi.org/10.1137/0215051.
http://dx.doi.org/10.1137/0215051
Bernard Chazelle. Lower bounds for orthogonal range searching: I. the reporting case. Journal of the ACM, 37(2):200-212, 1990.
Johannes Fischer. Optimal succinctness for range minimum queries. In Proc. 9th Latin American Theoretical Informatics Symposium, pages 158-169, 2010.
Kasper Green Larsen. Higher cell probe lower bounds for evaluating polynomials. In Proc. 53rd IEEE Symposium on Foundations of Computer Science, pages 293-301, 2012.
Yakov Nekrich. Orthogonal range searching in linear and almost-linear space. Computational Geometry: Theory and Applications, 42:342-351, 2009.
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Data Structure Lower Bounds for Document Indexing Problems
We study data structure problems related to document indexing and pattern matching queries and our main contribution is to show that the pointer machine model of computation can be extremely useful in proving high and unconditional lower bounds that cannot be obtained in any other known model of computation with the current techniques. Often our lower bounds match the known space-query time trade-off curve and in fact for all the problems considered, there is a very good and reasonable match between our lower bounds and the known upper bounds, at least for some choice of input parameters.
The problems that we consider are set intersection queries (both the reporting variant and the semi-group counting variant), indexing a set of documents for two-pattern queries, or forbidden-pattern queries, or queries with wild-cards, and indexing an input set of gapped-patterns (or two-patterns) to find those matching a document given at the query time.
Data Structure Lower Bounds
Pointer Machine
Set Intersection
Pattern Matching
93:1-93:15
Regular Paper
Peyman
Afshani
Peyman Afshani
Jesper Sindahl
Nielsen
Jesper Sindahl Nielsen
10.4230/LIPIcs.ICALP.2016.93
Amir Abboud and Virginia Vassilevska Williams. Popular conjectures imply strong lower bounds for dynamic problems. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 434-443, 2014.
Peyman Afshani. Improved pointer machine and I/O lower bounds for simplex range reporting and related problems. In Symposium on Computational Geometry (SoCG), pages 339-346, 2012.
Peyman Afshani, Lars Arge, and Kasper Dalgaard Larsen. Orthogonal range reporting: query lower bounds, optimal structures in 3-d, and higher-dimensional improvements. In Symposium on Computational Geometry (SoCG), pages 240-246, 2010.
Peyman Afshani, Lars Arge, and Kasper Green Larsen. Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model. In Symposium on Computational Geometry (SoCG), pages 323-332, 2012. URL: http://dx.doi.org/10.1145/2261250.2261299.
http://dx.doi.org/10.1145/2261250.2261299
Peyman Afshani and Jesper Sindahl Nielsen. Data structure lower bounds for document indexing problems. CoRR, abs/1604.06264, 2016. URL: http://arxiv.org/abs/1604.06264.
http://arxiv.org/abs/1604.06264
Amihood Amir, Tsvi Kopelowitz, Avivit Levy, Seth Pettie, Ely Porat, and B. Riva Shalom. Online Dictionary Matching with One Gap. CoRR, abs/1503.07563, 2015. URL: http://arxiv.org/abs/1503.07563.
http://arxiv.org/abs/1503.07563
Amihood Amir, Avivit Levy, Ely Porat, and B. Riva Shalom. Dictionary Matching with One Gap. In Annual Symposium on Combinatorial Pattern Matching (CPM), pages 11-20, 2014.
Philip Bille, Inge Li Gørtz, Hjalte Wedel Vildhøj, and Søren Vind. String indexing for patterns with wildcards. In Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pages 283-294, 2012.
Philip Bille, Anna Pagh, and Rasmus Pagh. Fast evaluation of union-intersection expressions. In Algorithms and Computation, volume 4835 of Lecture Notes in Computer Science, pages 739-750. Springer Berlin Heidelberg, 2007.
Sudip Biswas, Arnab Ganguly, Rahul Shah, and Sharma V. Thankachan. Ranked Document Retrieval with Forbidden Pattern. In Annual Symposium on Combinatorial Pattern Matching (CPM), pages 77-88, 2015.
Timothy M. Chan. Optimal partition trees. In Symposium on Computational Geometry (SoCG), pages 1-10. ACM, 2010.
Timothy M. Chan, Kasper Green Larsen, and Mihai Pǎtraşcu. Orthogonal range searching on the RAM, revisited. In Symposium on Computational Geometry (SoCG), pages 1-10, 2011.
Timothy M. Chan and Moshe Lewenstein. Clustered integer 3SUM via additive combinatorics. In Proceedings of ACM Symposium on Theory of Computing (STOC), pages 31-40, 2015.
Bernard Chazelle. Lower Bounds for Orthogonal Range Searching: I. The Reporting Case. J. ACM, 37(2):200-212, 1990.
Bernard Chazelle. Lower Bounds for Orthogonal Range Searching II. The Arithmetic Model. J. ACM, 37(3):439-463, 1990.
Bernard Chazelle and Burton Rosenberg. Simplex Range Reporting on a Pointer Machine. Comput. Geom., 5:237-247, 1995.
Bernard Chazelle, Micha Sharir, and Emo Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. Algorithmica, 8:407-429, December 1992.
Hagai Cohen and Ely Porat. Fast set intersection and two-patterns matching. Theor. Comput. Sci., 411(40-42):3795-3800, 2010.
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Richard Cole, Lee-Ad Gottlieb, and Moshe Lewenstein. Dictionary matching and indexing with errors and don't cares. In Proceedings of ACM Symposium on Theory of Computing (STOC), pages 91-100, 2004.
Paul F. Dietz, Kurt Mehlhorn, Rajeev Raman, and Christian Uhrig. Lower bounds for set intersection queries. Algorithmica, 14(2):154-168, 1995.
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Johannes Fischer, Travis Gagie, Tsvi Kopelowitz, Moshe Lewenstein, Veli Mäkinen, Leena Salmela, and Niko Välimäki. Forbidden patterns. In LATIN 2012: Theoretical Informatics - 10th Latin American Symposium, Arequipa, Peru, April 16-20, 2012. Proceedings, pages 327-337, 2012.
Wing-Kai Hon, Rahul Shah, Sharma V. Thankachan, and Jeffrey Scott Vitter. String Retrieval for Multi-pattern Queries. In String Processing and Information Retrieval - 17th International Symposium, SPIRE 2010, Los Cabos, Mexico, October 11-13, 2010. Proceedings, pages 55-66, 2010.
Wing-Kai Hon, Rahul Shah, Sharma V. Thankachan, and Jeffrey Scott Vitter. Document Listing for Queries with Excluded Pattern. In Combinatorial Pattern Matching - 23rd Annual Symposium, CPM 2012, Helsinki, Finland, July 3-5, 2012. Proceedings, pages 185-195, 2012.
Casper Kejlberg-Rasmussen, Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Word-packing algorithms for dynamic connectivity and dynamic sets. http://arxiv.org/abs/1407.6755, 2014.
Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3SUM conjecture. http://arxiv.org/abs/1407.6756, 2016.
Tsvi Kopelowitz, Seth Pettie, and Ely Porat. Higher lower bounds from the 3SUM conjecture. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1272-1287, 2016.
Kasper Green Larsen, J. Ian Munro, Jesper Sindahl Nielsen, and Sharma V. Thankachan. On Hardness of Several String Indexing Problems. In Combinatorial Pattern Matching - 25th Annual Symposium, CPM 2014, Moscow, Russia, June 16-18, 2014. Proceedings, pages 242-251, 2014.
Moshe Lewenstein, J. Ian Munro, Venkatesh Raman, and Sharma V. Thankachan. Less space: Indexing for queries with wildcards. Theoretical Computer Science, 557:120-127, 2014.
Moshe Lewenstein, Yakov Nekrich, and Jeffrey Scott Vitter. Space-efficient string indexing for wildcard pattern matching. CoRR, abs/1401.0625, 2014. URL: http://arxiv.org/abs/1401.0625.
http://arxiv.org/abs/1401.0625
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Proof Complexity Modulo the Polynomial Hierarchy: Understanding Alternation as a Source of Hardness
We present and study a framework in which one can present alternation-based lower bounds on proof length in proof systems for quantified Boolean formulas. A key notion in this framework is that of proof system ensemble, which is (essentially) a sequence of proof systems where, for each, proof checking can be performed in the polynomial hierarchy. We introduce a proof system ensemble called relaxing QU-res which is based on the established proof system QU-resolution.
Our main results include an exponential separation of the tree-like and general versions of relaxing QU-res, and an exponential lower bound for relaxing QU-res; these are analogs of classical results in propositional proof complexity.
proof complexity
polynomial hierarchy
quantified propositional logic
94:1-94:14
Regular Paper
Hubie
Chen
Hubie Chen
10.4230/LIPIcs.ICALP.2016.94
Albert Atserias, Johannes Klaus Fichte, and Marc Thurley. Clause-learning algorithms with many restarts and bounded-width resolution. J. Artif. Intell. Res. (JAIR), 40:353-373, 2011.
Valeriy Balabanov and Jie-Hong R. Jiang. Resolution proofs and skolem functions in QBF evaluation and applications. In Computer Aided Verification - 23rd International Conference, CAV 2011, Snowbird, UT, USA, July 14-20, 2011. Proceedings, pages 149-164, 2011.
Valeriy Balabanov, Magdalena Widl, and Jie-Hong R. Jiang. QBF resolution systems and their proof complexities. In Theory and Applications of Satisfiability Testing - SAT 2014 - 17th International Conference, Held as Part of the Vienna Summer of Logic, VSL 2014, Vienna, Austria, July 14-17, 2014. Proceedings, pages 154-169, 2014.
Paul Beame, Henry A. Kautz, and Ashish Sabharwal. Towards understanding and harnessing the potential of clause learning. J. Artif. Intell. Res. (JAIR), 22:319-351, 2004.
Paul Beame and Toniann Pitassi. Propositional proof complexity: Past, present and future. Bulletin of the EATCS, 65:66-89, 1998.
Eli Ben-Sasson, Russell Impagliazzo, and Avi Wigderson. Near optimal separation of tree-like and general resolution. Combinatorica, 24(4):585-603, 2004.
Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow - resolution made simple. J. ACM, 48(2):149-169, 2001.
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Olaf Beyersdorff, Leroy Chew, and Karteek Sreenivasaiah. A game characterisation of tree-like q-resolution size. Electronic Colloquium on Computational Complexity (ECCC), 2014.
Maria Luisa Bonet, Juan Luis Esteban, Nicola Galesi, and Jan Johannsen. On the relative complexity of resolution refinements and cutting planes proof systems. SIAM J. Comput., 30(5):1462-1484, 2000.
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Hubie Chen. Beyond q-resolution and prenex form: A proof system for quantified constraint satisfaction. CoRR, abs/1403.0222, 2014.
Hubie Chen. Proof complexity modulo the polynomial hierarchy: Understanding alternation as a source of hardness. CoRR, abs/1410.5369, 2014.
