46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), ICALP 2019, July 9-12, 2019, Patras, Greece
ICALP 2019
July 9-12, 2019
Patras, Greece
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
Christel
Baier
Christel Baier
TU Dresden, Germany
Ioannis
Chatzigiannakis
Ioannis Chatzigiannakis
Sapienza University of Rome, Italy
Paola
Flocchini
Paola Flocchini
University of Ottawa, Canada
Stefano
Leonardi
Stefano Leonardi
Sapienza University of Rome, Italy
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
132
2019
978-3-95977-109-2
https://www.dagstuhl.de/dagpub/978-3-95977-109-2
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter
Table of Contents
Preface
Conference Organization
Theory of computation
0:i-0:xxxviii
Front Matter
Christel
Baier
Christel Baier
TU Dresden, Germany
Ioannis
Chatzigiannakis
Ioannis Chatzigiannakis
Sapienza University of Rome, Italy
Paola
Flocchini
Paola Flocchini
University of Ottawa, Canada
Stefano
Leonardi
Stefano Leonardi
Sapienza University of Rome, Italy
10.4230/LIPIcs.ICALP.2019.0
Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Auction Design under Interdependent Values (Invited Talk)
We study combinatorial auctions with interdependent valuations. In such settings, every agent has a private signal, and every agent has a valuation function that depends on the private signals of all the agents. Interdependent valuations capture settings where agents lack information to determine their own valuations. Examples include auctions for artwork or oil drilling rights. For single item auctions and assume some restrictive conditions (the so-called single-crossing condition), full welfare can be achieved. However, in general, there are strong impossibility results on welfare maximization in the interdependent setting. This is in contrast to settings where agents are aware of their own valuations, where the optimal welfare can always be obtained by an incentive compatible mechanism.
Motivated by these impossibility results, we study welfare maximization for interdependent valuations through the lens of approximation. We introduce two valuation properties that enable positive results. The first is a relaxed, parameterized version of single crossing; the second is a submodularity condition over the signals. We obtain a host of approximation guarantees under these two notions for various scenarios.
Related publications: [Alon Eden et al., 2018; Alon Eden et al., 2019]
Combinatorial auctions
Interdependent values
Welfare approximation
Theory of computation~Algorithmic game theory and mechanism design
1:1-1:1
Invited Talk
Michal
Feldman
Michal Feldman
Blavatnik School of Computer Science, Tel-Aviv University, Israel
10.4230/LIPIcs.ICALP.2019.1
Alon Eden, Michal Feldman, Amos Fiat, and Kira Goldner. Interdependent Values without Single-Crossing. In Éva Tardos, Edith Elkind, and Rakesh Vohra, editors, Proceedings of the 2018 ACM Conference on Economics and Computation, Ithaca, NY, USA, June 18-22, 2018, page 369. ACM, 2018. URL: http://dx.doi.org/10.1145/3219166.3219173.
http://dx.doi.org/10.1145/3219166.3219173
Alon Eden, Michal Feldman, Amos Fiat, Kira Goldner, and Anna R. Karlin. Combinatorial Auctions with Interdependent Valuations: SOS to the Rescue. In Proceedings of the 2019 ACM Conference on Economics and Computation, Phoenix, AZ, June 24-28, 2019. ACM, 2019.
Michal Feldman
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Symmetry and Similarity (Invited Talk)
Deciding if two graphs are isomorphic, or equivalently, computing the symmetries of a graph, is a fundamental algorithmic problem. It has many interesting applications, and it is one of the few natural problems in the class NP whose complexity status is still unresolved. Three years ago, Babai (STOC 2016) gave a quasi-polynomial time isomorphism algorithm. Despite of this breakthrough, the question for a polynomial algorithm remains wide open.
Related to the isomorphism problem is the problem of determining the similarity between graphs. Variations of this problems are known as robust graph isomorphism or graph matching (the latter in the machine learning and computer vision literature). This problem is significantly harder than the isomorphism problem, both from a complexity theoretical and from a practical point of view, but for many applications it is the more relevant problem.
My talk will be a survey of recent progress on the isomorphism and on the similarity problem. I will focus on generic algorithmic strategies (as opposed to algorithms tailored towards specific graph classes) that have proved to be useful and interesting in various context, both theoretical and practical.
Graph Isomorphism
Graph Similarity
Graph Matching
Mathematics of computing~Graph theory
Theory of computation~Graph algorithms analysis
2:1-2:1
Invited Talk
Martin
Grohe
Martin Grohe
RWTH Aachen University, Lehrstuhl Informatik 7, Ahornstr. 55, 52074 Aachen, Germany
http://www.lics.rwth-aachen.de/~grohe
https://orcid.org/0000-0002-0292-9142
10.4230/LIPIcs.ICALP.2019.2
Martin Grohe
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Approximately Good and Modern Matchings (Invited Talk)
The matching problem is one of our favorite benchmark problems. Work on it has contributed to the development of many core concepts of computer science, including the equation of efficiency with polynomial time computation in the groundbreaking work by Edmonds in 1965.
However, half a century later, we still do not have full understanding of the complexity of the matching problem in several models of computation such as parallel, online, and streaming algorithms. In this talk we survey some of the major challenges and report some recent progress.
Algorithms
Matchings
Computational Complexity
Theory of computation~Design and analysis of algorithms
3:1-3:1
Invited Talk
Ola
Svensson
Ola Svensson
EPFL, Lausanne, Switzerland
https://theory.epfl.ch/osven/
https://orcid.org/0000-0003-2997-1372
10.4230/LIPIcs.ICALP.2019.3
Ola Svensson
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Automata Learning and Galois Connections (Invited Talk)
Automata learning is emerging as an effective technique for obtaining state machine models of software and hardware systems. I will present an overview of recent work in which we used active automata learning to find standard violations and security vulnerabilities in implementations of network protocols such as TCP and SSH. Also, I will discuss applications of automata learning to support refactoring of legacy control software and identifying job patterns in manufacturing systems. As a guiding theme in my presentation, I will show how Galois connections (adjunctions) help us to scale the application of learning algorithms to practical problems.
Automaton Learning
Model Learning
Protocol Verification
Applications of Automata Learning
Galois Connections
Theory of computation~Active learning
Theory of computation~Regular languages
Security and privacy~Logic and verification
Software and its engineering~Model-driven software engineering
Software and its engineering~Software testing and debugging
4:1-4:1
Invited Talk
NWO TOP project 612.001.852 Grey-box learning of Interfaces for Refactoring Legacy Software (GIRLS)
Frits
Vaandrager
Frits Vaandrager
Department of Software Science, Radboud University, The Netherlands
http://www.cs.ru.nl/~fvaan/
https://orcid.org/0000-0003-3955-1910
10.4230/LIPIcs.ICALP.2019.4
Frits Vaandrager
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Fixed Point Computation Problems and Facets of Complexity (Invited Talk)
Many problems from a wide variety of areas can be formulated mathematically as the problem of computing a fixed point of a suitable given multivariate function. Examples include a variety of problems from game theory, economics, optimization, stochastic analysis, verification, and others. In some problems there is a unique fixed point (for example if the function is a contraction); in others there may be multiple fixed points and any one of them is an acceptable solution; while in other cases the desired object is a specific fixed point (for example the least fixed point or greatest fixed point of a monotone function). In this talk we will discuss several types of fixed point computation problems, their complexity, and some of the common themes that have emerged: classes of problems for which there are efficient algorithms, and other classes for which there seem to be serious obstacles.
Fixed Point
Polynomial Time Algorithm
Computational Complexity
Theory of computation~Complexity theory and logic
5:1-5:1
Invited Talk
Mihalis
Yannakakis
Mihalis Yannakakis
Department of Computer Science, Columbia University, 455 Computer Science Building, 1214 Amsterdam Avenue, New York, NY 10027, USA
Supported by NSF Grants CCF-1703925, CCF-1763970.
10.4230/LIPIcs.ICALP.2019.5
Mihalis Yannakakis
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Complexity-Theoretic Limitations on Blind Delegated Quantum Computation
Blind delegation protocols allow a client to delegate a computation to a server so that the server learns nothing about the input to the computation apart from its size. For the specific case of quantum computation we know, from work over the past decade, that blind delegation protocols can achieve information-theoretic security (provided the client and the server exchange some amount of quantum information). In this paper we prove, provided certain complexity-theoretic conjectures are true, that the power of information-theoretically secure blind delegation protocols for quantum computation (ITS-BQC protocols) is in a number of ways constrained.
In the first part of our paper we provide some indication that ITS-BQC protocols for delegating polynomial-time quantum computations in which the client and the server interact only classically are unlikely to exist. We first show that having such a protocol in which the client and the server exchange O(n^d) bits of communication, implies that BQP subset MA/O(n^d). We conjecture that this containment is unlikely by proving that there exists an oracle relative to which BQP not subset MA/O(n^d). We then show that if an ITS-BQC protocol exists in which the client and the server interact only classically and which allows the client to delegate quantum sampling problems to the server (such as BosonSampling) then there exist non-uniform circuits of size 2^{n - Omega(n/log(n))}, making polynomially-sized queries to an NP^{NP} oracle, for computing the permanent of an n x n matrix.
The second part of our paper concerns ITS-BQC protocols in which the client and the server engage in one round of quantum communication and then exchange polynomially many classical messages. First, we provide a complexity-theoretic upper bound on the types of functions that could be delegated in such a protocol by showing that they must be contained in QCMA/qpoly cap coQCMA/qpoly. Then, we show that having such a protocol for delegating NP-hard functions implies coNP^{NP^{NP}} subseteq NP^{NP^{PromiseQMA}}.
Quantum cryptography
Complexity theory
Delegated quantum computation
Computing on encrypted data
Theory of computation~Quantum computation theory
Theory of computation~Quantum complexity theory
6:1-6:13
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1704.08482.
We would like to thank the following people for useful discussions and comments: Petros Wallden, Matty J Hoban, Kousha Etessami, Marc Kaplan, Ronald de Wolf, Urmila Mahadev, Umesh Vazirani, Chris Heunen, Thomas Vidick, Ashley Montanaro, Tina Zhang and Pia Kullik. A.G. is in particular grateful to Petros Wallden and Matty J Hoban for their patience and for their help in clarifying several technical issues.
Scott
Aaronson
Scott Aaronson
Department of Computer Science, University of Texas at Austin, USA
Vannevar Bush Fellowship from the US Department of Defense.
Alexandru
Cojocaru
Alexandru Cojocaru
School of Informatics, University of Edinburgh, UK
EPSRC grants EP/N003829/1, EP/M013243/1.
Alexandru
Gheorghiu
Alexandru Gheorghiu
Department of Computing and Mathematical Sciences, California Institute of Technology, USA
School of Informatics, University of Edinburgh, UK
https://orcid.org/0000-0001-6225-7168
MURI Grant FA9550-18-1-0161 and the IQIM, an NSF Physics Frontiers Center (NSF Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028).
Elham
Kashefi
Elham Kashefi
School of Informatics, University of Edinburgh, UK
CNRS LIP6, Université Pierre et Marie Curie, Paris, France
EPSRC grants EP/N003829/1, EP/M013243/1.
10.4230/LIPIcs.ICALP.2019.6
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http://dx.doi.org/10.1103/PhysRevA.90.050303
Scott Aaronson, Alexandru Cojocaru, Alexandru Gheorghiu, and Elham Kashefi
Creative Commons Attribution 3.0 Unported license
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Faster Algorithms for All-Pairs Bounded Min-Cuts
The All-Pairs Min-Cut problem (aka All-Pairs Max-Flow) asks to compute a minimum s-t cut (or just its value) for all pairs of vertices s,t. We study this problem in directed graphs with unit edge/vertex capacities (corresponding to edge/vertex connectivity). Our focus is on the k-bounded case, where the algorithm has to find all pairs with min-cut value less than k, and report only those. The most basic case k=1 is the Transitive Closure (TC) problem, which can be solved in graphs with n vertices and m edges in time O(mn) combinatorially, and in time O(n^{omega}) where omega<2.38 is the matrix-multiplication exponent. These time bounds are conjectured to be optimal.
We present new algorithms and conditional lower bounds that advance the frontier for larger k, as follows:
- A randomized algorithm for vertex capacities that runs in time {O}((nk)^{omega}). This is only a factor k^omega away from the TC bound, and nearly matches it for all k=n^{o(1)}.