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Horst Samulowitz and Fahiem Bacchus. Using SAT in QBF. In Principles and Practice of Constraint Programming - CP 2005, 11th International Conference, CP 2005, Sitges, Spain, October 1-5, 2005, Proceedings, pages 578-592, 2005.
N. Segerlind. The complexity of propositional proofs. Bull. Symbolic Logic, 13:417-626, 2007.
Yinlei Yu and Sharad Malik. Validating the result of a quantified boolean formula (QBF) solver: theory and practice. In Proceedings of the 2005 Conference on Asia South Pacific Design Automation, ASP-DAC 2005, Shanghai, China, January 18-21, 2005, pages 1047-1051, 2005.
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Past, Present, and Infinite Future
I was supposed to deliver one of the speeches at Wolfgang Thomas's retirement ceremony. Wolfgang had called me on the phone earlier and posed some questions about temporal logic, but I hadn't had good answers at the time. What I decided to do at the ceremony was to take up the conversation again and show how it could have evolved if only I had put more effort into answering his questions. Here is the imaginary conversation with Wolfgang.
The contributions are (1) the first direct translation from counter-free omega-automata into future temporal formulas, (2) a definition of bimachines for omega-words, (3) a translation from arbitrary temporal formulas (including both, future and past operators) into counter-free omega-bimachines, and (4) an automata-based proof of separation: every arbitrary temporal formula is equivalent to a boolean combination of pure future, present, and pure past formulas when interpreted in omega-words.
linear-time temporal logic
separation
backward deterministic omega-automata
counter freeness
95:1-95:14
Regular Paper
Thomas
Wilke
Thomas Wilke
10.4230/LIPIcs.ICALP.2016.95
Olivier Carton. Personal communication.
Olivier Carton. Right-sequential functions on infinite words. In Farid M. Ablayev and Ernst W. Mayr, editors, Computer Science - Theory and Applications, 5th International Computer Science Symposium in Russia, CSR 2010, Kazan, Russia, June 16-20, 2010. Proceedings, volume 6072 of Lecture Notes in Computer Science, pages 96-106. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-13182-0_9.
http://dx.doi.org/10.1007/978-3-642-13182-0_9
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Volker Diekert and Paul Gastin. First-order definable languages. In Jörg Flum, Erich Grädel, and Thomas Wilke, editors, Logic and Automata: History and Perspectives [in Honor of Wolfgang Thomas], volume 2 of Texts in Logic and Games, pages 261-306. Amsterdam University Press, 2008.
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Marcel Paul Schützenberger. A remark on finite transducers. Information and Control, 4(2-3):185-196, 1961. URL: http://dx.doi.org/10.1016/S0019-9958(61)80006-5.
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Marcel Paul Schützenberger. On finite monoids having only trivial subgroups. Information and Control, 8(2):190-194, 1965. URL: http://dx.doi.org/10.1016/S0019-9958(65)90108-7.
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Michael Sipser. Introduction to the Theory of Computation. Cengage Learning, Boston, Mass., 3rd edition, 2013.
Wolfgang Thomas. Star-free regular sets of omega-sequences. Information and Control, 42(2):148-156, 1979. URL: http://dx.doi.org/10.1016/S0019-9958(79)90629-6.
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Moshe Y. Vardi and Pierre Wolper. Reasoning about infinite computations. Inf. Comput., 115(1):1-37, 1994. URL: http://dx.doi.org/10.1006/inco.1994.1092.
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Thomas Wilke. Classifying discrete temporal properties. In Christoph Meinel and Sophie Tison, editors, STACS 99, 16th Annual Symposium on Theoretical Aspects of Computer Science, Trier, Germany, March 4-6, 1999, Proceedings, volume 1563 of Lecture Notes in Computer Science, pages 32-46. Springer, 1999. URL: http://dx.doi.org/10.1007/3-540-49116-3_3.
http://dx.doi.org/10.1007/3-540-49116-3_3
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Thin MSO with a Probabilistic Path Quantifier
This paper is about a variant of MSO on infinite trees where:
- there is a quantifier "zero probability of choosing a path pi in 2^{omega} which makes omega(pi) true";
- the monadic quantifiers range over sets with countable topological closure.
We introduce an automaton model, and show that it captures the logic.
Automata
mso
infinite trees
probabilistic temporal logics
96:1-96:13
Regular Paper
Mikolaj
Bojanczyk
Mikolaj Bojanczyk
10.4230/LIPIcs.ICALP.2016.96
Christel Baier, Marcus Größer, and Nathalie Bertrand. Probabilistic ω-automata. J. ACM, 59(1):1, 2012. URL: http://dx.doi.org/10.1145/2108242.2108243.
http://dx.doi.org/10.1145/2108242.2108243
Vince Bárány, Łukasz Kaiser, and Alexander Rabinovich. Cardinality quantifiers in MLO over trees. In Proc. of CSL, 2009.
Mikolaj Bojanczyk. U. ACM SIGLOG News, 2(4):2-15, 2015.
Tomás; Brázdil, Vojtech Forejt, Jan Kretínský, and Antonín Kucera. The satisfiability problem for probabilistic CTL. In Proc. of LICS, pages 391-402, 2008.
Julius R. Büchi. On a decision method in restricted second-order arithmetic. In Proc. 1960 Int. Congr. for Logic, Methodology and Philosophy of Science, pages 1-11, 1962.
Arnaud Carayol, Axel Haddad, and Olivier Serre. Randomization in automata on infinite trees. ACM Trans. Comput. Log., 15(3):24:1-24:33, 2014. URL: http://dx.doi.org/10.1145/2629336.
http://dx.doi.org/10.1145/2629336
Thomas Colcombet. Regular cost functions, part I: logic and algebra over words. Logical Methods in Computer Science, 9(3), 2013. URL: http://dx.doi.org/10.2168/LMCS-9(3:3)2013.
http://dx.doi.org/10.2168/LMCS-9(3:3)2013
Daniel Lehmann and Saharon Shelah. Reasoning with time and chance. Information and Control, 53(3):165-1983, 1982.
Henryk Michalewski and Matteo Mio. Baire category quantifier in monadic second order logic. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part II, pages 362-374, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47666-6_29.
http://dx.doi.org/10.1007/978-3-662-47666-6_29
Henryk Michalewski and Matteo Mio. Measure quantifier in monadic second order logic. In Logical Foundations of Computer Science - International Symposium, LFCS 2016, Deerfield Beach, FL, USA, January 4-7, 2016. Proceedings, pages 267-282, 2016. URL: http://dx.doi.org/10.1007/978-3-319-27683-0_19.
http://dx.doi.org/10.1007/978-3-319-27683-0_19
Michael O. Rabin. Decidability of second-order theories and automata on infinite trees. Transactions of American Mathematical Society, 141:1-35, 1969.
Alexander Rabinovich. On decidability of monadic logic of order over the naturals extended by monadic predicates. Inf. Comput., 205(6):870-889, 2007. URL: http://dx.doi.org/10.1016/j.ic.2006.12.004.
http://dx.doi.org/10.1016/j.ic.2006.12.004
Sergiu Hart Micha Sharir. Probabilistic propositional temporal logics. Information and Control, 70(2-3):97-155, 1986.
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Deciding Piecewise Testable Separability for Regular Tree Languages
The piecewise testable separability problem asks, given two input languages, whether there exists a piecewise testable language that contains the first input language and is disjoint from the second. We prove a general characterisation of piecewise testable separability on languages in a well-quasiorder, in terms of ideals of the ordering. This subsumes the known characterisations in the case of finite words. In the case of finite ranked trees ordered by homeomorphic embedding, we show using effective representations for tree ideals that it entails the decidability of piecewise testable separability when the input languages are regular. A final byproduct is a new proof of the decidability of whether an input regular language of ranked trees is piecewise testable, which was first shown in the unranked case by Bojanczyk, Segoufin, and Straubing [Log. Meth. in Comput. Sci., 8(3:26), 2012].
Well-quasi-order
ideal
tree languages
first-order logic
97:1-97:15
Regular Paper
Jean
Goubault-Larrecq
Jean Goubault-Larrecq
Sylvain
Schmitz
Sylvain Schmitz
10.4230/LIPIcs.ICALP.2016.97
Parosh Aziz Abdulla, Aurore Collomb-Annichini, Ahmed Bouajjani, and Bengt Jonsson. Using forward reachability analysis for verification of lossy channel systems. Formal Methods in System Design, 25(1):39-65, 2004. URL: http://dx.doi.org/10.1023/B:FORM.0000033962.51898.1a.
http://dx.doi.org/10.1023/B:FORM.0000033962.51898.1a
Jorge Almeida. Some algorithmic problems for pseudovarieties. Publicationes Mathematicae Debrecen, 54(suppl.):531-552, 1999.
Jorge Almeida and Marc Zeitoun. The pseudovariety 𝖩 is hyperdecidable. RAIRO Theoretical Informatics and Applications, 31:457-482, 1997.
Kazuyuki Asada and Naoki Kobayashi. On word and frontier languages of unsafe higher-order grammars. In ICALP 2016, Leibniz International Proceedings in Informatics, 2016. To appear. URL: https://arxiv.org/abs/1604.01595.
https://arxiv.org/abs/1604.01595
Leo Bachmair and David A. Plaisted. Termination orderings for associative-commutative rewriting systems. Journal of Logic and Computation, 1(4):329-349, 1985. URL: http://dx.doi.org/10.1016/S0747-7171(85)80019-5.
http://dx.doi.org/10.1016/S0747-7171(85)80019-5
Mikołaj Bojańczyk, Luc Segoufin, and Howard Straubing. Piecewise testable tree languages. Logical Methods in Computer Science, 8(3), 2012. URL: http://dx.doi.org/10.2168/LMCS-8(3:26)2012.
http://dx.doi.org/10.2168/LMCS-8(3:26)2012
Robert Bonnet. On the cardinality of the set of initial intervals of a partially ordered set. In Infinite and finite sets\string: to Paul Erdős on his 60th birthday, Vol. 1, Coll. Math. Soc. János Bolyai, pages 189-198. North-Holland, 1975.
Lorenzo Clemente, Paweł Parys, Sylvain Salvati, and Igor Walukiewicz. The diagonal problem for higher-order recursion schemes is decidable. In LICS 2016. ACM, 2016. To appear.
H. Comon, M. Dauchet, R. Gilleron, C. Löding, F. Jacquemard, D. Lugiez, S. Tison, and M. Tommasi. Tree Automata Techniques and Applications. Inria, 2007. URL: http://tata.gforge.inria.fr/.
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Bruno Courcelle. On constructing obstruction sets of words. Bulletin of the EATCS, 44:178-186, 1991.
Wojciech Czerwiński, Wim Martens, and Tomáš Masopust. Efficient separability of regular languages by subsequences and suffixes. In ICALP 2013, volume 7966 of Lecture Notes in Computer Science, pages 150-161. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-39212-2_16.
http://dx.doi.org/10.1007/978-3-642-39212-2_16
Wojciech Czerwiński, Wim Martens, Lorijn van Rooijen, Marc Zeitoun, and Georg Zetzsche. A characterization for decidable separability by piecewise testable languages. Preprint, 2015. An extended abstract appeared as: W. Czerwiński, W. Martens, L. van Rooijen, and M. Zeitoun. A note on decidable separability by piecewise testable languages. In FCT 2015, volume 9210 of LNCS, pages 173-185. Springer, 2015. URL: http://arxiv.org/abs/1410.1042v2.
http://arxiv.org/abs/1410.1042v2
Alain Finkel and Jean Goubault-Larrecq. Forward analysis for WSTS, part I: Completions. In STACS 2009, volume 3 of Leibniz International Proceedings in Informatics, pages 433-444. LZI, 2009. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2009.1844.
http://dx.doi.org/10.4230/LIPIcs.STACS.2009.1844
Alain Finkel and Jean Goubault-Larrecq. Forward analysis for WSTS, part II: Complete WSTS. Logical Methods in Computer Science, 8(3), 2012. URL: http://dx.doi.org/10.2168/LMCS-8(3:28)2012.
http://dx.doi.org/10.2168/LMCS-8(3:28)2012
Jean Goubault-Larrecq, Prateek Karandikar, K. Narayan Kumar, and Philippe Schnoebelen. The ideal approach to computing closed subsets in well-quasi-orderings. In preparation, 2016. See also an earlier version in: J. Goubault-Larrecq. On a generalization of a result by Valk and Jantzen. Research Report LSV-09-09, LSV, ENS Cachan, 2009. URL: http://www.lsv.ens-cachan.fr/Publis/RAPPORTS_LSV/PDF/rr-lsv-2009-09.pdf.
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Peter Habermehl, Roland Meyer, and Harro Wimmel. The downward-closure of Petri net languages. In ICALP 2010, volume 6199 of Lecture Notes in Computer Science, pages 466-477. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-14162-1_39.
http://dx.doi.org/10.1007/978-3-642-14162-1_39
Matthew Hague, Jonathan Kochems, and C.-H. Luke Ong. Unboundedness and downward closures of higher-order pushdown automata. In POPL 2016, pages 151-163. ACM, 2016. URL: http://dx.doi.org/10.1145/2837614.2837627.
http://dx.doi.org/10.1145/2837614.2837627
Graham Higman. Ordering by divisibility in abstract algebras. Proceedings of the London Mathematical Society, 3(2):326-336, 1952. URL: http://dx.doi.org/10.1112/plms/s3-2.1.326.
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Pierre Jullien. Contribution à l'étude des types d'ordres dispersés. Thèse de doctorat, Université de Marseille, 1969.
Joseph B. Kruskal. Well-quasi-ordering, the Tree Theorem, and Vazsonyi’s Conjecture. Transactions of the American Mathematical Society, 95(2):210-225, 1960. URL: http://dx.doi.org/10.2307/1993287.
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Ranko Lazić and Sylvain Schmitz. The ideal view on Rackoff’s coverability technique. In RP 2015, volume 9328 of Lecture Notes in Computer Science, pages 1-13. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-24537-9_8.
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Jérôme Leroux and Sylvain Schmitz. Demystifying reachability in vector addition systems. In LICS 2015, pages 56-67. IEEE Press, 2015. URL: http://dx.doi.org/10.1109/LICS.2015.16.
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Oded Padon, Neil Immerman, Sharon Shoham, Aleksandr Karbyshev, and Mooly Sagiv. Decidability of inferring inductive invariants. In POPL 2016, pages 217-231. ACM, 2016. URL: http://dx.doi.org/10.1145/2837614.2837640.
http://dx.doi.org/10.1145/2837614.2837640
Thomas Place, Lorijn van Rooijen, and Marc Zeitoun. Separating regular languages by piecewise testable and unambiguous languages. In MFCS 2013, volume 8087 of Lecture Notes in Computer Science, pages 729-740. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40313-2_64.
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Thomas Place and Marc Zeitoun. Automata column: The tale of the quantifier alternation hierarchy of first-order logic over words. SIGLOG News, 2(3):4-17, 2015. URL: http://dx.doi.org/10.1145/2815493.2815495.
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Georg Zetzsche. An approach to computing downward closures. In ICALP 2015, volume 9135 of Lecture Notes in Computer Science, pages 440-451. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47666-6_35.
http://dx.doi.org/10.1007/978-3-662-47666-6_35
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Computation Tree Logic for Synchronization Properties
We present a logic that extends CTL (Computation Tree Logic) with operators that express synchronization properties. A property is synchronized in a system if it holds in all paths of a certain length. The new logic is obtained by using the same path quantifiers and temporal operators as in CTL, but allowing a different order of the quantifiers. This small syntactic variation induces a logic that can express non-regular properties for which known extensions of MSO with equality of path length are undecidable. We show that our variant of CTL is decidable and that the model-checking problem is in Delta_3^P = P^{NP^{NP}}, and is hard for the class of problems solvable in polynomial time using a parallel access to an NP oracle. We analogously consider quantifier exchange in extensions of CTL, and we present operators defined using basic operators of CTL* that express the occurrence of infinitely many synchronization points. We show that the model-checking problem remains in Delta_3^P. The distinguishing power of CTL and of our new logic coincide if the Next operator is allowed in the logics, thus the classical bisimulation quotient can be used for state-space reduction before model checking.
Computation Tree Logic
Synchronization
model-checking
complexity
98:1-98:14
Regular Paper
Krishnendu
Chatterjee
Krishnendu Chatterjee
Laurent
Doyen
Laurent Doyen
10.4230/LIPIcs.ICALP.2016.98
R. Alur, T. A. Henzinger, and O. Kupferman. Alternating-time temporal logic. Journal of the ACM, 49:672-713, 2002.
E. Bach and J. Shallit. Algorithmic Number Theory, Vol. 1: Efficient Algorithms. MIT Press, 1996.
A. Blass and Y. Gurevich. Henkin quantifiers and complete problems. Ann. Pure Appl. Logic, 32:1-16, 1986.
M. C. Browne, E. M. Clarke, and O. Grumberg. Characterizing finite Kripke structures in propositional temporal logic. Theor. Comput. Sci., 59:115-131, 1988.
J. Černý. Poznámka k. homogénnym experimentom s konečnými automatmi. In Matematicko-fyzikálny Časopis, volume 14(3), pages 208-216, 1964.
K. Chatterjee and L. Doyen. Computation tree logic for synchronization properties. CoRR, arXiv:1604.06384, 2016.
K. Chatterjee, T. A. Henzinger, and N. Piterman. Strategy logic. Inf. Comput., 208(6):677-693, 2010.
D. Chistikov, P. Martyugin, and M. Shirmohammadi. Synchronizing automata over nested words. In Proc. of FOSSACS: Foundations of Software Science and Computation Structures, LNCS 9634, pages 252-268. Springer, 2016.
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E. M. Clarke, O. Grumberg, and D. Peled. Model checking. MIT Press, 2001.
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M. R. Clarkson and F. B. Schneider. Hyperproperties. Journal of Computer Security, 18(6):1157-1210, 2010.
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L. Doyen, L. Juhl, K. G. Larsen, N. Markey, and M. Shirmohammadi. Synchronizing words for weighted and timed automata. In Proc. of FSTTCS: Foundations of Software Technology and Theoretical Computer Science, LIPIcs, pages 121-132. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2014. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2014.121.
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L. Doyen, T. Massart, and M. Shirmohammadi. Limit synchronization in Markov decision processes. In Proc. of FoSSaCS: Foundations of Software Science and Computation Structures, LNCS 8412, pages 58-72. Springer-Verlag, 2014.
L. Doyen, T. Massart, and M. Shirmohammadi. Robust synchronization in Markov decision processes. In Proc. of CONCUR: Concurrency Theory, volume LNCS 8704, pages 234-248. Springer, 2014.
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http://dx.doi.org/10.4230/LIPIcs.CONCUR.2015.142
A. Kučera and Jan Strejček. The stuttering principle revisited. Acta Inf., 41(7-8):415-434, 2005.
K. G. Larsen, S. Laursen, and J. Srba. Synchronizing strategies under partial observability. In Proc. of CONCUR: Concurrency Theory, LNCS 8704, pages 188-202. Springer, 2014.
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H. Spakowski. Completeness for Parallel Access to NP and Counting Class Separations. PhD thesis, Heinrich-Heine-Universität Düsseldorf, 2005.
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Deciding the Topological Complexity of Büchi Languages
We study the topological complexity of languages of Büchi automata on infinite binary trees. We show that such a language is either Borel and WMSO-definable, or Sigma_1^1-complete and not WMSO-definable; moreover it can be algorithmically decided which of the two cases holds. The proof relies on a direct reduction to deciding the winner in a finite game with a regular winning condition.
tree automata
non-determinism
Borel sets
topological complexity
decidability
99:1-99:13
Regular Paper
Michal
Skrzypczak
Michal Skrzypczak
Igor
Walukiewicz
Igor Walukiewicz
10.4230/LIPIcs.ICALP.2016.99
Mikołaj Bojańczyk. Star height via games. In LICS, pages 214-219, 2015.
Mikołaj Bojańczyk and Tomasz Idziaszek. Algebra for infinite forests with an application to the temporal logic EF. In CONCUR, pages 131-145, 2009.
Mikołaj Bojańczyk, Damian Niwiński, Alexander Rabinovich, Adam Radziwończyk -Syta, and Michał Skrzypczak. On the Borel complexity of MSO definable sets of branches. Fundamenta Informaticae, 98(4):337-349, 2010.
Mikołaj Bojańczyk and Thomas Place. Regular languages of infinite trees that are Boolean combinations of open sets. In ICALP, pages 104-115, 2012.
Julian Bradfield. The modal mu-calculus alternation hierarchy is strict. Theoretical Computer Science, 195:133-153, 1997.
Jérémie Cabessa, Jacques Duparc, Alessandro Facchini, and Filip Murlak. The Wadge hierarchy of max-regular languages. In FSTTCS, pages 121-132, 2009.
Thomas Colcombet. Fonctions régulières de coût. Habilitation thesis, Université Paris Diderot - Paris 7, 2013.
Thomas Colcombet, Denis Kuperberg, Christof Löding, and Michael Vanden Boom. Deciding the weak definability of Büchi definable tree languages. In CSL, pages 215-230, 2013.
Thomas Colcombet and Christof Löding. The non-deterministic Mostowski hierarchy and distance-parity automata. In ICALP (2), pages 398-409, 2008.
Jacques Duparc, Olivier Finkel, and Jean-Pierre Ressayre. Computer science and the fine structure of Borel sets. Theoretical Computer Science, 257(1-2):85-105, 2001.