- Two deterministic algorithms for edge capacities (which is more general) that work in DAGs and further reports a minimum cut for each pair. The first algorithm is combinatorial (does not involve matrix multiplication) and runs in time {O}(2^{{O}(k^2)}* mn). The second algorithm can be faster on dense DAGs and runs in time {O}((k log n)^{4^{k+o(k)}}* n^{omega}). Previously, Georgiadis et al. [ICALP 2017], could match the TC bound (up to n^{o(1)} factors) only when k=2, and now our two algorithms match it for all k=o(sqrt{log n}) and k=o(log log n).
- The first super-cubic lower bound of n^{omega-1-o(1)} k^2 time under the 4-Clique conjecture, which holds even in the simplest case of DAGs with unit vertex capacities. It improves on the previous (SETH-based) lower bounds even in the unbounded setting k=n. For combinatorial algorithms, our reduction implies an n^{2-o(1)} k^2 conditional lower bound. Thus, we identify new settings where the complexity of the problem is (conditionally) higher than that of TC.
Our three sets of results are obtained via different techniques. The first one adapts the network coding method of Cheung, Lau, and Leung [SICOMP 2013] to vertex-capacitated digraphs. The second set exploits new insights on the structure of latest cuts together with suitable algebraic tools. The lower bounds arise from a novel reduction of a different structure than the SETH-based constructions.
All-pairs min-cut
k-reachability
network coding
Directed graphs
fine-grained complexity
Theory of computation~Graph algorithms analysis
Theory of computation~Network flows
7:1-7:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1807.05803.
We thank Paweł Gawrychowski, Mohsen Ghaffari, Atri Rudra and Peter Widmayer for the valuable discussions on this problem.
Amir
Abboud
Amir Abboud
IBM Almaden Research Center, California, USA
Loukas
Georgiadis
Loukas Georgiadis
University of Ioannina, Greece
Giuseppe F.
Italiano
Giuseppe F. Italiano
LUISS University, Rome, Italy
Robert
Krauthgamer
Robert Krauthgamer
Weizmann Institute of Science, Israel
Work supported in part by ONR Award N00014-18-1-2364, Israel Science Foundation grant #1086/18, a Minerva Foundation grant, and a Google Faculty Research Award.
Nikos
Parotsidis
Nikos Parotsidis
University of Copenhagen, Denmark
The author is supported by Grant Number 16582, Basic Algorithms Research Copenhagen (BARC), from the VILLUM Foundation.
Ohad
Trabelsi
Ohad Trabelsi
Weizmann Institute of Science, Israel
Work partly done at IBM Almaden Research Center, USA.
Przemysław
Uznański
Przemysław Uznański
University of Wrocław, Poland
Daniel
Wolleb-Graf
Daniel Wolleb-Graf
ETH Zürich, Switzerland
10.4230/LIPIcs.ICALP.2019.7
A. Abboud and V. V. Williams. Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems. In FOCS, pages 434-443, October 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.53.
http://dx.doi.org/10.1109/FOCS.2014.53
Amir Abboud, Arturs Backurs, Karl Bringmann, and Marvin Künnemann. Fine-Grained Complexity of Analyzing Compressed Data: Quantifying Improvements over Decompress-and-Solve. In FOCS, pages 192-203, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.12.
http://dx.doi.org/10.1109/FOCS.2017.12
Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. If the Current Clique Algorithms are Optimal, So is Valiant’s Parser. In FOCS, pages 98-117, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.16.
http://dx.doi.org/10.1109/FOCS.2015.16
Amir Abboud, Robert Krauthgamer, and Ohad Trabelsi. New Algorithms and Lower Bounds for All-Pairs Max-Flow in Undirected Graphs. CoRR, abs/1901.01412, 2019. URL: http://arxiv.org/abs/1901.01412.
http://arxiv.org/abs/1901.01412
Amir Abboud, Virginia Vassilevska Williams, and Huacheng Yu. Matching Triangles and Basing Hardness on an Extremely Popular Conjecture. SIAM J. Comput., 47(3):1098-1122, 2018. URL: http://dx.doi.org/10.1137/15M1050987.
http://dx.doi.org/10.1137/15M1050987
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http://dx.doi.org/10.1109/FOCS.2017.36
Karl Bringmann and Philip Wellnitz. Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars. In CPM, pages 12:1-12:14, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CPM.2017.12.
http://dx.doi.org/10.4230/LIPIcs.CPM.2017.12
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Amir Abboud, Loukas Georgiadis, Giuseppe F. Italiano, Robert Krauthgamer, Nikos Parotsidis, Ohad Trabelsi, Przemysław Uznański, and Daniel Wolleb-Graf
Creative Commons Attribution 3.0 Unported license
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Fine-Grained Reductions and Quantum Speedups for Dynamic Programming
This paper points at a connection between certain (classical) fine-grained reductions and the question: Do quantum algorithms offer an advantage for problems whose (classical) best solution is via dynamic programming?
A remarkable recent result of Ambainis et al. [SODA 2019] indicates that the answer is positive for some fundamental problems such as Set-Cover and Travelling Salesman. They design a quantum O^*(1.728^n) time algorithm whereas the dynamic programming O^*(2^n) time algorithms are conjectured to be classically optimal. In this paper, fine-grained reductions are extracted from their algorithms giving the first lower bounds for problems in P that are based on the intriguing Set-Cover Conjecture (SeCoCo) of Cygan et al. [CCC 2010].
In particular, the SeCoCo implies:
- a super-linear Omega(n^{1.08}) lower bound for 3-SUM on n integers,
- an Omega(n^{k/(c_k)-epsilon}) lower bound for k-SUM on n integers and k-Clique on n-node graphs, for any integer k >= 3, where c_k <= log_2{k}+1.4427.
While far from being tight, these lower bounds are significantly stronger than what is known to follow from the Strong Exponential Time Hypothesis (SETH); the well-known n^{Omega(k)} ETH-based lower bounds for k-Clique and k-SUM are vacuous when k is constant.
Going in the opposite direction, this paper observes that some "sequential" problems with previously known fine-grained reductions to a "parallelizable" core also enjoy quantum speedups over their classical dynamic programming solutions. Examples include RNA Folding and Least-Weight Subsequence.
Fine-Grained Complexity
Set-Cover
3-SUM
k-Clique
k-SUM
Dynamic Programming
Quantum Algorithms
Theory of computation~Problems, reductions and completeness
8:1-8:13
Track A: Algorithms, Complexity and Games
We acknowledge the support of the Quantum Computing Sciences program of the U.S. Air Force, Office of Scientific Research, administered through Air Force Research Laboratory contract FA8750-18-C-0098.
We thank Karl Bringmann and the anonymous reviewers for helpful feedback.
Amir
Abboud
Amir Abboud
IBM Almaden Research Center, San Jose, California, USA
10.4230/LIPIcs.ICALP.2019.8
Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight Hardness Results for LCS and other Sequence Similarity Measures. In Proc. of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 59-78, 2015.
Amir Abboud, Karl Bringmann, Danny Hermelin, and Dvir Shabtay. SETH-Based Lower Bounds for Subset Sum and Bicriteria Path. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 41-57, 2019.
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Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity.
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http://dx.doi.org/10.1145/3186893
Amir Abboud
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Geometric Multicut
We study the following separation problem: Given a collection of colored objects in the plane, compute a shortest "fence" F, i.e., a union of curves of minimum total length, that separates every two objects of different colors. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem as GEOMETRIC k-CUT, where k is the number of different colors, as it can be seen as a geometric analogue to the well-studied multicut problem on graphs. We first give an O(n^4 log^3 n)-time algorithm that computes an optimal fence for the case where the input consists of polygons of two colors and n corners in total. We then show that the problem is NP-hard for the case of three colors. Finally, we give a (2-4/3k)-approximation algorithm.
multicut
clustering
Steiner tree
Theory of computation~Design and analysis of algorithms
Theory of computation~Computational geometry
9:1-9:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1902.04045.
This work was initiated at the workshop on Fixed-Parameter Computational Geometry at the Lorentz Center in Leiden in May 2018. We thank the organizers and the Lorentz Center for a nice workshop and Michael Hoffmann for useful discussions during the workshop.
Mikkel
Abrahamsen
Mikkel Abrahamsen
BARC, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen, Denmark
https://orcid.org/0000-0003-2734-4690
Supported by the Innovation Fund Denmark through the DABAI project. MA is also a part of BARC, Basic Algorithms Research Copenhagen, supported by the VILLUM Foundation grant 16582.
Panos
Giannopoulos
Panos Giannopoulos
giCenter, Department of Computer Science, City University of London, EC1V 0HB, London, UK
Maarten
Löffler
Maarten Löffler
Department of Information and Computing Sciences, Utrecht University, The Netherlands
Partially supported by the Netherlands Organisation for Scientific Research (NWO); 614.001.504.
Günter
Rote
Günter Rote
Institut für Informatik, Freie Universität Berlin, Takustraße 9, 14195 Berlin, Germany
https://orcid.org/0000-0002-0351-5945
10.4230/LIPIcs.ICALP.2019.9
Mikkel Abrahamsen, Anna Adamaszek, Karl Bringmann, Vincent Cohen-Addad, Mehran Mehr, Eva Rotenberg, Alan Roytman, and Mikkel Thorup. Fast fencing. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC 2018), pages 564-573, 2018. URL: http://dx.doi.org/10.1145/3188745.3188878.
http://dx.doi.org/10.1145/3188745.3188878
Mikkel Abrahamsen, Panos Giannopoulos, Maarten Löffler, and Günter Rote. Geometric multicut, 2019. URL: http://arxiv.org/abs/1902.04045.
http://arxiv.org/abs/1902.04045
Oswin Aichholzer and Franz Aurenhammer. Straight skeletons for general polygonal figures in the plane. In Jin-Yi Cai and Chak Kuen Wong, editors, Computing and Combinatorics (COCOON 1996), volume 1090 of Lecture Notes in Computer Science, pages 117-126. Springer-Verlag, 1996. URL: http://dx.doi.org/10.1007/3-540-61332-3_144.
http://dx.doi.org/10.1007/3-540-61332-3_144
Oswin Aichholzer and Franz Aurenhammer. Straight skeletons for general polygonal figures in the plane. In A. Samoilenko, editor, Voronoi’s Impact on Modern Sciences, Vol. II, volume 21 of Proceedings of the Institute of Mathematics of the National Academy of Sciences of Ukraine, pages 7-21. Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiew, 1998.
Glencora Borradaile, Philip N. Klein, Shay Mozes, Yahav Nussbaum, and Christian Wulff-Nilsen. Multiple-source multiple-sink maximum flow in directed planar graphs in near-linear time. SIAM Journal on Computing, 46(4):1280-1303, 2017. URL: http://dx.doi.org/10.1137/15M1042929.
http://dx.doi.org/10.1137/15M1042929
Gruia Călinescu, Howard Karloff, and Yuval Rabani. An improved approximation algorithm for Multiway Cut. Journal of Computer and System Sciences, 60(3):564-574, 2000. URL: http://dx.doi.org/10.1006/jcss.1999.1687.
http://dx.doi.org/10.1006/jcss.1999.1687
Bernard Chazelle and Herbert Edelsbrunner. An optimal algorithm for intersecting line segments in the plane. J. ACM, 39(1):1-54, 1992. URL: http://dx.doi.org/10.1145/147508.147511.
http://dx.doi.org/10.1145/147508.147511
Elias Dahlhaus, David S. Johnson, Christos H. Papadimitriou, Paul D. Seymour, and Mihalis Yannakakis. The complexity of multiterminal cuts. SIAM Journal on Computing, 23(4):864-894, 1994. URL: http://dx.doi.org/10.1137/S0097539792225297.
http://dx.doi.org/10.1137/S0097539792225297
Paweł Gawrychowski and Adam Karczmarz. Improved bounds for shortest paths in dense distance graphs. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018), volume 107 of Leibniz International Proceedings in Informatics (LIPIcs), pages 61:1-61:15, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.61.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.61
Edgar N. Gilbert and Henry O. Pollak. Steiner minimal trees. SIAM Journal on Applied Mathematics, 16(1):1-29, 1968. URL: http://dx.doi.org/10.1137/0116001.
http://dx.doi.org/10.1137/0116001
Dorothy M. Greig, Bruce T. Porteous, and Allan H. Seheult. Exact maximum a posteriori estimation for binary images. Journal of the Royal Statistical Society. Series B (Methodological), 51(2):271-279, 1989. URL: http://dx.doi.org/10.1111/j.2517-6161.1989.tb01764.x.
http://dx.doi.org/10.1111/j.2517-6161.1989.tb01764.x
David R. Karger, Philip Klein, Cliff Stein, Mikkel Thorup, and Neal E. Young. Rounding algorithms for a geometric embedding of minimum multiway cut. Mathematics of Operations Research, 29(3):436-461, 2004. URL: http://dx.doi.org/10.1287/moor.1030.0086.
http://dx.doi.org/10.1287/moor.1030.0086
Wolfgang Mulzer and Günter Rote. Minimum-weight triangulation is NP-hard. Journal of the ACM, 55:Article 11, 29 pp., May 2008. URL: http://dx.doi.org/10.1145/1346330.1346336.
http://dx.doi.org/10.1145/1346330.1346336
Mikkel Abrahamsen, Panos Giannopoulos, Maarten Löffler, and Günter Rote
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Lower Bounds for Multiplication via Network Coding
Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, very recently proved by Harvey and van der Hoeven (2019), shows that two n-bit numbers can be multiplied via a boolean circuit of size O(n lg n). In this work, we prove that if a central conjecture in the area of network coding is true, then any constant degree boolean circuit for multiplication must have size Omega(n lg n), thus almost completely settling the complexity of multiplication circuits. We additionally revisit classic conjectures in circuit complexity, due to Valiant, and show that the network coding conjecture also implies one of Valiant’s conjectures.