Jacques Duparc, Olivier Finkel, and Jean-Pierre Ressayre. The Wadge hierarchy of Petri nets w-languages. In LFCS, pages 179-193, 2013.
Alessandro Facchini, Filip Murlak, and Michał Skrzypczak. Rabin-Mostowski index problem: A step beyond deterministic automata. In LICS, pages 499-508, 2013.
Olivier Finkel. Borel ranks and Wadge degrees of context free omega-languages. Mathematical Structures in Computer Science, 16(5):813-840, 2006.
Tomasz Idziaszek, Michał Skrzypczak, and Mikołaj Bojańczyk. Regular languages of thin trees. Theory of Computing Systems, pages 1-50, 2015.
Alexander Kechris. Classical descriptive set theory. Springer-Verlag, New York, 1995.
Denis Kuperberg and Michael Vanden Boom. Quasi-weak cost automata: A new variant of weakness. In FSTTCS, volume 13 of LIPIcs, pages 66-77, 2011.
Ralf Küsters and Thomas Wilke. Deciding the first level of the μ-calculus alternation hierarchy. In FST TCS 2002:, volume 2556 of LNCS, pages 241-252, 2002.
Filip Murlak. The Wadge hierarchy of deterministic tree languages. Logical Methods in Logical Methods in Comput. Sci., 4(4), 2008.
Filip Murlak. Weak index versus Borel rank. In STACS'08, LIPIcs, pages 573-584, 2008.
Damian Niwiński and Igor Walukiewicz. A gap property of deterministic tree languages. Theor. Comput. Sci., 1(303):215-231, 2003.
Damian Niwiński and Igor Walukiewicz. Deciding nondeterministic hierarchy of deterministic tree automata. Electronic Notes in Theoretical Computer Science, 123:195-208, 2005.
Michael Oser Rabin. Weakly definable relations and special automata. In Proceedings of the Symposium on Mathematical Logic and Foundations of Set Theory, pages 1-23. North-Holland, 1970.
Jerzy Skurczyński. The Borel hierarchy is infinite in the class of regular sets of trees. Theoretical Computer Science, 112(2):413-418, 1993.
Wolfgang Thomas and Helmut Lescow. Logical specifications of infinite computations. In REX School/Symposium, pages 583-621, 1993.
Igor Walukiewicz. Deciding low levels of tree-automata hierarchy. In Workshop on Logic, Language, Information and Computation, volume 67 of Electronic Notes in Theoretical Computer Science, pages 61-75, 2002.
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On the Skolem Problem for Continuous Linear Dynamical Systems
The Continuous Skolem Problem asks whether a real-valued function satisfying a linear differential equation has a zero in a given interval of real numbers. This is a fundamental reachability problem for continuous linear dynamical systems, such as linear hybrid automata and continuoustime Markov chains. Decidability of the problem is currently open — indeed decidability is open even for the sub-problem in which a zero is sought in a bounded interval. In this paper we show decidability of the bounded problem subject to Schanuel's Conjecture, a unifying conjecture in transcendental number theory. We furthermore analyse the unbounded problem in terms of the frequencies of the differential equation, that is, the imaginary parts of the characteristic roots.
We show that the unbounded problem can be reduced to the bounded problem if there is at most one rationally linearly independent frequency, or if there are two rationally linearly independent frequencies and all characteristic roots are simple. We complete the picture by showing that decidability of the unbounded problem in the case of two (or more) rationally linearly independent frequencies would entail a major new effectiveness result in Diophantine approximation, namely computability of the Diophantine-approximation types of all real algebraic numbers.
differential equations
reachability
Baker’s Theorem
Schanuel’s Conjecture
semi-algebraic sets
100:1-100:13
Regular Paper
Ventsislav
Chonev
Ventsislav Chonev
Joël
Ouaknine
Joël Ouaknine
James
Worrell
James Worrell
10.4230/LIPIcs.ICALP.2016.100
Rajeev Alur. Principles of Cyber-Physical Systems. MIT Press, 2015.
Alan Baker. Transcendental number theory. Cambridge University Press, Cambridge, 1975.
Saugata Basu, Richard Pollack, and Marie-Françoise Roy. Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics). Springer-Verlag, 2006.
Paul C. Bell, Jean-Charles Delvenne, Raphaël M. Jungers, and Vincent D. Blondel. The Continuous Skolem-Pisot Problem. Theoretical Computer Science (TCS), 411(40-42):3625-3634, 2010.
Edward Bierstone and Pierre D. Milman. Semianalytic and subanalytic sets. Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 67(1):5-42, 1988.
Richard P. Brent. Fast multiple-precision evaluation of elementary functions. Journal of the ACM (JACM), 23(2):242-251, 1976.
Ventsislav Chonev, Joël Ouaknine, and James Worrell. On the skolem problem for continuous linear dynamical systems. CoRR, abs/1506.00695, 2015.
Ventsislav Chonev, Joël Ouaknine, and James Worrell. On recurrent reachability for continuous linear dynamical systems. Logic in Computer Science (LICS), 2016.
Henri Cohen. A Course in Computational Algebraic Number Theory. Springer-Verlag, 1993.
Paul M. Cohn. Basic Algebra: Groups, Rings and Fields. Springer, 2002.
David A. Cox, John Little, and Donal O'Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, 2007.
Vesa Halava, Tero Harju, Mika Hirvensalo, and Juhani Karhumäki. Skolem’s Problem - on the Border between Decidability and Undecidability. TUCS Technical Reports, 683, 2005.
Godfrey H. Hardy and Edward M. Wright. An Introduction to the Theory of Numbers, volume 1. Oxford, 1999.
Bettina Just. Integer relations among algebraic numbers. In Mathematical Foundations of Computer Science (MFCS), volume 379, pages 314-320. Springer, 1989.
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Arjen K. Lenstra. Factoring multivariate polynomials over algebraic number fields. SIAM Journal on Computing, 16(3):591-598, 1987.
Angus Macintyre. Turing meets Schanuel. Preprint, to appear in the proceedings of Logic Colloquium, 2012.
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Boris Zilber. Exponential sums equations and the Schanuel conjecture. Journal of the London Mathematical Society, 65:27-44, 2002.
Boris Zilber. Pseudo-exponentiation on algebraically closed fields of characteristic zero. Annals of Pure and Applied Logic, 132(1):67-95, 2005.
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Analysing Decisive Stochastic Processes
In 2007, Abdulla et al. introduced the elegant concept of decisive Markov chain. Intuitively, decisiveness allows one to lift the good properties of finite Markov chains to infinite Markov chains. For instance, the approximate quantitative reachability problem can be solved for decisive Markov chains (enjoying reasonable effectiveness assumptions) including probabilistic lossy channel systems and probabilistic vector addition systems with states. In this paper, we extend the concept of decisiveness to more general stochastic processes. This extension is non trivial as we consider stochastic processes with a potentially continuous set of states and uncountable branching (common features of real-time stochastic processes). This allows us to obtain decidability results for both qualitative and quantitative verification problems on some classes of real-time stochastic processes, including generalized semi-Markov processes and stochastic timed
automata
Real-time stochastic processes
Decisiveness
Approximation Scheme
101:1-101:14
Regular Paper
Nathalie
Bertrand
Nathalie Bertrand
Patricia
Bouyer
Patricia Bouyer
Thomas
Brihaye
Thomas Brihaye
Pierre
Carlier
Pierre Carlier
10.4230/LIPIcs.ICALP.2016.101
Parosh Aziz Abdulla, Mohamed Faouzi Atig, and Jonathan Cederberg. Timed lossy channel systems. In Proc. 31sth Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS'12), volume 18 of LIPIcs, pages 374-386. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2012. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2012.374.
http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2012.374
Parosh Aziz Abdulla, Noomene Ben Henda, and Richard Mayr. Decisive Markov chains. Logical Methods in Computer Science, 3(4), 2007.
Rajeev Alur and Mikhail Bernadsky. Bounded model checking for GSMP models of stochastic real-time systems. In Proc. 9th International Workshop on Hybrid Systems: Computation and Control (HSCC'06), volume 3927 of Lecture Notes in Computer Science, pages 19-33. Springer, 2006.
Rajeev Alur and David L. Dill. A theory of timed automata. Theoretical Computer Science, 126(2):183-235, 1994.
Christel Baier. Reasoning about cost-utility constraints in probabilistic models. In Proc. 9th Workshop on Reachability Problems in Computational Models (RP'15), volume 9328 of Lecture Notes in Computer Science, pages 1-6. Springer, 2015.
Christel Baier, Boudewijn Haverkort, Holger Hermanns, and Joost-Pieter Katoen. Model-checking algorithms for continuous-time Markov chains. IEEE Transactions on Software Engineering, 29(7):524-541, 2003.
Mikhail Bernadsky and Rajeev Alur. Symbolic analysis for GSMP models with one stateful clock. In Proc. 10th International Workshop on Hybrid Systems: Computation and Control (HSCC'07), volume 4416 of Lecture Notes in Computer Science, pages 90-103. Springer, 2007.
Nathalie Bertrand, Patricia Bouyer, Thomas Brihaye, and Nicolas Markey. Quantitative model-checking of one-clock timed automata under probabilistic semantics. In Proc. 5th International Conference on Quantitative Evaluation of Systems (QEST'08). IEEE Computer Society Press, 2008.
Nathalie Bertrand, Patricia Bouyer, Thomas Brihaye, Quentin Menet, Christel Baier, Marcus Größer, and Marcin Jurdziński. Stochastic timed automata. Logical Methods in Computer Science, 10(4):1-73, 2014.
Tomáš Brázdil, Jan Krčál, Jan Křetínský, and Vojtěch Řehák. Fixed-delay events in generalized semi-Markov processes revisited. In Proc. 22nd International Conference on Concurrency Theory (CONCUR'11), volume 6901 of Lecture Notes in Computer Science, pages 140-155. Springer, 2011.
Lorenzo Clemente, Frédéric Herbreteau, Amélie Stainer, and Grégoire Sutre. Reachability of communicating timed processes. In Proc. 16th International Conference on Foundations of Software Science and Computation Structures (FoSSaCS'13), volume 7794 of Lecture Notes in Computer Science, pages 81-96. Springer, 2013.
Josée Desharnais and Prakash Panangaden. Continuous stochastic logic characterizes bisimulation of continuous-time Markov processes. Journal of Logic and Algebraic Programming, 56:99-115, 2003.
Peter W. Glynn. A GSMP formalism for discrete event systems. Proceedings of the IEEE, 77(1):14-23, 1989.
Geoffrey R. Grimmett and David R. Stirzaker. Probability and Random Processes. Oxford University Press, 1992.
Ernst Moritz Hahn, Holger Hermanns, Björn Wachter, and Lijun Zhang. Time-bounded model checking of infinite-state continuous-time Markov chains. Fundamenta Informaticae, 95(1):129-155, 2009.
Ronald A. Howard. Dynamic Probabilistic Systems, Volume II: Semi-Markov and Decision Processes. John Wiley &Sons, 1971.
S. Purushothaman Iyer and Murali Narasimha. Probabilistic lossy channel systems. In Proc. 7th International Joint Conference on Theory and Practice of Software Development (TAPSOFT'97), volume 1214 of Lecture Notes in Computer Science, pages 667-681. Springer, 1997.
Sadegh Soudjani, Rupak Majumdar, and Alessandro Abate. Safety verification of continuous-space pure jump Markov processes. In Proc. 22nd International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS'16), volume 9636 of Lecture Notes in Computer Science, pages 147-163. Springer, 2016.
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Composition of Stochastic Transition Systems Based on Spans and Couplings
Conventional approaches for parallel composition of stochastic systems relate probability measures of the individual components in terms of product measures. Such approaches rely on the assumption that components interact stochastically independent, which might be too rigid for modeling real world systems. In this paper, we introduce a parallel-composition operator for stochastic transition systems that is based on couplings of probability measures and does not impose any stochastic assumptions. When composing systems within our framework, the intended dependencies between components can be determined by providing so-called spans and span couplings. We present a congruence result for our operator with respect to a standard notion of bisimilarity and develop a general theory for spans, exploiting deep results from descriptive set theory. As an application of our general approach, we propose a model for stochastic hybrid systems called stochastic hybrid motion automata.
Stochastic Transition System
Composition
Stochastic Hybrid Motion Automata
Stochastically Independent
Coupling
Span
Bisimulation
Congruence
Po
102:1-102:15
Regular Paper
Daniel
Gburek
Daniel Gburek
Christel
Baier
Christel Baier
Sascha
Klüppelholz
Sascha Klüppelholz
10.4230/LIPIcs.ICALP.2016.102
L. de Alfaro. Formal Verification of Probabilistic Systems. PhD thesis, University of Stanford, 1997.
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R. Alur, C. Courcoubetis, N. Halbwachs, T. A. Henzinger, P. Ho, X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. The algorithmic analysis of hybrid systems. Theoretical Computer Science, 138(1):3-34, 1995.
R. Alur and D. L. Dill. A theory of timed automata. Theoretical Computer Science, 126:183-235, 1994.
R. Alur and T. A. Henzinger. Modularity for timed and hybrid systems. In 8th International Conference on Concurrency Theory (CONCUR), LNCS 1243, pages 74-88. Springer, 1997.
G. Bacci, G. Bacci, K. G. Larsen, and R. Mardare. Bisimulation on Markov processes over arbitrary measurable spaces. In Horizons of the Mind. A Tribute to P. Panangaden, LNCS 8464, pages 76-95. Springer, 2014.
P. Billingsley. Convergence of Probability Measures. Wiley-Interscience, 2 edition, 1999.
V. I. Bogachev. Measure Theory Volume, volume 1 and 2. Springer, 2007.
H. Bohnenkamp, P. R. D'Argenio, H. Hermanns, and J.-P. Katoen. MoDeST: A compositional modeling formalism for real-time and stochastic systems. IEEE Transactions on Software Engineering, 32(10):812-830, 2006.
S. Bornot and J. Sifakis. On the composition of hybrid systems. In 1st International Workshop on Hybrid Systems: Computation and Control (HSCC), volume 1386 of Lecture Notes in Computer Science, pages 49-63. Springer, 1998.
P. Bouyer, T. Brihaye, P. Carlier, and Q. Menet. Compositional design of stochastic timed automata. In 11th International Computer Science Symposium in Russia (CSR), LNCS, 2016 (to appear).
M. Bravetti. Real time and stochastic time. In Formal Methods for the Design of Real-Time Systems, International School on Formal Methods for the Design of Computer, Communication and Software Systems (SFM-RT), LNCS 3185, pages 132-180. Springer, 2004.
M. Bravetti and P. R. D'Argenio. Tutte le algebre insieme: Concepts, discussions and relations of stochastic process algebras with general distributions. In Validation of Stochastic Systems - A Guide to Current Research, LNCS 2925, pages 44-88. Springer, 2004.
M. Bravetti and R. Gorrieri. The theory of interactive generalized semi-Markov processes. Theoretical Computer Science, 282(1):5-32, 2002.
M. L. Bujorianu and John Lygeros. Towards a general theory of stochastic hybrid systems. In Stochastic Hybrid Systems, volume 337 of Lecture Notes in Control and Information Science, pages 3-30. Springer, 2006.
S. Cattani, R. Segala, M. Z. Kwiatkowska, and G. Norman. Stochastic transition systems for continuous state spaces and non-determinism. In 8th International Conference on Foundations of Software Science and Computational Structures (FOSSACS), LNCS 3441, pages 125-139. Springer, 2005.
V. Danos, J. Desharnais, F. Laviolette, and P. Panangaden. Bisimulation and cocongruence for probabilistic systems. Information and Computation, 204(4):503-523, 2006.
P. R. D'Argenio. Algebras and Automata for Timed and Stochastic Systems. PhD thesis, University of Twente, 1999.
P. R. D'Argenio and J.-P. Katoen. A theory of stochastic systems part I: Stochastic automata and part II: Process algebra. Information and Computation, 203(1):1-74, 2005.
P. R. D'Argenio, P. Sánchez Terraf, and N. Wolovick. Bisimulations for non-deterministic labelled Markov processes. Mathematical Structures in Computer Science, 22:43-68, 2012.
E. P. de Vink and J.J.M.M. Rutten. Bisimulation for probabilistic transition systems: a coalgebraic approach. Theoretical Computer Science, 221:271-293, 1999.
J. Desharnais, A. Edalat, and P. Panangaden. Bisimulation for labelled Markov processes. Information and Computation, 179(2):163-193, 2002.
C. Eisentraut, H. Hermanns, and L. Zhang. On probabilistic automata in continuous time. In 25th Annual IEEE Symposium on Logic in Computer Science (LICS), pages 342-351. IEEE Computer Society, 2010.
M. Fränzle, E. M. Hahn, H. Hermanns, N. Wolovick, and L. Zhang. Measurability and safety verification for stochastic hybrid systems. In 14th International Conference on Hybrid Systems: Computation and Control (HSCC), pages 43-52. ACM, 2011.
D. Gburek, C. Baier, and S. Klüppelholz. Composition of stochastic transition systems based on spans and couplings. Technical report, Technische Universität Dresden, 2016. URL: http://wwwtcs.inf.tu-dresden.de/ALGI/PUB/ICALP16/.
http://wwwtcs.inf.tu-dresden.de/ALGI/PUB/ICALP16/
E. M. Hahn. Model checking stochastic hybrid systems. PhD thesis, Universität des Saarlandes, 2013.
E. M. Hahn, A. Hartmanns, H. Hermanns, and J.-P. Katoen. A compositional modelling and analysis framework for stochastic hybrid systems. Formal Methods in System Design, 2012.
A. Hartmanns and H. Hermanns. In the quantitative automata zoo. Science of Computer Programming, 112:3-23, 2015.
T. A. Henzinger. The theory of hybrid automata. In 11th Annual IEEE Symposium on Logic in Computer Science (LICS), pages 278-292. IEEE Computer Society, 1996.
H. Hermanns. Interactive Markov Chains: And the Quest for Quantified Quality. Springer, 2002.
H. Hermanns, J. Krcál, and J. Kretínský. Probabilistic bisimulation: Naturally on distributions. In 25th International Conference on Concurrency Theory (CONCUR), LNCS 8704, pages 249-265. Springer, 2014.
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H. Kerstan and B. König. Coalgebraic trace semantics for probabilistic transition systems based on measure theory. In 23th International Conference on Concurrency Theory (CONCUR), LNCS 7454, pages 410-424. Springer, 2012.
N. Lynch, R. Segala, and F. Vaandrager. Hybrid I/O automata. Information and Computation, 185(1):105-157, 2003.
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On Restricted Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n*m matrix M into a product of a nonnegative n*d matrix W and a nonnegative d*m matrix H. Restricted NMF requires in addition that the column spaces of M and W coincide.
Finding the minimal inner dimension d is known to be NP-hard, both for NMF and restricted NMF. We show that restricted NMF is closely related to a question about the nature of minimal probabilistic automata, posed by Paz in his seminal 1971 textbook. We use this connection to answer Paz's question negatively, thus falsifying a positive answer claimed in 1974.
Furthermore, we investigate whether a rational matrix M always has a restricted NMF of minimal inner dimension whose factors W and H are also rational. We show that this holds for matrices M of rank at most 3 and we exhibit a rank-4 matrix for which W and H require irrational entries.
nonnegative matrix factorization
nonnegative rank
probabilistic automata
labelled Markov chains
minimization
103:1-103:14
Regular Paper
Dmitry
Chistikov
Dmitry Chistikov
Stefan
Kiefer
Stefan Kiefer
Ines
Marusic
Ines Marusic
Mahsa
Shirmohammadi
Mahsa Shirmohammadi
James
Worrell
James Worrell
10.4230/LIPIcs.ICALP.2016.103
A. Aggarwal, H. Booth, J. O'Rourke, S. Suri, and C. K. Yap. Finding minimal convex nested polygons. Information and Computation, 83(1):98-110, 1989.
S. Arora, R. Ge, R. Kannan, and A. Moitra. Computing a nonnegative matrix factorization - provably. In Proceedings of the 44th Symposium on Theory of Computing (STOC), pages 145-162, 2012.
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M. W. Berry, N. Gillis, and F. Glineur. Document classification using nonnegative matrix factorization and underapproximation. In International Symposium on Circuits and Systems (ISCAS), pages 2782-2785. IEEE, 2009.
S. S. Bucak and B. Günsel. Video content representation by incremental non-negative matrix factorization. In Proceedings of the International Conference on Image Processing (ICIP), pages 113-116. IEEE, 2007.
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A. Cichocki, R. Zdunek, A. H. Phan, and S.-i. Amari. Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation. Wiley Publishing, 2009.
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N. Gillis and F. Glineur. On the geometric interpretation of the nonnegative rank. Linear Algebra and its Applications, 437(11):2685-2712, 2012.
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A. Paz. Introduction to probabilistic automata. Academic Press, New York, 1971.
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Y. Shitov. Nonnegative rank depends on the field. Technical report, arxiv.org, 2015. Available at ěrb|http://arxiv.org/abs/1505.01893|.
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S. A. Vavasis. On the complexity of nonnegative matrix factorization. SIAM Journal on Optimization, 20(3):1364-1377, 2009.
T. Yokota, R. Zdunek, A. Cichocki, and Y. Yamashita. Smooth nonnegative matrix and tensor factorizations for robust multi-way data analysis. Signal Processing, 113:234-249, 2015.
S. Zhang, W. Wang, J. Ford, and F. Makedon. Learning from incomplete ratings using non-negative matrix factorization. In Proceedings of the 6th SIAM International Conference on Data Mining, pages 549-553. SIAM, 2006.