Circuit Complexity
Circuit Lower Bounds
Multiplication
Network Coding
Fine-Grained Complexity
Theory of computation
Theory of computation~Circuit complexity
10:1-10:12
Track A: Algorithms, Complexity and Games
Peyman Afshani is supported by DFF (Det Frie Forskningsraad) of Danish Council for Independent Research under grant ID DFF-7014-00404.
Casper Benjamin Freksen and Lior Kamma are supported by Villum Young Investigator Grant 13163.
Kasper Green Larsen is supported by Villum Young Investigator Grant 13163 and an AUFF Starting Grant.
Peyman
Afshani
Peyman Afshani
Computer Science Department, Aarhus University, Denmark
Casper Benjamin
Freksen
Casper Benjamin Freksen
Computer Science Department, Aarhus University, Denmark
Lior
Kamma
Lior Kamma
Computer Science Department, Aarhus University, Denmark
Kasper Green
Larsen
Kasper Green Larsen
Computer Science Department, Aarhus University, Denmark
10.4230/LIPIcs.ICALP.2019.10
Micah Adler, Nicholas J. A. Harvey, Kamal Jain, Robert Kleinberg, and April Rasala Lehman. On the Capacity of Information Networks. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, SODA '06, pages 241-250. Society for Industrial and Applied Mathematics, 2006. URL: http://dl.acm.org/citation.cfm?id=1109557.1109585.
http://dl.acm.org/citation.cfm?id=1109557.1109585
Mark Braverman, Sumegha Garg, and Ariel Schvartzman. Coding in Undirected Graphs Is Either Very Helpful or Not Helpful at All. In 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, January 9-11, 2017, Berkeley, CA, USA, pages 18:1-18:18, 2017.
Raphaël Clifford and Markus Jalsenius. Lower Bounds for Online Integer Multiplication and Convolution in the Cell-Probe Model. In Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Zurich, Switzerland, July 4-8, 2011, Proceedings, Part I, pages 593-604, 2011.
Stephen Arthur Cook. On the minimum computation time of functions. PhD thesis, Harvard University, 1966.
Alireza Farhadi, MohammadTaghi Hajiaghayi, Kasper Green Larsen, and Elaine Shi. Lower Bounds for External Memory Integer Sorting via Network Coding. In Proceedings of the 52st Symposium on Theory of Computing, STOC 2019, 2019. To appear.
Martin Fürer. Faster Integer Multiplication. SIAM Journal on Computing, 39(3):979-1005, 2009. URL: http://dx.doi.org/10.1137/070711761.
http://dx.doi.org/10.1137/070711761
D. Harvey and J. van der Hoeven. Integer multiplication in time O(n log n). Technical report, HAL, 2019. ěrb|http://hal.archives-ouvertes.fr/hal-02070778|.
David Harvey and Joris van der Hoeven. Faster integer multiplication using short lattice vectors. CoRR, 2018. URL: http://arxiv.org/abs/1802.07932.
http://arxiv.org/abs/1802.07932
Anatoly Alekseevich Karatsuba and Yurii Petrovich Ofman. Multiplication of Many-Digital Numbers by Automatic Computers. Proceedings of the USSR Academy of Sciences, 145:293-294, 1962.
Zongpeng Li and Baochun Li. Network Coding: The Case of Multiple Unicast Sessions. In Proceedings of the 42nd Annual Allerton Conference on Communication, Control, and Computing, 2004.
Jacques Morgenstern. Note on a Lower Bound on the Linear Complexity of the Fast Fourier Transform. Journal of the ACM, 20(2):305-306, 1973. URL: http://dx.doi.org/10.1145/321752.321761.
http://dx.doi.org/10.1145/321752.321761
Stephen Ponzio. A Lower Bound for Integer Multiplication with Read-Once Branching Programs. SIAM J. Comput., 28(3):798-815, 1998.
Søren Riis. Information flows, graphs and their guessing numbers. The Electronic Journal of Combinatorics, 14(1), 2007.
Arnold Schönhage and Volker Strassen. Schnelle Multiplikation großer Zahlen. Computing, 7(3):281-292, September 1971. URL: http://dx.doi.org/10.1007/BF02242355.
http://dx.doi.org/10.1007/BF02242355
Andrei Leonovich Toom. The complexity of a scheme of functional elements realizing the multiplication of integers. Proceedings of the USSR Academy of Sciences, 150(3):496-498, 1963.
Leslie G. Valiant. Graph-theoretic arguments in low-level complexity. In Mathematical Foundations of Computer Science 1977, pages 162-176, 1977.
Leslie G. Valiant. Why is Boolean Complexity Theory Difficult? In Proceedings of the London Mathematical Society Symposium on Boolean Function Complexity, pages 84-94, 1992.
Peyman Afshani, Casper Freksen, Lior Kamma, and Kasper G. Larsen
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Path Contraction Faster Than 2^n
A graph G is contractible to a graph H if there is a set X subseteq E(G), such that G/X is isomorphic to H. Here, G/X is the graph obtained from G by contracting all the edges in X. For a family of graphs F, the F-Contraction problem takes as input a graph G on n vertices, and the objective is to output the largest integer t, such that G is contractible to a graph H in F, where |V(H)|=t. When F is the family of paths, then the corresponding F-Contraction problem is called Path Contraction. The problem Path Contraction admits a simple algorithm running in time 2^n * n^{O(1)}. In spite of the deceptive simplicity of the problem, beating the 2^n * n^{O(1)} bound for Path Contraction seems quite challenging. In this paper, we design an exact exponential time algorithm for Path Contraction that runs in time 1.99987^n * n^{O(1)}. We also define a problem called 3-Disjoint Connected Subgraphs, and design an algorithm for it that runs in time 1.88^n * n^{O(1)}. The above algorithm is used as a sub-routine in our algorithm for Path Contraction.
path contraction
exact exponential time algorithms
graph algorithms
enumerating connected sets
3-disjoint connected subgraphs
Mathematics of computing~Graph algorithms
Theory of computation~Graph algorithms analysis
Theory of computation~Parameterized complexity and exact algorithms
11:1-11:13
Track A: Algorithms, Complexity and Games
Akanksha
Agrawal
Akanksha Agrawal
Ben-Gurion University of the Negev, Beersheba, Israel
During some part of the work, the author was supported by ERC Consolidator Grant SYSTEMATIC-GRAPH (No. 725978).
Fedor V.
Fomin
Fedor V. Fomin
University of Bergen, Bergen, Norway
Daniel
Lokshtanov
Daniel Lokshtanov
University of California Santa Barbara, Santa Barbara, California
Saket
Saurabh
Saket Saurabh
Institute of Mathematical Sciences, HBNI and UMI ReLaX Chennai, India
University of Bergen, Bergen, Norway
This work is supported by the European Research Council (ERC) via grant LOPPRE, reference no. 819416.
Prafullkumar
Tale
Prafullkumar Tale
Institute of Mathematical Sciences, HBNI, Chennai, India
10.4230/LIPIcs.ICALP.2019.11
Takao Asano and Tomio Hirata. Edge-Contraction Problems. Journal of Computer and System Sciences, 26(2):197-208, 1983.
Andreas Björklund. Determinant Sums for Undirected Hamiltonicity. SIAM J. Comput., 43(1):280-299, 2014.
Andries Evert Brouwer and Hendrik Jan Veldman. Contractibility and NP-completeness. Journal of Graph Theory, 11(1):71-79, 1987.
Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, and Magnus Wahlström. On Problems as Hard as CNF-SAT. ACM Trans. Algorithms, 12(3):41:1-41:24, 2016.
Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, and Jakub Onufry Wojtaszczyk. Solving the 2-disjoint connected subgraphs problem faster than 2ⁿ. Algorithmica, 70(2):195-207, 2014.
Konrad K Dabrowski and Daniël Paulusma. Contracting bipartite graphs to paths and cycles. Information Processing Letters, 127:37-42, 2017.
Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
Jiří Fiala, Marcin Kamiński, and Daniël Paulusma. A note on contracting claw-free graphs. Discrete Mathematics and Theoretical Computer Science, 15(2):223-232, 2013.
Fedor V. Fomin and Yngve Villanger. Treewidth computation and extremal combinatorics. Combinatorica, 32(3):289-308, 2012.
Pinar Heggernes, Pim van 't Hof, Benjamin Lévêque, and Christophe Paul. Contracting chordal graphs and bipartite graphs to paths and trees. Discrete Applied Mathematics, 164:444-449, 2014.
Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001.
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001.
Walter Kern and Daniel Paulusma. Contracting to a Longest Path in H-Free Graphs. arXiv preprint, 2018. URL: http://arxiv.org/abs/1810.01542.
http://arxiv.org/abs/1810.01542
Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Known Algorithms on Graphs of Bounded Treewidth Are Probably Optimal. ACM Trans. Algorithms, 14(2):13:1-13:30, 2018.
Robert Endre Tarjan and Anthony E. Trojanowski. Finding a Maximum Independent Set. SIAM J. Comput., 6(3):537-546, 1977.
Jan Arne Telle and Yngve Villanger. Connecting terminals and 2-disjoint connected subgraphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 418-428. Springer, 2013.
Pim van't Hof, Daniël Paulusma, and Gerhard J Woeginger. Partitioning graphs into connected parts. Theoretical Computer Science, 410(47-49):4834-4843, 2009.
Toshimasa Watanabe, Tadashi Ae, and Akira Nakamura. On the removal of forbidden graphs by edge-deletion or by edge-contraction. Discrete Applied Mathematics, 3(2):151-153, 1981.
Toshimasa Watanabe, Tadashi Ae, and Akira Nakamura. On the NP-hardness of Edge-Deletion and Contraction Problems. Discrete Applied Mathematics, 6(1):63-78, 1983.
Akanksha Agrawal, Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Prafullkumar Tale
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Deterministic Combinatorial Replacement Paths and Distance Sensitivity Oracles
In this work we derandomize two central results in graph algorithms, replacement paths and distance sensitivity oracles (DSOs) matching in both cases the running time of the randomized algorithms.
For the replacement paths problem, let G = (V,E) be a directed unweighted graph with n vertices and m edges and let P be a shortest path from s to t in G. The replacement paths problem is to find for every edge e in P the shortest path from s to t avoiding e. Roditty and Zwick [ICALP 2005] obtained a randomized algorithm with running time of O~(m sqrt{n}). Here we provide the first deterministic algorithm for this problem, with the same O~(m sqrt{n}) time. Due to matching conditional lower bounds of Williams et al. [FOCS 2010], our deterministic combinatorial algorithm for the replacement paths problem is optimal up to polylogarithmic factors (unless the long standing bound of O~(mn) for the combinatorial boolean matrix multiplication can be improved). This also implies a deterministic algorithm for the second simple shortest path problem in O~(m sqrt{n}) time, and a deterministic algorithm for the k-simple shortest paths problem in O~(k m sqrt{n}) time (for any integer constant k > 0).
For the problem of distance sensitivity oracles, let G = (V,E) be a directed graph with real-edge weights. An f-Sensitivity Distance Oracle (f-DSO) gets as input the graph G=(V,E) and a parameter f, preprocesses it into a data-structure, such that given a query (s,t,F) with s,t in V and F subseteq E cup V, |F| <=f being a set of at most f edges or vertices (failures), the query algorithm efficiently computes the distance from s to t in the graph G \ F (i.e., the distance from s to t in the graph G after removing from it the failing edges and vertices F).