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Proving the Herman-Protocol Conjecture
Herman's self-stabilization algorithm, introduced 25 years ago, is a well-studied synchronous randomized protocol for enabling a ring of N processes collectively holding any odd number of tokens to reach a stable state in which a single token remains. Determining the worst-case expected time to stabilization is the central outstanding open problem about this protocol. It is known that there is a constant h such that any initial configuration has expected stabilization time at most hN2. Ten years ago, McIver and Morgan established a lower bound of 4/27 ~ 0.148 for h, achieved with three equally-spaced tokens, and conjectured this to be the optimal value of h. A series of papers over the last decade gradually reduced the upper bound on h, with the present record (achieved in 2014) standing at approximately 0.156. In this paper, we prove McIver and Morgan's conjecture and establish that h = 4/27 is indeed optimal.
randomized protocols
self-stabilization
Lyapunov function
expected time
104:1-104:12
Regular Paper
Maria
Bruna
Maria Bruna
Radu
Grigore
Radu Grigore
Stefan
Kiefer
Stefan Kiefer
Joël
Ouaknine
Joël Ouaknine
James
Worrell
James Worrell
10.4230/LIPIcs.ICALP.2016.104
D. Aldous and J. A. Fill. Reversible Markov chains and random walks on graphs, 2002. Unfinished monograph, recompiled 2014, available at URL: http://www.stat.berkeley.edu/~aldous/RWG/book.html.
http://www.stat.berkeley.edu/~aldous/RWG/book.html
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M. Bruna, R. Grigore, S. Kiefer, J. Ouaknine, and J. Worrell. Proving the Herman-Protocol Conjecture. Technical report, arxiv.org, 2015. Available at ěrb|http://arxiv.org/abs/1504.01130|.
PRISM case studies. Randomised self-stabilising algorithms. ěrb|http://www.prismmodelchecker.org/casestudies/self-stabilisation.php|.
C. Cooper, R. Elsässer, H. Ono, and T. Radzik. Coalescing random walks and voting on graphs. In Proc. PODC, pages 47-56. ACM, 2012.
D. Coppersmith, P. Tetali, and P. Winkler. Collisions among random walks on a graph. SIAM Journal on Discrete Mathematics, 6(3):363-374, 1993.
J.T. Cox. Coalescing random walks and voter model consensus times on the torus in Z^d. The Annals of Probability, 17(4):1333-1366, 1989.
E. Csóka and S. Mészáros. Generalized solution for the Herman protocol conjecture. Technical report, arxiv.org, 2015. Available at ěrb|http://arxiv.org/abs/1504.06963|.
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Y. Feng and L. Zhang. A nearly optimal upper bound for the self-stabilization time in Herman’s algorithm. Dist. Comp., pages 1-12, 2015.
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S. Kiefer, A. Murawski, J. Ouaknine, J. Worrell, and L. Zhang. On stabilization in Herman’s algorithm. In Proc. ICALP, volume 6756 of LNCS. Springer, 2011.
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A. McIver and C. Morgan. An elementary proof that Herman’s ring is Θ(N²). Inf. Process. Lett., 94(2):79-84, 2005.
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A Polynomial-Time Algorithm for Reachability in Branching VASS in Dimension One
Branching VASS (BVASS) generalise vector addition systems with states by allowing for special branching transitions that can non-deterministically distribute a counter value between two control states. A run of a BVASS consequently becomes a tree, and reachability is to decide whether a given configuration is the root of a reachability tree. This paper shows P-completeness of reachability in BVASS in dimension one, the first decidability result for reachability in a subclass of BVASS known so far. Moreover, we show that coverability and boundedness in BVASS in dimension one are P-complete as well.
branching vector addition systems
reachability
coverability
boundedness
105:1-105:13
Regular Paper
Stefan
Göller
Stefan Göller
Christoph
Haase
Christoph Haase
Ranko
Lazic
Ranko Lazic
Patrick
Totzke
Patrick Totzke
10.4230/LIPIcs.ICALP.2016.105
A.K. Chandra, D. Kozen, and L.J. Stockmeyer. Alternation. J. ACM, 28(1):114-133, 1981. URL: http://dx.doi.org/10.1145/322234.322243.
http://dx.doi.org/10.1145/322234.322243
Ph. de Groote, B. Guillaume, and S. Salvati. Vector addition tree automata. In Logic in Computer Science, LICS, pages 64-73. IEEE Computer Society, 2004. URL: http://dx.doi.org/10.1109/LICS.2004.1319601.
http://dx.doi.org/10.1109/LICS.2004.1319601
S. Demri, M. Jurdziński, O. Lachish, and R. Lazić. The covering and boundedness problems for branching vector addition systems. J. Comput. Syst. Sci., 79(1):23-38, 2013. URL: http://dx.doi.org/10.1016/j.jcss.2012.04.002.
http://dx.doi.org/10.1016/j.jcss.2012.04.002
M. Englert, R. Lazić, and P. Totzke. Reachability in two-dimensional unary vector addition systems with states is NL-complete. In Logic in Computer Science, LICS, 2016. To appear.
P. Ganty and R. Majumdar. Algorithmic verification of asynchronous programs. ACM Trans. Program. Lang. Syst., 34(1):6, 2012. URL: http://dx.doi.org/10.1145/2160910.2160915.
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S.M. German and A.P. Sistla. Reasoning about systems with many processes. J. ACM, 39(3):675-735, 1992. URL: http://dx.doi.org/10.1145/146637.146681.
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S. Göller, C. Haase, R. Lazić, and P. Totzke. A polynomial-time algorithm for reachability in branching VASS in dimension one. CoRR, abs/1602.05547, 2016. URL: http://arxiv.org/abs/1602.05547.
http://arxiv.org/abs/1602.05547
P. Jančar and Z. Sawa. A note on emptiness for alternating finite automata with a one-letter alphabet. Inf. Process. Lett., 104(5):164-167, 2007. URL: http://dx.doi.org/10.1016/j.ipl.2007.06.006.
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http://dx.doi.org/10.1007/s00224-008-9139-5
R. Lazić and S. Schmitz. Nonelementary complexities for branching VASS, MELL, and Extensions. ACM Trans. Comput. Log., 16(3):20, 2015. URL: http://dx.doi.org/10.1145/2733375.
http://dx.doi.org/10.1145/2733375
J. Leroux and S. Schmitz. Demystifying reachability in vector addition systems. In Logic in Computer Science, LICS, pages 56-67. IEEE, 2015. URL: http://dx.doi.org/10.1109/LICS.2015.16.
http://dx.doi.org/10.1109/LICS.2015.16
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http://dx.doi.org/10.1016/S0022-0000(75)80005-5
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Reachability in Networks of Register Protocols under Stochastic Schedulers
We study the almost-sure reachability problem in a distributed system obtained as the asynchronous composition of N copies (called processes) of the same automaton (called protocol), that can communicate via a shared register with finite domain. The automaton has two types of transitions: write-transitions update the value of the register, while read-transitions move to a new state depending on the content of the register. Non-determinism is resolved by a stochastic scheduler. Given a protocol, we focus on almost-sure reachability of a target state by one of the processes. The answer to this problem naturally depends on the number N of processes. However, we prove that our setting has a cut-off property: the answer to the almost-sure reachability problem is constant when N is large enough; we then develop an EXPSPACE algorithm deciding whether this constant answer is positive or negative.
Networks of Processes
Parametrized Systems
Stochastic Scheduler
Almost-sure Reachability
Cut-Off Property
106:1-106:14
Regular Paper
Patricia
Bouyer
Patricia Bouyer
Nicolas
Markey
Nicolas Markey
Mickael
Randour
Mickael Randour
Arnaud
Sangnier
Arnaud Sangnier
Daniel
Stan
Daniel Stan
10.4230/LIPIcs.ICALP.2016.106
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A Program Logic for Union Bounds
We propose a probabilistic Hoare logic aHL based on the union bound, a tool from basic probability theory. While the union bound is simple, it is an extremely common tool for analyzing randomized algorithms. In formal verification terms, the union bound allows flexible and compositional reasoning over possible ways an algorithm may go wrong. It also enables a clean separation between reasoning about probabilities and reasoning about events, which are expressed as standard first-order formulas in our logic. Notably, assertions in our logic are non-probabilistic, even though we can conclude probabilistic facts from the judgments.
Our logic can also prove accuracy properties for interactive programs, where the program must produce intermediate outputs as soon as pieces of the input arrive, rather than accessing the entire input at once. This setting also enables adaptivity, where later inputs may depend on earlier intermediate outputs. We show how to prove accuracy for several examples from the differential privacy literature, both interactive and non-interactive.
Probabilistic Algorithms
Accuracy
Formal Verification
Hoare Logic
Union Bound
107:1-107:15
Regular Paper
Gilles
Barthe
Gilles Barthe
Marco
Gaboardi
Marco Gaboardi
Benjamin
Grégoire
Benjamin Grégoire
Justin
Hsu
Justin Hsu
Pierre-Yves
Strub
Pierre-Yves Strub
10.4230/LIPIcs.ICALP.2016.107
Philippe Audebaud and Christine Paulin-Mohring. Proofs of randomized algorithms in Coq. Science of Computer Programming, 74(8):568-589, 2009. URL: https://www.lri.fr/~paulin/ALEA/random-scp.pdf.
https://www.lri.fr/~paulin/ALEA/random-scp.pdf
Christel Baier and Joost-Pieter Katoen. Principles of model checking. MIT Press, 2008.