For weighted graphs with real edge weights, Weimann and Yuster [FOCS 2010] presented several randomized f-DSOs. In particular, they presented a combinatorial f-DSO with O~(mn^{4-alpha}) preprocessing time and subquadratic O~(n^{2-2(1-alpha)/f}) query time, giving a tradeoff between preprocessing and query time for every value of 0 < alpha < 1. We derandomize this result and present a combinatorial deterministic f-DSO with the same asymptotic preprocessing and query time.
replacement paths
distance sensitivity oracles
derandomization
Theory of computation~Design and analysis of algorithms
Theory of computation~Dynamic graph algorithms
12:1-12:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at \cite{AlChCoArxiv}, https://arxiv.org/abs/1905.07483.
Noga
Alon
Noga Alon
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
Schools of Mathematics and Computer Science, Tel Aviv University, Tel Aviv 69978, Israel
Research supported in part by NSF grant DMS-1855464, ISF grant 281/17 and GIF grant G-1347-304.6/2016.
Shiri
Chechik
Shiri Chechik
Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel
Research supported in part by the Israel Science Foundation grant No. 1528/15 and the Blavatnik Fund.
Sarel
Cohen
Sarel Cohen
Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel
Research supported in part by the Israel Science Foundation grant No. 1528/15 and the Blavatnik Fund.
10.4230/LIPIcs.ICALP.2019.12
Udit Agarwal, Vijaya Ramachandran, Valerie King, and Matteo Pontecorvi. A Deterministic Distributed Algorithm for Exact Weighted All-Pairs Shortest Paths in Õ(n 3/2 ) Rounds. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC 2018, Egham, United Kingdom, July 23-27, 2018, pages 199-205, 2018.
N. Alon and J.H. Spencer. The Probabilistic Method. Fourth Edition. Wiley, 2016.
Noga Alon, Shiri Chechik, and Sarel Cohen. Deterministic combinatorial replacement paths and distance sensitivity oracles. CoRR, abs/1905.07483, 2019. URL: http://arxiv.org/abs/1905.07483.
http://arxiv.org/abs/1905.07483
Grey Ballard, James Demmel, Olga Holtz, and Oded Schwartz. Graph Expansion and Communication Costs of Fast Matrix Multiplication. J. ACM, 59(6):32:1-32:23, January 2013. URL: http://dx.doi.org/10.1145/2395116.2395121.
http://dx.doi.org/10.1145/2395116.2395121
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http://dx.doi.org/10.1145/2858788.2688513
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Aaron Bernstein and David Karger. Improved Distance Sensitivity Oracles via Random Sampling. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 34-43, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347087.
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Aaron Bernstein and David Karger. A Nearly Optimal Oracle for Avoiding Failed Vertices and Edges. In Proceedings of the Forty-first Annual ACM Symposium on Theory of Computing (STOC), pages 101-110, 2009. URL: http://dx.doi.org/10.1145/1536414.1536431.
http://dx.doi.org/10.1145/1536414.1536431
Shiri Chechik, Sarel Cohen, Amos Fiat, and Haim Kaplan. (1 + ε) approximate f-sensitive Distance Oracles. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, pages 1479-1496, 2017. URL: http://dl.acm.org/citation.cfm?id=3039686.3039782.
http://dl.acm.org/citation.cfm?id=3039686.3039782
Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. f-Sensitivity Distance Oracles and Routing Schemes. Algorithmica, 63(4):861-882, 2012. URL: http://dx.doi.org/10.1007/s00453-011-9543-0.
http://dx.doi.org/10.1007/s00453-011-9543-0
Camil Demetrescu and Mikkel Thorup. Oracles for Distances Avoiding a Link-failure. In Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 838-843, 2002. URL: http://dl.acm.org/citation.cfm?id=545381.545490.
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http://dx.doi.org/10.1137/120897146
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http://dx.doi.org/10.1016/0167-6377(89)90065-5
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http://dx.doi.org/10.1016/S0020-0190(00)00175-7
Liam Roditty and Uri Zwick. Replacement Paths and k Simple Shortest Paths in Unweighted Directed Graphs. In Automata, Languages and Programming, 32nd International Colloquium, ICALP, 2005, pages 249-260. See also ACM Trans. Algorithms, 8(4):33:1-11, 2012, 2005. URL: http://dx.doi.org/10.1007/11523468_21.
http://dx.doi.org/10.1007/11523468_21
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http://dx.doi.org/10.1109/FOCS.2010.68
Virginia Vassilevska Williams. Faster Replacement Paths. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, 23-25, 2011, pages 1337-1346, 2011. URL: http://dx.doi.org/10.1137/1.9781611973082.102.
http://dx.doi.org/10.1137/1.9781611973082.102
Virginia Vassilevska Williams and Ryan Williams. Subcubic Equivalences between Path, Matrix and Triangle Problems. In 51st Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, pages 645-654, 2010. URL: http://dx.doi.org/10.1109/FOCS.2010.67.
http://dx.doi.org/10.1109/FOCS.2010.67
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http://dx.doi.org/10.1287/mnsc.17.11.712
Noga Alon, Shiri Chechik, and Sarel Cohen
Creative Commons Attribution 3.0 Unported license
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Algorithms and Hardness for Diameter in Dynamic Graphs
The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported.
This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include:
- Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP.
- Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly (3/2+epsilon)-approximation to Diameter in directed or undirected n-vertex, m-edge graphs can be maintained decrementally in total time m^{1+o(1)}sqrt{n}/epsilon^2. This nearly matches the static 3/2-approximation algorithm for the problem that is known to be conditionally optimal.
fine-grained complexity
graph algorithms
dynamic algorithms
Mathematics of computing~Graph algorithms
13:1-13:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1811.12527.
The authors would like to thank Roei Tov for discussions.
Bertie
Ancona
Bertie Ancona
MIT, Cambridge, MA, USA
Monika
Henzinger
Monika Henzinger
University of Vienna, Austria
The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement No. 340506.
Liam
Roditty
Liam Roditty
Bar Ilan University, Ramat Gan, Israel
Virginia Vassilevska
Williams
Virginia Vassilevska Williams
MIT, Cambridge, MA, USA
Supported by an NSF CAREER Award, NSF Grants CCF-1417238, CCF-1528078 and CCF-1514339, a BSF Grant BSF:2012338 and a Sloan Research Fellowship.
Nicole
Wein
Nicole Wein
MIT, Cambridge, MA, USA
Supported by an NSF Graduate Fellowship and NSF Grant CCF-1514339.
10.4230/LIPIcs.ICALP.2019.13
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.
Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 377-391, 2016.
R. Albert, H. Jeong, and A.L. Barabasi. Diameter of the world wide web. Nature, 401:130-131, 1999.
N. Alon, R. Yuster, and U. Zwick. Finding and counting given length cycles. Algorithmica, 17:209-223, 1997.
Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, and Nicole Wein. Algorithms and Hardness for Diameter in Dynamic Graphs. arXiv preprint, 2018. URL: http://arxiv.org/abs/1811.12527.
http://arxiv.org/abs/1811.12527
Arturs Backurs, Liam Roditty, Gilad Segal, Virginia Vassilevska Williams, and Nicole Wein. Towards Tight Approximation Bounds for Graph Diameter and Eccentricities. In Proceedings of STOC'18, page to appear, 2018.
Massimo Cairo, Roberto Grossi, and Romeo Rizzi. New Bounds for Approximating Extremal Distances in Undirected Graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 363-376, 2016.
Shiri Chechik, Daniel H. Larkin, Liam Roditty, Grant Schoenebeck, Robert Endre Tarjan, and Virginia Vassilevska Williams. Better Approximation Algorithms for the Graph Diameter. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1041-1052, 2014.
Mina Dalirrooyfard, Thuy Duong Vuong, and Virginia Vassilevska Williams. Graph pattern detection: Hardness for all induced patterns and faster non-induced cycles. In STOC, page to appear, 2019.
Camil Demetrescu and Giuseppe F. Italiano. A New Approach to Dynamic All Pairs Shortest Paths. Journal of the ACM, 51(6):968-992, 2004. Announced at STOC'03. URL: http://dx.doi.org/10.1145/1039488.1039492.
http://dx.doi.org/10.1145/1039488.1039492
Shimon Even and Yossi Shiloach. An On-Line Edge-Deletion Problem. Journal of the ACM, 28(1):1-4, 1981. URL: http://dx.doi.org/10.1145/322234.322235.
http://dx.doi.org/10.1145/322234.322235
Monika Henzinger and Valerie King. Fully Dynamic Biconnectivity and Transitive Closure. In Symposium on Foundations of Computer Science (FOCS), pages 664-672, 1995. URL: http://dx.doi.org/10.1109/SFCS.1995.492668.
http://dx.doi.org/10.1109/SFCS.1995.492668
Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. Decremental Single-Source Shortest Paths on Undirected Graphs in Near-Linear Total Update Time. In Symposium on Foundations of Computer Science (FOCS), pages 146-155, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.24.
http://dx.doi.org/10.1109/FOCS.2014.24
Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai, and Thatchaphol Saranurak. Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture. In Symposium on Theory of Computing (STOC), pages 21-30, 2015. URL: http://dx.doi.org/10.1145/2746539.2746609.
http://dx.doi.org/10.1145/2746539.2746609
Monika Henzinger, Andrea Lincoln, Stefan Neumann, and Virginia Vassilevska Williams. Conditional Hardness for Sensitivity Problems. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), volume 67 of Leibniz International Proceedings in Informatics (LIPIcs), pages 26:1-26:31, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ITCS.2017.26.
http://dx.doi.org/10.4230/LIPIcs.ITCS.2017.26
R. Impagliazzo, R. Paturi, and F. Zane. Which Problems Have Strongly Exponential Complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001.
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Moshe Lewenstein, Seth Pettie, and Virginia Vassilevska Williams. Open Problems from Dagstuhl Seminar 16451: Structure and Hardness in P, 2016.
S. Pettie. A new approach to all-pairs shortest paths on real-weighted graphs. Theor. Comput. Sci., 312(1):47-74, 2004.
Seth Pettie and Vijaya Ramachandran. A Shortest Path Algorithm for Real-Weighted Undirected Graphs. SIAM J. Comput., 34(6):1398-1431, 2005.
Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, STOC '13, pages 515-524, 2013.
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Virginia Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 887-898. ACM, 2012.
Virginia Vassilevska Williams. On some fine-grained questions in algorithms and complexity. In Proceedings of the International Congress of Mathematicians, page to appear, 2018.
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Uri Zwick. All Pairs Shortest Paths using Bridging Sets and Rectangular Matrix Multiplication. Journal of the ACM, 49(3):289-317, 2002. Announced at FOCS'98. URL: http://dx.doi.org/10.1145/567112.567114.
http://dx.doi.org/10.1145/567112.567114
Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, and Nicole Wein
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Log Diameter Rounds Algorithms for 2-Vertex and 2-Edge Connectivity
Many modern parallel systems, such as MapReduce, Hadoop and Spark, can be modeled well by the MPC model. The MPC model captures well coarse-grained computation on large data - data is distributed to processors, each of which has a sublinear (in the input data) amount of memory and we alternate between rounds of computation and rounds of communication, where each machine can communicate an amount of data as large as the size of its memory. This model is stronger than the classical PRAM model, and it is an intriguing question to design algorithms whose running time is smaller than in the PRAM model.
In this paper, we study two fundamental problems, 2-edge connectivity and 2-vertex connectivity (biconnectivity). PRAM algorithms which run in O(log n) time have been known for many years. We give algorithms using roughly log diameter rounds in the MPC model. Our main results are, for an n-vertex, m-edge graph of diameter D and bi-diameter D', 1) a O(log D log log_{m/n} n) parallel time 2-edge connectivity algorithm, 2) a O(log D log^2 log_{m/n}n+log D'log log_{m/n}n) parallel time biconnectivity algorithm, where the bi-diameter D' is the largest cycle length over all the vertex pairs in the same biconnected component. Our results are fully scalable, meaning that the memory per processor can be O(n^{delta}) for arbitrary constant delta>0, and the total memory used is linear in the problem size. Our 2-edge connectivity algorithm achieves the same parallel time as the connectivity algorithm of [Andoni et al., 2018]. We also show an Omega(log D') conditional lower bound for the biconnectivity problem.
parallel algorithms
biconnectivity
2-edge connectivity
the MPC model
Theory of computation~MapReduce algorithms
Mathematics of computing~Paths and connectivity problems
14:1-14:16
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1905.00850.
Alexandr
Andoni
Alexandr Andoni
Columbia University, New York City, NY, USA
Research partly supported by NSF Grants (CCF-1617955 and CCF-1740833), Simons Foundation (#491119) and Google Research Award.