Gilles Barthe, Thomas Espitau, Marco Gaboardi, Benjamin Grégoire, Justin Hsu, and Pierre-Yves Strub. Formal certification of randomized algorithms. 2015. URL: http://justinh.su/files/docs/BEGGHS15paper.pdf.
http://justinh.su/files/docs/BEGGHS15paper.pdf
Gilles Barthe, Thomas Espitau, Benjamin Grégoire, Justin Hsu, Léo Stefanesco, and Pierre-Yves Strub. Relational reasoning via probabilistic coupling. In International Conference on Logic for Programming, Artificial Intelligence and Reasoning (LPAR), Suva, Fiji, volume 9450, pages 387-401, 2015. URL: http://arxiv.org/abs/1509.03476.
http://arxiv.org/abs/1509.03476
Gilles Barthe, Marco Gaboardi, Emilio Jesús Gallego Arias, Justin Hsu, César Kunz, and Pierre-Yves Strub. Proving differential privacy in Hoare logic. In IEEE Computer Security Foundations Symposium (CSF), Vienna, Austria, 2014. URL: http://arxiv.org/abs/1407.2988, URL: http://arxiv.org/abs/Yes.
http://arxiv.org/abs/Yes
Gilles Barthe, Marco Gaboardi, Benjamin Grégoire, Justin Hsu, and Pierre-Yves Strub. Proving differential privacy via probabilistic couplings. In IEEE Symposium on Logic in Computer Science (LICS), New York, New York, 2016. To appear. URL: http://arxiv.org/abs/1601.05047, URL: http://arxiv.org/abs/Yes.
http://arxiv.org/abs/Yes
Gilles Barthe, Benjamin Grégoire, and Santiago Zanella-Béguelin. Formal certification of code-based cryptographic proofs. In ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), Savannah, Georgia, pages 90-101, New York, 2009. URL: http://research.microsoft.com/pubs/185309/Zanella.2009.POPL.pdf.
http://research.microsoft.com/pubs/185309/Zanella.2009.POPL.pdf
Gilles Barthe, Boris Köpf, Federico Olmedo, and Santiago Zanella-Béguelin. Probabilistic relational reasoning for differential privacy. In ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), Philadelphia, Pennsylvania, pages 97-110, 2012. URL: http://certicrypt.gforge.inria.fr/2012.POPL.pdf.
http://certicrypt.gforge.inria.fr/2012.POPL.pdf
Sooraj Bhat, Ashish Agarwal, Richard Vuduc, and Alexander Gray. A type theory for probability density functions. In ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), Philadelphia, Pennsylvania, pages 545-556, 2012. URL: http://dx.doi.org/10.1145/2103656.2103721.
http://dx.doi.org/10.1145/2103656.2103721
Johannes Borgström, Andrew D. Gordon, Michael Greenberg, James Margetson, and Jurgen Van Gael. Measure transformer semantics for Bayesian machine learning. Logical Methods in Computer Science, 9(3), 2013. URL: https://doi.org/10.2168/LMCS-9(3:11)2013.
https://doi.org/10.2168/LMCS-9(3:11)2013
Aleksandar Chakarov and Sriram Sankaranarayanan. Probabilistic program analysis with martingales. In International Conference on Computer Aided Verification (CAV), Saint Petersburg, Russia, pages 511-526, 2013. URL: https://www.cs.colorado.edu/~srirams/papers/cav2013-martingales.pdf.
https://www.cs.colorado.edu/~srirams/papers/cav2013-martingales.pdf
Aleksandar Chakarov and Sriram Sankaranarayanan. Expectation invariants as fixed points of probabilistic programs. In International Symposium on Static Analysis (SAS), Munich, Germany, volume 8723 of Lecture Notes in Computer Science, pages 85-100. Springer-Verlag, 2014. URL: https://www.cs.colorado.edu/~srirams/papers/sas14-expectations.pdf.
https://www.cs.colorado.edu/~srirams/papers/sas14-expectations.pdf
Patrick Cousot and Michael Monerau. Probabilistic abstract interpretation. In Helmut Seidl, editor, European Symposium on Programming (ESOP), Tallinn, Estonia, volume 7211 of Lecture Notes in Computer Science, pages 169-193. Springer, 2012. URL: http://www.di.ens.fr/~cousot/publications.www/Cousot-Monerau-ESOP2012-extended.pdf.
http://www.di.ens.fr/~cousot/publications.www/Cousot-Monerau-ESOP2012-extended.pdf
Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In IACR Theory of Cryptography Conference (TCC), New York, New York, pages 265-284, 2006. URL: http://dx.doi.org/10.1007/11681878_14.
http://dx.doi.org/10.1007/11681878_14
Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. Foundations and Trends in Theoretical Computer Science, 9(3-4):211-407, 2014. URL: http://dx.doi.org/10.1561/0400000042.
http://dx.doi.org/10.1561/0400000042
Luis María Ferrer Fioriti and Holger Hermanns. Probabilistic termination: Soundness, completeness, and compositionality. In ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), Mumbai, India, pages 489-501. ACM, 2015. URL: http://www.ae-info.org/attach/User/Hermanns_Holger/Publications/FH-POPL15.pdf.
http://www.ae-info.org/attach/User/Hermanns_Holger/Publications/FH-POPL15.pdf
Nate Foster, Dexter Kozen, Konstantinos Mamouras, Mark Reitblatt, and Alexandra Silva. Probabilistic NetKAT. In European Symposium on Programming (ESOP), Eindhoven, The Netherlands, Lecture Notes in Computer Science, 2016.
Marco Gaboardi, Andreas Haeberlen, Justin Hsu, Arjun Narayan, and Benjamin C Pierce. Linear dependent types for differential privacy. In ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), Rome, Italy, pages 357-370, 2013. URL: http://dl.acm.org/citation.cfm?id=2429113.
http://dl.acm.org/citation.cfm?id=2429113
Friedrich Gretz, Joost-Pieter Katoen, and Annabelle McIver. Prinsys - on a quest for probabilistic loop invariants. In International Conference on Quantitative Evaluation of Systems (QEST), pages 193-208, 2013.
Anupam Gupta, Aaron Roth, and Jonathan Ullman. Iterative constructions and private data release. In IACR Theory of Cryptography Conference (TCC), Taormina, Italy, pages 339-356, 2012. URL: http://arxiv.org/abs/1107.3731.
http://arxiv.org/abs/1107.3731
Moritz Hardt and Guy N Rothblum. A multiplicative weights mechanism for privacy-preserving data analysis. In IEEE Symposium on Foundations of Computer Science (FOCS), Las Vegas, Nevada, pages 61-70, 2010. URL: http://www.mit.edu/~rothblum/papers/pmw.pdf.
http://www.mit.edu/~rothblum/papers/pmw.pdf
Sergiu Hart, Micha Sharir, and Amir Pnueli. Termination of probabilistic concurrent programs. In ACM Symposium on Principles of Programming Languages (POPL), Albuquerque, New Mexico, pages 1-6, 1982. URL: http://dx.doi.org/10.1145/582153.582154.
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Joe Hurd, Annabelle McIver, and Carroll Morgan. Probabilistic guarded commands mechanized in HOL. Theoretical Computer Science, 346(1):96-112, 2005.
C. Jones and Gordon D. Plotkin. A probabilistic powerdomain of evaluations. In IEEE Symposium on Logic in Computer Science (LICS), Asilomar, California, pages 186-195, 1989. URL: http://dx.doi.org/10.1109/LICS.1989.39173.
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Benjamin Lucien Kaminski, Joost-Pieter Katoen, Christoph Matheja, and Federico Olmedo. Weakest precondition reasoning for expected run-times of probabilistic programs. In European Symposium on Programming (ESOP), Eindhoven, The Netherlands, Lecture Notes in Computer Science, 2016.
Joost-Pieter Katoen. Perspectives in probabilistic verification. In IEEE/IFIP International Symposium on Theoretical Aspects of Software Engineering (TASE), pages 3-10, 2008.
Joost-Pieter Katoen, Annabelle McIver, Larissa Meinicke, and Carroll Morgan. Linear-invariant generation for probabilistic programs. In Radhia Cousot and Matthieu Martel, editors, International Symposium on Static Analysis (SAS), Perpignan, France, volume 6337 of Lecture Notes in Computer Science, pages 390-406. Springer, 2010.
Dexter Kozen. Semantics of probabilistic programs. In IEEE Symposium on Foundations of Computer Science (FOCS), San Juan, Puerto Rico, pages 101-114, 1979.
Dexter Kozen. A probabilistic PDL. J. Comput. Syst. Sci., 30(2):162-178, 1985.
Marta Z. Kwiatkowska, Gethin Norman, and David Parker. Probabilistic symbolic model checking with PRISM: A hybrid approach. In International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS), Tallinn, Estonia, pages 52-66, 2002.
A. McIver and C. Morgan. Abstraction, Refinement, and Proof for Probabilistic Systems. Monographs in Computer Science. Springer, 2005.
Frank McSherry and Kunal Talwar. Mechanism design via differential privacy. In IEEE Symposium on Foundations of Computer Science (FOCS), Providence, Rhode Island, pages 94-103, 2007. URL: http://dx.doi.org/10.1109/FOCS.2007.41.
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Carroll Morgan, Annabelle McIver, and Karen Seidel. Probabilistic predicate transformers. ACM Transactions on Programming Languages and Systems, 18(3):325-353, 1996.
Lyle Harold Ramshaw. Formalizing the Analysis of Algorithms. PhD thesis, Stanford University, 1979.
Robert Rand and Steve Zdancewic. VPHL: A verified partial-correctness logic for probabilistic programs. In Mathematical Foundations of Program Semantics (MFPS), 2015.
Jason Reed and Benjamin C Pierce. Distance makes the types grow stronger: A calculus for differential privacy. In ACM SIGPLAN International Conference on Functional Programming (ICFP), Baltimore, Maryland, 2010. URL: http://dl.acm.org/citation.cfm?id=1863568.
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Adrian Sampson, Pavel Panchekha, Todd Mytkowicz, Kathryn S McKinley, Dan Grossman, and Luis Ceze. Expressing and verifying probabilistic assertions. In ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), Edinburgh, Scotland, page 14, 2014. URL: http://research.microsoft.com/pubs/211410/passert-pldi2014.pdf.
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Micha Sharir, Amir Pnueli, and Sergiu Hart. Verification of probabilistic programs. SIAM Journal on Computing, 13(2):292-314, 1984. URL: http://dx.doi.org/10.1137/0213021.
http://dx.doi.org/10.1137/0213021
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The Decidable Properties of Subrecursive Functions
What can be decided or semidecided about a primitive recursive function, given a definition of that function by primitive recursion? What about subrecursive classes other than primitive recursive functions? We provide a complete and explicit characterization of the decidable and semidecidable properties. This characterization uses a variant of Kolmogorov complexity where only programs in a subrecursive programming language are allowed. More precisely, we prove that all the decidable and semidecidable properties can be obtained as combinations of two classes of basic decidable properties: (i) the function takes some particular values on a finite set of inputs, and (ii) every finite part of the function can be compressed to some extent.
Rice theorem
subrecursive class
decidable property
Kolmogorov complexity
compressibility
108:1-108:13
Regular Paper
Mathieu
Hoyrup
Mathieu Hoyrup
10.4230/LIPIcs.ICALP.2016.108
Andrea Asperti. The intensional content of rice’s theorem. In George C. Necula and Philip Wadler, editors, Proceedings of the 35th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2008, San Francisco, California, USA, January 7-12, 2008, pages 113-119. ACM, 2008. URL: http://dx.doi.org/10.1145/1328438.1328455.
http://dx.doi.org/10.1145/1328438.1328455
G. S. Ceitin. Algorithmic operators in constructive metric spaces. Trudy Matematiki Instituta Steklov, 67:295-361, 1962. English translation: American Mathematical Society Translations, series 2, 64:1-80, 1967.
Johanna N. Y. Franklin, Noam Greenberg, Frank Stephan, and Guohua Wu. Anti-complex sets and reducibilities with tiny use. J. Symb. Log., 78(4):1307-1327, 2013. URL: http://dx.doi.org/10.2178/jsl.7804170.
http://dx.doi.org/10.2178/jsl.7804170
Richard M. Friedberg. Un contre-exemple relatif aux fonctionnelles récursives. Comptes Rendus de l'Académie des Sciences, 247:852-854, 1958.