Clifford
Stein
Clifford Stein
Columbia University, New York City, NY, USA
Research partly supported by NSF Grants CCF-1714818 and CCF-1822809.
Peilin
Zhong
Peilin Zhong
Columbia University, New York City, NY, USA
Research partly supported by NSF Grants (CCF-1703925, CCF-1421161, CCF-1714818, CCF-1617955 and CCF-1740833), Simons Foundation (#491119) and Google Research Award.
10.4230/LIPIcs.ICALP.2019.14
Kook Jin Ahn and Sudipto Guha. Access to data and number of iterations: Dual primal algorithms for maximum matching under resource constraints. ACM Transactions on Parallel Computing (TOPC), 4(4):17, 2018.
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Alexandr Andoni, Zhao Song, Clifford Stein, Zhengyu Wang, and Peilin Zhong. Parallel Graph Connectivity in Log Diameter Rounds, 2018. In FOCS 2018. URL: http://arxiv.org/abs/1805.03055.
http://arxiv.org/abs/1805.03055
Sepehr Assadi, MohammadHossein Bateni, Aaron Bernstein, Vahab Mirrokni, and Cliff Stein. Coresets meet EDCS: algorithms for matching and vertex cover on massive graphs. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1616-1635. SIAM, 2019.
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Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, and Richard M Karp. Massively parallel symmetry breaking on sparse graphs: MIS and maximal matching. arXiv preprint, 2018. URL: http://arxiv.org/abs/1807.06701.
http://arxiv.org/abs/1807.06701
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http://arxiv.org/abs/1807.05374
Artur Czumaj, Jakub Łącki, Aleksander Mądry, Slobodan Mitrović, Krzysztof Onak, and Piotr Sankowski. Round compression for parallel matching algorithms. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 471-484. ACM, 2018.
Jeffrey Dean and Sanjay Ghemawat. MapReduce: Simplified Data Processing on Large Clusters. To appear in OSDI, page 1, 2004.
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Howard Karloff, Siddharth Suri, and Sergei Vassilvitskii. A model of computation for MapReduce. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 938-948. Society for Industrial and Applied Mathematics, 2010.
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Sixue Liu and Robert E Tarjan. Simple Concurrent Labeling Algorithms for Connected Components. arXiv preprint, 2018. URL: http://arxiv.org/abs/1812.06177.
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Alexandr Andoni, Clifford Stein, and Peilin Zhong
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Two Party Distribution Testing: Communication and Security
We study the problem of discrete distribution testing in the two-party setting. For example, in the standard closeness testing problem, Alice and Bob each have t samples from, respectively, distributions a and b over [n], and they need to test whether a=b or a,b are epsilon-far (in the l_1 distance). This is in contrast to the well-studied one-party case, where the tester has unrestricted access to samples of both distributions. Despite being a natural constraint in applications, the two-party setting has previously evaded attention.
We address two fundamental aspects of the two-party setting: 1) what is the communication complexity, and 2) can it be accomplished securely, without Alice and Bob learning extra information about each other’s input. Besides closeness testing, we also study the independence testing problem, where Alice and Bob have t samples from distributions a and b respectively, which may be correlated; the question is whether a,b are independent or epsilon-far from being independent. Our contribution is three-fold: 1) We show how to gain communication efficiency given more samples, beyond the information-theoretic bound on t. The gain is polynomially better than what one would obtain via adapting one-party algorithms. 2) We prove tightness of our trade-off for the closeness testing, as well as that the independence testing requires tight Omega(sqrt{m}) communication for unbounded number of samples. These lower bounds are of independent interest as, to the best of our knowledge, these are the first 2-party communication lower bounds for testing problems, where the inputs are a set of i.i.d. samples. 3) We define the concept of secure distribution testing, and provide secure versions of the above protocols with an overhead that is only polynomial in the security parameter.
distribution testing
communication complexity
security
Mathematics of computing~Hypothesis testing and confidence interval computation
15:1-15:16
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1811.04065.
We thank Devanshi Nishit Vyas for her contribution to some of the initial work which led to this paper. We thank Clement Canonne for invaluable comments on an early draft of the manuscript. We thank Yuval Ishai for helpful discussions. Work supported in part by Simons Foundation (#491119), NSF grants CCF-1617955 and CCF-1740833.
Alexandr
Andoni
Alexandr Andoni
Columbia University, New York City, NY, USA
Tal
Malkin
Tal Malkin
Columbia University, New York City, NY, USA
Negev Shekel
Nosatzki
Negev Shekel Nosatzki
Columbia University, New York City, NY, USA
10.4230/LIPIcs.ICALP.2019.15
Jayadev Acharya, Clément L. Canonne, Cody Freitag, and Himanshu Tyagi. Test without Trust: Optimal Locally Private Distribution Testing. CoRR, abs/1808.02174, 2018.
Jayadev Acharya, Clément L. Canonne, and Himanshu Tyagi. Distributed Simulation and Distributed Inference. CoRR, abs/1804.06952, 2018. URL: http://arxiv.org/abs/1804.06952.
http://arxiv.org/abs/1804.06952
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Jayadev Acharya, Ziteng Sun, and Huanyu Zhang. Differentially Private Testing of Identity and Closeness of Discrete Distributions. CoRR, abs/1707.05128, 2017. URL: http://arxiv.org/abs/1707.05128.
http://arxiv.org/abs/1707.05128
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http://arxiv.org/abs/1707.05497
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Alexandr Andoni, Tal Malkin, and Negev Shekel Nosatzki
Creative Commons Attribution 3.0 Unported license
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Two New Results About Quantum Exact Learning
We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k-Fourier-sparse n-bit Boolean function from O(k^{1.5}(log k)^2) uniform quantum examples for that function. This improves over the bound of Theta~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Our main tool is an improvement of Chang’s lemma for sparse Boolean functions. Second, we show that if a concept class {C} can be exactly learned using Q quantum membership queries, then it can also be learned using O ({Q^2}/{log Q} * log|C|) classical membership queries. This improves the previous-best simulation result (Servedio-Gortler, SICOMP'04) by a log Q-factor.
quantum computing
exact learning
analysis of Boolean functions
Fourier sparse Boolean functions
Hardware~Quantum computation
Theory of computation~Sample complexity and generalization bounds
Theory of computation~Boolean function learning
16:1-16:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1810.00481.
Srinivasan
Arunachalam
Srinivasan Arunachalam
Center for Theoretical Physics, MIT, Cambridge, MA, USA
Work done when at QuSoft, CWI, Amsterdam, the Netherlands. Supported by ERC Consolidator Grant 615307 QPROGRESS and MIT-IBM Watson AI Lab under the project Machine Learning in Hilbert space.
Sourav
Chakraborty
Sourav Chakraborty
Indian Statistical Institute, Kolkata, India
Work done while on sabbatical at CWI, supported by ERC QPROGRESS.
Troy
Lee
Troy Lee
Centre for Quantum Software and Information, School of Software, Faculty of Engineering and Information Technology, University of Technology Sydney, Australia
Part of this work was done while at the School for Physical and Mathematical Sciences, Nanyang Technological University and the Centre for Quantum Technologies, Singapore, supported by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13.
Manaswi
Paraashar
Manaswi Paraashar
Indian Statistical Institute, Kolkata, India
Ronald
de Wolf
Ronald de Wolf
QuSoft, CWI and University of Amsterdam, The Netherlands
https://homepages.cwi.nl/~rdewolf/
Supported by ERC QPROGRESS, and QuantERA project QuantAlgo 680-91-034, and NWO Gravitation project QSC.
10.4230/LIPIcs.ICALP.2019.16
J. Adcock, E. Allen, M. Day, S. Frick, J. Hinchliff, M. Johnson, S. Morley-Short, S. Pallister, A. Price, and S. Stanisic. Advances in quantum machine learning, 9 Dec 2015. URL: http://arxiv.org/abs/1512.02900.
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S. Arunachalam, S. Chakraborty, T. Lee, M. Paraashar, and R. de Wolf. Two new results about quantum exact learning. URL: http://arxiv.org/abs/1810.00481.
http://arxiv.org/abs/1810.00481
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Srinivasan Arunchalam, Sourav Chakraborty, Troy Lee, Manaswi Paraashar, and Ronald de Wolf
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
When Algorithms for Maximal Independent Set and Maximal Matching Run in Sublinear Time
Maximal independent set (MIS), maximal matching (MM), and (Delta+1)-(vertex) coloring in graphs of maximum degree Delta are among the most prominent algorithmic graph theory problems. They are all solvable by a simple linear-time greedy algorithm and up until very recently this constituted the state-of-the-art. In SODA 2019, Assadi, Chen, and Khanna gave a randomized algorithm for (Delta+1)-coloring that runs in O~(n sqrt{n}) time, which even for moderately dense graphs is sublinear in the input size. The work of Assadi et al. however contained a spoiler for MIS and MM: neither problems provably admits a sublinear-time algorithm in general graphs. In this work, we dig deeper into the possibility of achieving sublinear-time algorithms for MIS and MM.
The neighborhood independence number of a graph G, denoted by beta(G), is the size of the largest independent set in the neighborhood of any vertex. We identify beta(G) as the "right" parameter to measure the runtime of MIS and MM algorithms: Although graphs of bounded neighborhood independence may be very dense (clique is one example), we prove that carefully chosen variants of greedy algorithms for MIS and MM run in O(n beta(G)) and O(n log{n} * beta(G)) time respectively on any n-vertex graph G. We complement this positive result by observing that a simple extension of the lower bound of Assadi et al. implies that Omega(n beta(G)) time is also necessary for any algorithm to either problem for all values of beta(G) from 1 to Theta(n). We note that our algorithm for MIS is deterministic while for MM we use randomization which we prove is unavoidable: any deterministic algorithm for MM requires Omega(n^2) time even for beta(G) = 2.
Graphs with bounded neighborhood independence, already for constant beta = beta(G), constitute a rich family of possibly dense graphs, including line graphs, proper interval graphs, unit-disk graphs, claw-free graphs, and graphs of bounded growth. Our results suggest that even though MIS and MM do not admit sublinear-time algorithms in general graphs, one can still solve both problems in sublinear time for a wide range of beta(G) << n.
Finally, by observing that the lower bound of Omega(n sqrt{n}) time for (Delta+1)-coloring due to Assadi et al. applies to graphs of (small) constant neighborhood independence, we unveil an intriguing separation between the time complexity of MIS and MM, and that of (Delta+1)-coloring: while the time complexity of MIS and MM is strictly higher than that of (Delta+1) coloring in general graphs, the exact opposite relation holds for graphs with small neighborhood independence.
Maximal Independent Set
Maximal Matching
Sublinear-Time Algorithms
Bounded Neighborhood Independence
Theory of computation~Graph algorithms analysis
17:1-17:17
Track A: Algorithms, Complexity and Games
Sepehr
Assadi
Sepehr Assadi
Department of Computer Science, Princeton University, NJ, USA
Research supported in part by the Simons Collaboration on Algorithms and Geometry.
Shay
Solomon
Shay Solomon
School of Electrical Engineering, Tel Aviv University, Israel
10.4230/LIPIcs.ICALP.2019.17
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Sepehr Assadi and Shay Solomon
Creative Commons Attribution 3.0 Unported license
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Robust Communication-Optimal Distributed Clustering Algorithms
In this work, we study the k-median and k-means clustering problems when the data is distributed across many servers and can contain outliers. While there has been a lot of work on these problems for worst-case instances, we focus on gaining a finer understanding through the lens of beyond worst-case analysis. Our main motivation is the following: for many applications such as clustering proteins by function or clustering communities in a social network, there is some unknown target clustering, and the hope is that running a k-median or k-means algorithm will produce clusterings which are close to matching the target clustering. Worst-case results can guarantee constant factor approximations to the optimal k-median or k-means objective value, but not closeness to the target clustering.
Our first result is a distributed algorithm which returns a near-optimal clustering assuming a natural notion of stability, namely, approximation stability [Awasthi and Balcan, 2014], even when a constant fraction of the data are outliers. The communication complexity is O~(sk+z) where s is the number of machines, k is the number of clusters, and z is the number of outliers. Next, we show this amount of communication cannot be improved even in the setting when the input satisfies various non-worst-case assumptions. We give a matching Omega(sk+z) lower bound on the communication required both for approximating the optimal k-means or k-median cost up to any constant, and for returning a clustering that is close to the target clustering in Hamming distance. These lower bounds hold even when the data satisfies approximation stability or other common notions of stability, and the cluster sizes are balanced. Therefore, Omega(sk+z) is a communication bottleneck, even for real-world instances.
robust distributed clustering
communication complexity
Theory of computation~Unsupervised learning and clustering
18:1-18:16
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1703.00830.
This work was supported in part by NSF grants CCF-1422910, CCF-1535967, IIS-1618714, an Office of Naval Research (ONR) grant N00014-18-1-2562, an Amazon Research Award, a Microsoft Research Faculty Fellowship, and a National Defense Science & Engineering Graduate (NDSEG) fellowship. Part of this work was done while Ainesh Bakshi and David Woodruff were visiting the Simons Institute for the Theory of Computing.
Pranjal
Awasthi
Pranjal Awasthi
Rutgers University, Piscataway, NJ, USA
Ainesh
Bakshi
Ainesh Bakshi
Carnegie Mellon University, Pittsburgh, PA, USA
Maria-Florina
Balcan
Maria-Florina Balcan
Carnegie Mellon University, Pittsburgh, PA, USA
Colin
White
Colin White
Carnegie Mellon University, Pittsburgh, PA, USA
David P.
Woodruff
David P. Woodruff
Carnegie Mellon University, Pittsburgh, PA, USA
10.4230/LIPIcs.ICALP.2019.18
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http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.69
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Maria-Florina Balcan, Avrim Blum, and Anupam Gupta. Clustering under approximation stability. Journal of the ACM (JACM), 60(2):8, 2013.
Maria-Florina Balcan, Avrim Blum, and Santosh Vempala. A discriminative framework for clustering via similarity functions. In Proceedings of the Annual Symposium on Theory of Computing (STOC), pages 671-680, 2008.
Maria-Florina Balcan and Mark Braverman. Finding Low Error Clusterings. In COLT, volume 3, pages 3-4, 2009.
Maria-Florina Balcan, Nika Haghtalab, and Colin White. k-center Clustering under Perturbation Resilience. In Proceedings of the Annual International Colloquium on Automata, Languages, and Programming (ICALP), 2016.
Maria Florina Balcan and Yingyu Liang. Clustering under perturbation resilience. SIAM Journal on Computing, 45(1):102-155, 2016.
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Pranjal Awasthi, Ainesh Bakshi, Maria-Florina Balcan, Colin White, and David Woodruff
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Capacitated Dynamic Programming: Faster Knapsack and Graph Algorithms
One of the most fundamental problems in Computer Science is the Knapsack problem. Given a set of n items with different weights and values, it asks to pick the most valuable subset whose total weight is below a capacity threshold T. Despite its wide applicability in various areas in Computer Science, Operations Research, and Finance, the best known running time for the problem is O(T n). The main result of our work is an improved algorithm running in time O(TD), where D is the number of distinct weights. Previously, faster runtimes for Knapsack were only possible when both weights and values are bounded by M and V respectively, running in time O(nMV) [Pisinger, 1999]. In comparison, our algorithm implies a bound of O(n M^2) without any dependence on V, or O(n V^2) without any dependence on M. Additionally, for the unbounded Knapsack problem, we provide an algorithm running in time O(M^2) or O(V^2). Both our algorithms match recent conditional lower bounds shown for the Knapsack problem [Marek Cygan et al., 2017; Marvin Künnemann et al., 2017].
We also initiate a systematic study of general capacitated dynamic programming, of which Knapsack is a core problem. This problem asks to compute the maximum weight path of length k in an edge- or node-weighted directed acyclic graph. In a graph with m edges, these problems are solvable by dynamic programming in time O(k m), and we explore under which conditions the dependence on k can be eliminated. We identify large classes of graphs where this is possible and apply our results to obtain linear time algorithms for the problem of k-sparse Delta-separated sequences. The main technical innovation behind our results is identifying and exploiting concavity that appears in relaxations and subproblems of the tasks we consider.
Knapsack
Fine-Grained Complexity
Dynamic Programming
Theory of computation~Algorithm design techniques
19:1-19:13
Track A: Algorithms, Complexity and Games
We are grateful to Arturs Backurs for insightful discussions that helped us improve this work.
Kyriakos
Axiotis
Kyriakos Axiotis
MIT, Cambridge, MA, USA
Christos
Tzamos
Christos Tzamos
University of Wisconsin-Madison, USA
10.4230/LIPIcs.ICALP.2019.19
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http://arxiv.org/abs/1803.04744
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Marvin Künnemann, Ramamohan Paturi, and Stefan Schneider. On the Fine-Grained Complexity of One-Dimensional Dynamic Programming. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 21:1-21:15, 2017.
Aleksander Mądry, Slobodan Mitrović, and Ludwig Schmidt. A Fast Algorithm for Separated Sparsity via Perturbed Lagrangians. In International Conference on Artificial Intelligence and Statistics, AISTATS 2018, 9-11 April 2018, Playa Blanca, Lanzarote, Canary Islands, Spain, pages 20-28, 2018.
David Pisinger. Linear time algorithms for knapsack problems with bounded weights. Journal of Algorithms, 33(1):1-14, 1999.
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Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 664-673. ACM, 2014.
Kyriakos Axiotis and Christos Tzamos
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Covering Metric Spaces by Few Trees
A tree cover of a metric space (X,d) is a collection of trees, so that every pair x,y in X has a low distortion path in one of the trees. If it has the stronger property that every point x in X has a single tree with low distortion paths to all other points, we call this a Ramsey tree cover. Tree covers and Ramsey tree covers have been studied by [Yair Bartal et al., 2005; Anupam Gupta et al., 2004; T-H. Hubert Chan et al., 2005; Gupta et al., 2006; Mendel and Naor, 2007], and have found several important algorithmic applications, e.g. routing and distance oracles. The union of trees in a tree cover also serves as a special type of spanner, that can be decomposed into a few trees with low distortion paths contained in a single tree; Such spanners for Euclidean pointsets were presented by [S. Arya et al., 1995].
In this paper we devise efficient algorithms to construct tree covers and Ramsey tree covers for general, planar and doubling metrics. We pay particular attention to the desirable case of distortion close to 1, and study what can be achieved when the number of trees is small. In particular, our work shows a large separation between what can be achieved by tree covers vs. Ramsey tree covers.
tree cover
Ramsey tree cover
probabilistic hierarchical family
Mathematics of computing~Graph algorithms
Theory of computation~Sparsification and spanners
20:1-20:16
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1905.07559.
We are grateful to Michael Elkin and Shay Solomon for fruitful discussions.
Yair
Bartal
Yair Bartal
Department of Computer Science, Hebrew University of Jerusalem, Israel
Supported in part by a grant from the Israeli Science Foundation (1817/17).
Nova
Fandina
Nova Fandina
Department of Computer Science, Hebrew University of Jerusalem, Israel
Supported in part by a grant from the Israeli Science Foundation (1817/17).
Ofer
Neiman
Ofer Neiman
Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Supported in part by a grant from the Israeli Science Foundation (1817/17) and in part by BSF grant 2015813.
10.4230/LIPIcs.ICALP.2019.20
Ittai Abraham, Yair Bartal, and Ofer Neiman. Advances in metric embedding theory. Advances in Mathematics, 228(6):3026-3126, 2011. URL: http://dx.doi.org/10.1016/j.aim.2011.08.003.
http://dx.doi.org/10.1016/j.aim.2011.08.003
Ittai Abraham, Yair Bartal, and Ofer Neiman. Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion. SIAM J. Comput., 44(1):160-192, 2015. URL: http://dx.doi.org/10.1137/120884390.
http://dx.doi.org/10.1137/120884390
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http://dx.doi.org/10.1137/1.9781611975031.108
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Yair Bartal, Nova Fandina, and Ofer Neiman
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Even Faster Elastic-Degenerate String Matching via Fast Matrix Multiplication
An elastic-degenerate (ED) string is a sequence of n sets of strings of total length N, which was recently proposed to model a set of similar sequences. The ED string matching (EDSM) problem is to find all occurrences of a pattern of length m in an ED text. The EDSM problem has recently received some attention in the combinatorial pattern matching community, and an O(nm^{1.5}sqrt{log m} + N)-time algorithm is known [Aoyama et al., CPM 2018]. The standard assumption in the prior work on this question is that N is substantially larger than both n and m, and thus we would like to have a linear dependency on the former. Under this assumption, the natural open problem is whether we can decrease the 1.5 exponent in the time complexity, similarly as in the related (but, to the best of our knowledge, not equivalent) word break problem [Backurs and Indyk, FOCS 2016].
Our starting point is a conditional lower bound for the EDSM problem. We use the popular combinatorial Boolean matrix multiplication (BMM) conjecture stating that there is no truly subcubic combinatorial algorithm for BMM [Abboud and Williams, FOCS 2014]. By designing an appropriate reduction we show that a combinatorial algorithm solving the EDSM problem in O(nm^{1.5-epsilon} + N) time, for any epsilon>0, refutes this conjecture. Of course, the notion of combinatorial algorithms is not clearly defined, so our reduction should be understood as an indication that decreasing the exponent requires fast matrix multiplication.
Two standard tools used in algorithms on strings are string periodicity and fast Fourier transform. Our main technical contribution is that we successfully combine these tools with fast matrix multiplication to design a non-combinatorial O(nm^{1.381} + N)-time algorithm for EDSM. To the best of our knowledge, we are the first to do so.
string algorithms
pattern matching
elastic-degenerate string
matrix multiplication
fast Fourier transform
Theory of computation~Pattern matching
21:1-21:15
Track A: Algorithms, Complexity and Games
GR and NP are partially supported by MIUR-SIR project CMACBioSeq "Combinatorial methods for analysis and compression of biological sequences" grant n. RBSI146R5L.
A full version of the paper is available at https://arxiv.org/abs/1905.02298.
Giulia
Bernardini
Giulia Bernardini
Department of Informatics, Systems and Communication, University of Milano - Bicocca, Italy
Paweł
Gawrychowski
Paweł Gawrychowski
Institute of Computer Science, University of Wrocław, Poland
Nadia
Pisanti
Nadia Pisanti
Department of Computer Science, University of Pisa, Italy
ERABLE Team, INRIA, France
Solon P.
Pissis
Solon P. Pissis
CWI, Amsterdam, The Netherlands
Giovanna
Rosone
Giovanna Rosone
Department of Computer Science, University of Pisa, Italy
10.4230/LIPIcs.ICALP.2019.21
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Giulia Bernardini, Paweł Gawrychowski, Nadia Pisanti, Solon P. Pissis, and Giovanna Rosone
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Complexity of Approximating the Matching Polynomial in the Complex Plane
We study the problem of approximating the value of the matching polynomial on graphs with edge parameter gamma, where gamma takes arbitrary values in the complex plane.
When gamma is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of gamma, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Delta as long as gamma is not a negative real number less than or equal to -1/(4(Delta-1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Delta >= 3 and all real gamma less than -1/(4(Delta-1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Delta with edge parameter gamma is #P-hard.
We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real gamma it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of gamma values on the negative real axis. Nevertheless, we show that the result does extend for any complex value gamma that does not lie on the negative real axis. Our analysis accounts for complex values of gamma using geodesic distances in the complex plane in the metric defined by an appropriate density function.
matchings
partition function
correlation decay
connective constant
Theory of computation~Approximation algorithms analysis
Theory of computation~Problems, reductions and completeness
22:1-22:13
Track A: Algorithms, Complexity and Games
A full version of the paper is available at http://arxiv.org/abs/1807.04930. The theorem numbering here matches the full version.
Ivona
Bezáková
Ivona Bezáková
Department of Computer Science, Rochester Institute of Technology, Rochester, NY, USA
Research supported by NSF grant CCF-1319987.
Andreas
Galanis
Andreas Galanis
Department of Computer Science, University of Oxford, UK
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.
Leslie Ann
Goldberg
Leslie Ann Goldberg
Department of Computer Science, University of Oxford, UK
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.
Daniel
Štefankovič
Daniel Štefankovič
Department of Computer Science, University of Rochester, Rochester, NY, USA
Research supported by NSF grant CCF-1563757.