David Gajser. Verifying time complexity of turing machines. Theoretical Computer Science, 600:86-97, 2015. URL: http://dx.doi.org/10.1016/j.tcs.2015.07.028.
http://dx.doi.org/10.1016/j.tcs.2015.07.028
Mathieu Hoyrup and Cristóbal Rojas. On the information carried by programs about the objects they compute. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 447-459. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.447.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.447
Dexter Kozen. Indexings of subrecursive classes. Theoretical Computer Science, 11(3):277-301, 1980. URL: http://dx.doi.org/http://dx.doi.org/10.1016/0304-3975(80)90017-1.
http://dx.doi.org/http://dx.doi.org/10.1016/0304-3975(80)90017-1
Georg Kreisel, Daniel Lacombe, and Joseph R. Schœ nfield. Fonctionnelles récursivement définissables et fonctionnelles récursives. Comptes Rendus de l'Académie des Sciences, 245:399-402, 1957.
Ming Li and Paul M. B. Vitanyi. An Introduction to Kolmogorov Complexity and Its Applications. Springer-Verlag, Berlin, 1993.
Albert R. Meyer and Dennis M. Ritchie. The complexity of loop programs. In Proceedings of the 1967 22Nd National Conference, ACM'67, pages 465-469, New York, NY, USA, 1967. ACM. URL: http://dx.doi.org/10.1145/800196.806014.
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Norman Shapiro. Degrees of computability. Transactions of the American Mathematical Society, 82:281-299, 1956.
Klaus Weihrauch. Computable Analysis. Springer, Berlin, 2000.
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Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length: The General Purpose Analog Computer and Computable Analysis Are Two Efficiently Equivalent Models of Computations
The outcomes of this paper are twofold.
Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in terms of differential equations with polynomial right-hand side.
This result gives a purely continuous (time and space) elegant and simple characterization of P. We believe it is the first time such classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of computable analysis.
Our results may provide a new perspective on classical complexity, by giving a way to define complexity classes, like P, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations.
Continuous-Time Models of Computation. Our results can also be interpreted in terms of analog computers or analog model of computation: As a side effect, we get that the 1941 General Purpose Analog Computer (GPAC) of Claude Shannon is provably equivalent to Turing machines both at the computability and complexity level, a fact that has never been established before. This result provides arguments in favour of a generalised form of the Church-Turing Hypothesis, which states that any physically realistic (macroscopic) computer is equivalent to Turing machines both at a computability and at a computational complexity level.
Analog Models of Computation
Continuous-Time Models of Computation
Computable Analysis
Implicit Complexity
Computational Complexity
Ordinary Diff
109:1-109:15
Regular Paper
Olivier
Bournez
Olivier Bournez
Daniel S.
Graça
Daniel S. Graça
Amaury
Pouly
Amaury Pouly
10.4230/LIPIcs.ICALP.2016.109
Rajeev Alur and David L. Dill. Automata for modeling real-time systems. In Mike Paterson, editor, Automata, Languages and Programming, 17th International Colloquium, ICALP90, Warwick University, England, July 16-20, 1990, Proceedings, volume 443 of Lecture Notes in Computer Science, pages 322-335. Springer, 1990.
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Algorithmic Complexity for the Realization of an Effective Subshift By a Sofic
Realization of d-dimensional effective subshifts as projective sub-actions of d + d'-dimensional sofic subshifts for d' >= 1 is now well known [Hochman, 2009; Durand/Romashchenko/Shen, 2012; Aubrun/Sablik, 2013]. In this paper we are interested in qualitative aspects of this realization. We introduce a new topological conjugacy invariant for effective subshifts, the speed of convergence, in view to exhibit algorithmic properties of these subshifts in contrast to the usual framework that focuses on undecidable properties.
Subshift
computability
time complexity
space complexity
tilings
110:1-110:14
Regular Paper
Mathieu
Sablik
Mathieu Sablik
Michael
Schraudner
Michael Schraudner
10.4230/LIPIcs.ICALP.2016.110
Nathalie Aubrun and Mathieu Sablik. An order on sets of tilings corresponding to an order on languages. In Susanne Albers, Susanne Albers, Susanne Albers, Susanne Albers, and Jean-Yves Marion, editors, 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009, February 26-28, 2009, Freiburg, Germany, Proceedings, volume 3 of LIPIcs, pages 99-110. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany, 2009. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2009.1833.
http://dx.doi.org/10.4230/LIPIcs.STACS.2009.1833
Nathalie Aubrun and Mathieu Sablik. Simulation of Effective Subshifts by Two-dimensional Subshifts of Finite Type. Acta Appl. Math., 126:35-63, 2013. URL: http://dx.doi.org/10.1007/s10440-013-9808-5.
http://dx.doi.org/10.1007/s10440-013-9808-5
Nathalie Aubrun and Mathieu Sablik. Multidimensional effective s-adic systems are sofic. Uniform Distribution Theory, 9(2), 2014.
Bruno Durand, Andrei Romashchenko, and Alexander Shen. Fixed-point tile sets and their applications. J. Comput. System Sci., 78(3):731-764, 2012. URL: http://dx.doi.org/10.1016/j.jcss.2011.11.001.
http://dx.doi.org/10.1016/j.jcss.2011.11.001
Thomas Fernique and Mathieu Sablik. Local rules for computable planar tilings. In Proceedings 18th international workshop on Cellular Automata and Discrete Complex Systems and 3rd international symposium Journées Automates Cellulaires, pages 133-141, 2012. URL: http://dx.doi.org/10.4204/EPTCS.90.11.
http://dx.doi.org/10.4204/EPTCS.90.11
Michael Hochman. On the dynamics and recursive properties of multidimensional symbolic systems. Invent. Math., 176(1):131-167, 2009. URL: http://dx.doi.org/10.1007/s00222-008-0161-7.
http://dx.doi.org/10.1007/s00222-008-0161-7
Michael Hochman and Tom Meyerovitch. A characterization of the entropies of multidimensional shifts of finite type. Annals of Mathematics, 171(3):2011-2038, 2010. URL: http://dx.doi.org/10.4007/annals.2010.171.2011.
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On Word and Frontier Languages of Unsafe Higher-Order Grammars
Higher-order grammars are an extension of regular and context-free grammars, where nonterminals may take parameters. They have been extensively studied in 1980's, and restudied recently in the context of model checking and program verification. We show that the class of unsafe order-(n+1) word languages coincides with the class of frontier languages of unsafe order-n tree languages. We use intersection types for transforming an order-(n+1) word grammar to a corresponding order-n tree grammar. The result has been proved for safe languages by Damm in 1982, but it has been open for unsafe languages, to our knowledge. Various known results on higher-order grammars can be obtained as almost immediate corollaries of our result.
intersection types
higher-order grammars
111:1-111:13
Regular Paper
Kazuyuki
Asada
Kazuyuki Asada
Naoki
Kobayashi
Naoki Kobayashi
10.4230/LIPIcs.ICALP.2016.111
Klaus Aehlig, Jolie G. de Miranda, and C.-H. Luke Ong. Safety is not a restriction at level 2 for string languages. In Proceedings of FoSSaCS 2005, volume 3441 of LNCS, pages 490-504. Springer, 2005.
Kazuyuki Asada and Naoki Kobayashi. On word and frontier languages of unsafe higher-order grammars. CoRR, abs/1604.01595, 2016.
William Blum and C.-H. Luke Ong. The safe lambda calculus. Logical Methods in Computer Science, 5(1), 2009.
Lorenzo Clemente, Pawel Parys, Sylvain Salvati, and Igor Walukiewicz. The diagonal problem for higher-order recusion schemes is decidable. In Proceedings of LICS 2016, 2016.
Wojciech Czerwinski and Wim Martens. A note on decidable separability by piecewise testable languages. CoRR, abs/1410.1042, 2014. URL: http://arxiv.org/abs/1410.1042.
http://arxiv.org/abs/1410.1042
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Matthew Hague, Jonathan Kochems, and C.-H. Luke Ong. Unboundedness and downward closures of higher-order pushdown automata. In Proceedings of POPL 2016, pages 151-163, 2016. URL: http://dx.doi.org/10.1145/2837614.2837627.
http://dx.doi.org/10.1145/2837614.2837627
Teodor Knapik, Damian Niwinski, and Pawel Urzyczyn. Deciding monadic theories of hyperalgebraic trees. In TLCA 2001, volume 2044 of LNCS, pages 253-267. Springer, 2001.
Naoki Kobayashi. Model checking higher-order programs. Journal of the ACM, 60(3), 2013.
Naoki Kobayashi. Pumping by typing. In Proceedings of LICS 2013, pages 398-407. IEEE Computer Society, 2013.
Naoki Kobayashi, Kazuhiro Inaba, and Takeshi Tsukada. Unsafe order-2 tree languages are context-sensitive. In Proceedings of FoSSaCS 2014, volume 8412 of LNCS, pages 149-163. Springer, 2014.
Naoki Kobayashi, Kazutaka Matsuda, Ayumi Shinohara, and Kazuya Yaguchi. Functional programs as compressed data. Higher-Order and Symbolic Computation, 2013.
Naoki Kobayashi and C.-H. Luke Ong. A type system equivalent to the modal mu-calculus model checking of higher-order recursion schemes. In Proceedings of LICS 2009, pages 179-188. IEEE Computer Society Press, 2009.
Gregory M. Kobele and Sylvain Salvati. The IO and OI hierarchies revisited. Inf. Comput., 243:205-221, 2015.
C.-H. Luke Ong. On model-checking trees generated by higher-order recursion schemes. In LICS 2006, pages 81-90. IEEE Computer Society Press, 2006.
Pawel Parys. How many numbers can a lambda-term contain? In Proceedings of FLOPS 2014, volume 8475 of LNCS, pages 302-318. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-07151-0_19.
http://dx.doi.org/10.1007/978-3-319-07151-0_19
Sylvain Salvati and Igor Walukiewicz. Typing weak MSOL properties. In Andrew M. Pitts, editor, Proceedings of FoSSaCS 2015, volume 9034 of LNCS, pages 343-357. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-46678-0_22.
http://dx.doi.org/10.1007/978-3-662-46678-0_22
Takeshi Tsukada and C.-H. Luke Ong. Compositional higher-order model checking via ω-regular games over böhm trees. In Proceedings of CSL-LICS'14, pages 78:1-78:10. ACM, 2014. URL: http://dx.doi.org/10.1145/2603088.2603133.
http://dx.doi.org/10.1145/2603088.2603133
Georg Zetzsche. An approach to computing downward closures. In Proceedings of ICALP 2015, volume 9135 of LNCS, pages 440-451. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47666-6_35.
http://dx.doi.org/10.1007/978-3-662-47666-6_35
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The Schützenberger Product for Syntactic S