10.4230/LIPIcs.ICALP.2019.22
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Ivona Bezáková, Andreas Galanis, Leslie A. Goldberg, and Daniel Štefankovič
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Finding Tutte Paths in Linear Time
It is well-known that every planar graph has a Tutte path, i.e., a path P such that any component of G-P has at most three attachment points on P. However, it was only recently shown that such Tutte paths can be found in polynomial time. In this paper, we give a new proof that 3-connected planar graphs have Tutte paths, which leads to a linear-time algorithm to find Tutte paths. Furthermore, our Tutte path has special properties: it visits all exterior vertices, all components of G-P have exactly three attachment points, and we can assign distinct representatives to them that are interior vertices. Finally, our running time bound is slightly stronger; we can bound it in terms of the degrees of the faces that are incident to P. This allows us to find some applications of Tutte paths (such as binary spanning trees and 2-walks) in linear time as well.
planar graph
Tutte path
Hamiltonian path
2-walk
linear time
Mathematics of computing~Graph algorithms
23:1-23:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at [Therese Biedl and Philipp Kindermann, 2018]: http://arxiv.org/abs/1812.04543.
Therese
Biedl
Therese Biedl
David R. Cheriton School of Computer Science, University of Waterloo, Canada
https://cs.uwaterloo.ca/~biedl/
https://orcid.org/0000-0002-9003-3783
Supported by NSERC.
Philipp
Kindermann
Philipp Kindermann
Lehrstuhl für Informatik I, Universität Würzburg, Germany
https://go.uniwue.de/pkinderm
https://orcid.org/0000-0001-5764-7719
10.4230/LIPIcs.ICALP.2019.23
Takao Asano, Shunji Kikuchi, and Nobuji Saito. A Linear Algorithm for Finding Hamiltonian Cycles in 4-Connected Maximal Planar Graphs. Discrete Applied Mathematics, 7:1-15, 1985. URL: http://dx.doi.org/10.1016/0166-218X(84)90109-4.
http://dx.doi.org/10.1016/0166-218X(84)90109-4
David W. Barnette. Trees in Polyhedral Graphs. Canadian Journal of Mathematics, 18:731-736, 1966. URL: http://dx.doi.org/10.4153/CJM-1966-073-4.
http://dx.doi.org/10.4153/CJM-1966-073-4
Therese Biedl. Trees and Co-trees with Bounded Degrees in Planar 3-Connected Graphs. In R. Ravi and Inge Li Gørtz, editors, Scandinavian Symposium and Workshops on Algorithms Theory (SWAT'14), volume 8503 of Lecture Notes in Computer Science, pages 62-73. Springer-Verlag, 2014. URL: http://dx.doi.org/10.1007/978-3-319-08404-6_6.
http://dx.doi.org/10.1007/978-3-319-08404-6_6
Therese Biedl and Martin Derka. 1-String B₂-VPG-Representations of Planar Graphs. Journal on Computational Geometry, 7(2):191-215, 2016. URL: http://dx.doi.org/10.20382/jocg.v7i2a8.
http://dx.doi.org/10.20382/jocg.v7i2a8
Therese Biedl and Philipp Kindermann. Finding Tutte Paths in Linear Time. Arxiv report, abs/1812.04543, 2018. URL: http://arxiv.org/abs/1812.04543.
http://arxiv.org/abs/1812.04543
Giuseppe Di Battista and Roberto Tamassia. Incremental Planarity Testing. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science (FOCS'89), pages 436-441. IEEE Computer Society, 1989. URL: http://dx.doi.org/10.1109/SFCS.1989.63515.
http://dx.doi.org/10.1109/SFCS.1989.63515
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http://diestel-graph-theory.com/
Zhicheng Gao and R. Bruce Richter. 2-Walks in Circuit Graphs. Journal of Combinatorial Theory, Series B, 62(2):259-267, 1994. URL: http://dx.doi.org/10.1006/jctb.1994.1068.
http://dx.doi.org/10.1006/jctb.1994.1068
Carsten Gutwenger and Petra Mutzel. A Linear Time Implementation of SPQR-Trees. In Joe Marks, editor, Proceedings of the 8th International Symposium on Graph Drawing (GD'00), volume 1984 of Lecture Notes in Computer Science, pages 77-90. Springer, 2000. URL: http://dx.doi.org/10.1007/3-540-44541-2_8.
http://dx.doi.org/10.1007/3-540-44541-2_8
John E. Hopcroft and Robert E. Tarjan. Dividing a graph into triconnected components. SIAM Journal on Computing, 2(3):135-158, 1973. URL: http://dx.doi.org/10.1137/0202012.
http://dx.doi.org/10.1137/0202012
Ken-ichi Kawarabayashi and Kenta Ozeki. 4-Connected Projective-Planar Graphs Are Hamiltonian-Connected. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'13), pages 378-395. Society for Industrial and Applied Mathematics, 2013. URL: http://dx.doi.org/10.1016/j.jctb.2012.11.004.
http://dx.doi.org/10.1016/j.jctb.2012.11.004
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http://dx.doi.org/10.1002/(SICI)1097-0118(199704)24:4<341::AID-JGT6>3.0.CO;2-O
Andreas Schmid and Jens M. Schmidt. Computing 2-Walks in Polynomial Time. In Ernst W. Mayr and Nicolas Ollinger, editors, Proceedings of the 32nd International Symposium on Theoretical Aspects of Computer Science (STACS'15), volume 30 of LIPIcs, pages 676-688. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.676.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.676
Andreas Schmid and Jens M. Schmidt. Computing 2-Walks in Polynomial Time. ACM Transactions on Algorithms, 14(2):22:1-22:18, 2018. URL: http://dx.doi.org/10.1145/3183368.
http://dx.doi.org/10.1145/3183368
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http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.98
Willy-Bernhard Strothmann. Bounded-Degree Spanning Trees. Ph.d. thesis, Universität Paderborn, Heinz Nixdorf Institut, Theoretische Informatik, Paderborn, 1997. ISBN 3-931466-34-5. URL: https://www.hni.uni-paderborn.de/pub/468.
https://www.hni.uni-paderborn.de/pub/468
Robin Thomas and Xingxing Yu. 4-Connected Projective-Planar Graphs Are Hamiltonian. Journal of Combinatorial Theory, Series B, 62:114-132, 1994. URL: http://dx.doi.org/10.1006/jctb.1994.1058.
http://dx.doi.org/10.1006/jctb.1994.1058
Carsten Thomassen. A Theorem on Paths in Planar Graphs. Journal of Graph Theory, 7(2):169-176, 1983. URL: http://dx.doi.org/10.1002/jgt.3190070205.
http://dx.doi.org/10.1002/jgt.3190070205
William T. Tutte. Bridges and Hamiltonian Circuits in Planar Graphs. Aequationes Mathematicae, 15(1):1-33, 1977. URL: http://dx.doi.org/10.1007/BF01837870.
http://dx.doi.org/10.1007/BF01837870
Therese Biedl and Philipp Kindermann
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Approximate Counting of k-Paths: Deterministic and in Polynomial Space
A few years ago, Alon et al. [ISMB 2008] gave a simple randomized O((2e)^km epsilon^{-2})-time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 +/- epsilon. Shortly afterwards, Alon and Gutner [IWPEC 2009, TALG 2010] gave a deterministic exponential-space algorithm with running time (2e)^{k+O(log^3k)}m log n whenever epsilon^{-1}=k^{O(1)}. Recently, Brand et al. [STOC 2018] provided a speed-up at the cost of reintroducing randomization. Specifically, they gave a randomized O(4^km epsilon^{-2})-time exponential-space algorithm. In this article, we revisit the algorithm by Alon and Gutner. We modify the foundation of their work, and with a novel twist, obtain the following results.
- We present a deterministic 4^{k+O(sqrt{k}(log^2k+log^2 epsilon^{-1}))}m log n-time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices.
- Additionally, we present a randomized 4^{k+O(log k(log k + log epsilon^{-1}))}m log n-time polynomial-space algorithm. While Brand et al. make non-trivial use of exterior algebra, our algorithm is very simple; we only make elementary use of the probabilistic method.
Thus, the algorithm by Brand et al. runs in time 4^{k+o(k)}m whenever epsilon^{-1}=2^{o(k)}, while our deterministic and randomized algorithms run in time 4^{k+o(k)}m log n whenever epsilon^{-1}=2^{o(k^{1/4})} and epsilon^{-1}=2^{o(k/(log k))}, respectively. Prior to our work, no 2^{O(k)}n^{O(1)}-time polynomial-space algorithm was known. Additionally, our approach is embeddable in the classic framework of divide-and-color, hence it immediately extends to approximate counting of graphs of bounded treewidth; in comparison, Brand et al. note that their approach is limited to graphs of bounded pathwidth.
parameterized complexity
approximate counting
{ k}-Path
Theory of computation~Fixed parameter tractability
24:1-24:15
Track A: Algorithms, Complexity and Games
Andreas
Björklund
Andreas Björklund
Lund University, Lund, Sweden
Daniel
Lokshtanov
Daniel Lokshtanov
University of California, Bergen, Santa Barbara, USA
Saket
Saurabh
Saket Saurabh
The Institute of Mathematical Sciences, HBNI, Chennai, India
This work is supported by the European Research Council (ERC) via grant LOPPRE, reference 819416.
Meirav
Zehavi
Meirav Zehavi
Ben-Gurion University, Beersheba, Israel
This work is supported by the Israel Science Foundation individual research grant no. 1176/18.
10.4230/LIPIcs.ICALP.2019.24
Randomization in Parameterized Complexity. URL: https://www.dagstuhl.de/de/programm/kalender/semhp/?semnr=17041.
https://www.dagstuhl.de/de/programm/kalender/semhp/?semnr=17041
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Andreas Björklund. Determinant Sums for Undirected Hamiltonicity. SIAM J. Comput., 43(1):280-299, 2014. URL: http://dx.doi.org/10.1137/110839229.
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Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Counting Paths and Packings in Halves. In Algorithms - ESA 2009, 17th Annual European Symposium, Copenhagen, Denmark, September 7-9, 2009. Proceedings, pages 578-586, 2009. URL: http://dx.doi.org/10.1007/978-3-642-04128-0_52.
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Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Narrow sieves for parameterized paths and packings. J. Comput. Syst. Sci., 87:119-139, 2017. URL: http://dx.doi.org/10.1016/j.jcss.2017.03.003.
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Andreas Björklund, Vikram Kamat, Lukasz Kowalik, and Meirav Zehavi. Spotting Trees with Few Leaves. SIAM J. Discrete Math., 31(2):687-713, 2017. URL: http://dx.doi.org/10.1137/15M1048975.
http://dx.doi.org/10.1137/15M1048975
Andreas Björklund, Petteri Kaski, and Lukasz Kowalik. Counting Thin Subgraphs via Packings Faster than Meet-in-the-Middle Time. ACM Trans. Algorithms, 13(4):48:1-48:26, 2017. URL: http://dx.doi.org/10.1145/3125500.
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Cornelius Brand, Holger Dell, and Thore Husfeldt. Extensor-coding. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 151-164, 2018. URL: http://dx.doi.org/10.1145/3188745.3188902.
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Radu Curticapean, Holger Dell, and Dániel Marx. Homomorphisms Are a Good Basis for Counting Small Subgraphs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pages 210-223, New York, NY, USA, 2017. ACM. URL: http://dx.doi.org/10.1145/3055399.3055502.
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Fedor V. Fomin, Petteri Kaski, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Parameterized Single-Exponential Time Polynomial Space Algorithm for Steiner Tree. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 494-505, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_40.
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Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, 2018.
Gregory Z. Gutin, Felix Reidl, Magnus Wahlström, and Meirav Zehavi. Designing deterministic polynomial-space algorithms by color-coding multivariate polynomials. J. Comput. Syst. Sci., 95:69-85, 2018. URL: http://dx.doi.org/10.1016/j.jcss.2018.01.004.
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http://dx.doi.org/10.1007/978-3-662-48350-3_86
Andreas Björklund, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi
Creative Commons Attribution 3.0 Unported license
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Computing Permanents and Counting Hamiltonian Cycles by Listing Dissimilar Vectors
We show that the permanent of an n x n matrix over any finite ring of r <= n elements can be computed with a deterministic 2^{n-Omega(n/r)} time algorithm. This improves on a Las Vegas algorithm running in expected 2^{n-Omega(n/(r log r))} time, implicit in [Björklund, Husfeldt, and Lyckberg, IPL 2017]. For the permanent over the integers of a 0/1-matrix with exactly d ones per row and column, we provide a deterministic 2^{n-Omega(n/(d^{3/4)})} time algorithm. This improves on a 2^{n-Omega(n/d)} time algorithm in [Cygan and Pilipczuk ICALP 2013]. We also show that the number of Hamiltonian cycles in an n-vertex directed graph of average degree delta can be computed by a deterministic 2^{n-Omega(n/(delta))} time algorithm. This improves on a Las Vegas algorithm running in expected 2^{n-Omega(n/poly(delta))} time in [Björklund, Kaski, and Koutis, ICALP 2017].
A key tool in our approach is a reduction from computing the permanent to listing pairs of dissimilar vectors from two sets of vectors, i.e., vectors over a finite set that differ in each coordinate, building on an observation of [Bax and Franklin, Algorithmica 2002]. We propose algorithms that can be used both to derandomise the construction of Bax and Franklin, and efficiently list dissimilar pairs using several algorithmic tools. We also give a simple randomised algorithm resulting in Monte Carlo algorithms within the same time bounds.
Our new fast algorithms for listing dissimilar vector pairs from two sets of vectors are inspired by recent algorithms for detecting and counting orthogonal vectors by [Abboud, Williams, and Yu, SODA 2015] and [Chan and Williams, SODA 2016].
permanent
Hamiltonian cycle
orthogonal vectors
Mathematics of computing~Combinatorial algorithms
25:1-25:14
Track A: Algorithms, Complexity and Games
Andreas
Björklund
Andreas Björklund
Department of Computer Science, Lund University, Sweden
The Swedish Research Council grant VR 2016-03855 "Algebraic Graph Algorithms".
Ryan
Williams
Ryan Williams
Department of Electrical Engineering and Computer Science & CSAIL, MIT, Cambridge, MA, USA
The U.S. National Science Foundation grant CCF-1741615 "CAREER: Common Links in Algorithms and Complexity".
10.4230/LIPIcs.ICALP.2019.25
Amir Abboud, Ryan Williams, and Huacheng Yu. More Applications of the Polynomial Method to Algorithm Design. In Proceedings of the Twenty-sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '15, pages 218-230, Philadelphia, PA, USA, 2015. Society for Industrial and Applied Mathematics.
Arturs Backurs and Piotr Indyk. Edit Distance Cannot Be Computed in Strongly Subquadratic Time (Unless SETH is False). In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 51-58, New York, NY, USA, 2015. ACM.
Bax and Franklin. A Permanent Algorithm with exp[Ω (N^1/3/2 ln N)] Expected Speedup for 0-1 Matrices. Algorithmica, 32(1):157-162, January 2002.
Andreas Björklund. Determinant Sums for Undirected Hamiltonicity. SIAM J. Comput., 43(1):280-299, 2014. URL: http://dx.doi.org/10.1137/110839229.
http://dx.doi.org/10.1137/110839229
Andreas Björklund and Thore Husfeldt. The Parity of Directed Hamiltonian Cycles. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 727-735, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.83.
http://dx.doi.org/10.1109/FOCS.2013.83
Andreas Björklund, Thore Husfeldt, and Isak Lyckberg. Computing the permanent modulo a prime power. Inf. Process. Lett., 125:20-25, 2017. URL: http://dx.doi.org/10.1016/j.ipl.2017.04.015.
http://dx.doi.org/10.1016/j.ipl.2017.04.015
Andreas Björklund, Petteri Kaski, and Ioannis Koutis. Directed Hamiltonicity and Out-Branchings via Generalized Laplacians. CoRR, abs/1607.04002, 2016. URL: http://arxiv.org/abs/1607.04002.
http://arxiv.org/abs/1607.04002
Andreas Björklund, Petteri Kaski, and Ioannis Koutis. Directed Hamiltonicity and Out-Branchings via Generalized Laplacians. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 91:1-91:14, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.91.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.91
Andreas Björklund, Petteri Kaski, and Ryan Williams. Generalized Kakeya sets for polynomial evaluation and faster computation of fermionants. Algorithmica, September 2018.
Hans L. Bodlaender, Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Deterministic Single Exponential Time Algorithms for Connectivity Problems Parameterized by Treewidth. Inf. Comput., 243(C):86-111, August 2015.
Timothy M. Chan and Ryan Williams. Deterministic APSP, Orthogonal Vectors, and More: Quickly Derandomizing Razborov-smolensky. In Proceedings of the Twenty-seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 1246-1255, Philadelphia, PA, USA, 2016. Society for Industrial and Applied Mathematics.
Don Coppersmith. Rapid Multiplication of Rectangular Matrices. SIAM J. Comput., 11(3):467-471, 1982. URL: http://dx.doi.org/10.1137/0211037.
http://dx.doi.org/10.1137/0211037
Marek Cygan and Marcin Pilipczuk. Faster Exponential-time Algorithms in Graphs of Bounded Average Degree. Inf. Comput., 243(C):75-85, August 2015.
Taisuke Izumi and Tadashi Wadayama. A New Direction for Counting Perfect Matchings. In Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, FOCS '12, pages 591-598, Washington, DC, USA, 2012. IEEE Computer Society.
Kasper Green Larsen and Ryan Williams. Faster Online Matrix-vector Multiplication. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, pages 2182-2189, Philadelphia, PA, USA, 2017. Society for Industrial and Applied Mathematics.
Maciej Liśkiewicz, Mitsunori Ogihara, and Seinosuke Toda. The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theoretical Computer Science, 304(1):129-156, 2003.
Prabhakar Raghavan. Probabilistic Construction of Deterministic Algorithms: Approximating Packing Integer Programs. J. Comput. Syst. Sci., 37(2):130-143, October 1988.
Herbert John Ryser. Combinatorial Mathematics. Mathematical Association of America, 1963.
Rocco A. Servedio and Andrew Wan. Computing sparse permanents faster. Information Processing Letters, 96(3):89-92, 2005.
W. T. Tutte. The dissection of equilateral triangles into equilateral triangles. Mathematical Proceedings of the Cambridge Philosophical Society, 44(4):463-482, 1948.
Leslie G. Valiant. The Complexity of Enumeration and Reliability Problems. SIAM J. Comput., 8:410-421, 1979.
L.G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189-201, 1979.
Andreas Björklund and Ryan Williams
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Solving Systems of Polynomial Equations over GF(2) by a Parity-Counting Self-Reduction
We consider the problem of finding solutions to systems of polynomial equations over a finite field. Lokshtanov et al. [SODA'17] recently obtained the first worst-case algorithms that beat exhaustive search for this problem. In particular for degree-d equations modulo two in n variables, they gave an O^*(2^{(1-1/(5d))n}) time algorithm, and for the special case d=2 they gave an O^*(2^{0.876n}) time algorithm.
We modify their approach in a way that improves these running times to O^*(2^{(1-1/(2.7d))n}) and O^*{2^{0.804n}), respectively. In particular, our latter bound - that holds for all systems of quadratic equations modulo 2 - comes close to the O^*(2^{0.792n}) expected time bound of an algorithm empirically found to hold for random equation systems in Bardet et al. [J. Complexity, 2013]. Our improvement involves three observations:
1) The Valiant-Vazirani lemma can be used to reduce the solution-finding problem to that of counting solutions modulo 2.
2) The monomials in the probabilistic polynomials used in this solution-counting modulo 2 have a special form that we exploit to obtain better bounds on their number than in Lokshtanov et al. [SODA'17].
3) The problem of solution-counting modulo 2 can be "embedded" in a smaller instance of the original problem, which enables us to apply the algorithm as a subroutine to itself.
equation systems
polynomial method
Mathematics of computing~Combinatorial algorithms
26:1-26:13
Track A: Algorithms, Complexity and Games
Andreas
Björklund
Andreas Björklund
Department of Computer Science, Lund University, Sweden
The Swedish Research Council grant VR 2016-03855 "Algebraic Graph Algorithms".
Petteri
Kaski
Petteri Kaski
Department of Computer Science, Aalto University, Finland
The European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement 338077 "Theory and Practice of Advanced Search and Enumeration".
Ryan
Williams
Ryan Williams
Department of Electrical Engineering and Computer Science & CSAIL, MIT, Cambridge, MA, USA
The U.S. National Science Foundation under grants CCF-1741638 and CCF-1741615.
10.4230/LIPIcs.ICALP.2019.26
Magali Bardet, Jean-Charles Faugère, Bruno Salvy, and Pierre-Jean Spaenlehauer. On the complexity of solving quadratic Boolean systems. J. Complexity, 29(1):53-75, 2013. URL: http://dx.doi.org/10.1016/j.jco.2012.07.001.
http://dx.doi.org/10.1016/j.jco.2012.07.001
Andreas Björklund, Thore Husfeldt, Petteri Kaski, Mikko Koivisto, Jesper Nederlof, and Pekka Parviainen. Fast zeta transforms for lattices with few irreducibles. ACM Trans. Algorithms, 12(1):Art. 4, 19, 2016.
Bruno Buchberger. Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. PhD thesis, Department of Mathematics, University of Innsbruck, 1965.
David A. Cox, John Little, and Donal O'Shea. Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer, Cham, fourth edition, 2015. URL: http://dx.doi.org/10.1007/978-3-319-16721-3.
http://dx.doi.org/10.1007/978-3-319-16721-3
Aviezri S. Fraenkel and Yaacov Yesha. Complexity of problems in games, graphs and algebraic equations. Discrete Applied Mathematics, 1(1-2):15-30, 1979. URL: http://dx.doi.org/10.1016/0166-218X(79)90012-X.
http://dx.doi.org/10.1016/0166-218X(79)90012-X
Antoine Joux and Vanessa Vitse. A crossbred algorithm for solving Boolean polynomial systems. Cryptology ePrint Archive, Report 2017/372, 2017. URL: https://eprint.iacr.org/2017/372.
https://eprint.iacr.org/2017/372
Petteri Kaski, Jukka Kohonen, and Thomas Westerbäck. Fast Möbius inversion in semimodular lattices and ER-labelable posets. Electron. J. Combin., 23(3):Paper 3.26, 13, 2016.
Aviad Kipnis, Jacques Patarin, and Louis Goubin. Unbalanced Oil and Vinegar Signature Schemes. In Jacques Stern, editor, Advances in Cryptology - EUROCRYPT '99, International Conference on the Theory and Application of Cryptographic Techniques, Prague, Czech Republic, May 2-6, 1999, Proceeding, volume 1592 of Lecture Notes in Computer Science, pages 206-222. Springer, 1999. URL: http://dx.doi.org/10.1007/3-540-48910-X_15.
http://dx.doi.org/10.1007/3-540-48910-X_15
Daniel Lokshtanov, Ramamohan Paturi, Suguru Tamaki, R. Ryan Williams, and Huacheng Yu. Beating Brute Force for Systems of Polynomial Equations over Finite Fields. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2190-2202. SIAM, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.143.
http://dx.doi.org/10.1137/1.9781611974782.143
Yuri V. Matiyasevich. Hilbert’s Tenth Problem. Foundations of Computing Series. MIT Press, Cambridge, MA, 1993.
Ernst W. Mayr. Some complexity results for polynomial ideals. J. Complexity, 13(3):303-325, 1997. URL: http://dx.doi.org/10.1006/jcom.1997.0447.
http://dx.doi.org/10.1006/jcom.1997.0447
Michael Mitzenmacher and Eli Upfal. Probability and Computing. Cambridge University Press, Cambridge, second edition, 2017.
Robin A. Moser and Dominik Scheder. A full derandomization of Schöning’s k-SAT algorithm. In Lance Fortnow and Salil P. Vadhan, editors, Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 245-252. ACM, 2011. URL: http://dx.doi.org/10.1145/1993636.1993670.
http://dx.doi.org/10.1145/1993636.1993670
Ruben Niederhagen, Kai-Chun Ning, and Bo-Yin Yang. Implementing Joux-Vitse’s Crossbred Algorithm for Solving MQ Systems over GF(2) on GPUs. Cryptology ePrint Archive, Report 2017/1181, 2017. URL: https://eprint.iacr.org/2017/1181.
https://eprint.iacr.org/2017/1181
Jacques Patarin. Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms. In Ueli M. Maurer, editor, Advances in Cryptology - EUROCRYPT '96, International Conference on the Theory and Application of Cryptographic Techniques, Saragossa, Spain, May 12-16, 1996, Proceeding, volume 1070 of Lecture Notes in Computer Science, pages 33-48. Springer, 1996. URL: http://dx.doi.org/10.1007/3-540-68339-9_4.
http://dx.doi.org/10.1007/3-540-68339-9_4
A. A. Razborov. Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Mat. Zametki, 41(4):598-607, 623, 1987.
Roman Smolensky. Algebraic Methods in the Theory of Lower Bounds for Boolean Circuit Complexity. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 77-82. ACM, 1987. URL: http://dx.doi.org/10.1145/28395.28404.
http://dx.doi.org/10.1145/28395.28404
Leslie G. Valiant and Vijay V. Vazirani. NP is as easy as detecting unique solutions. Theor. Comput. Sci., 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
F. Yates. The Design and Analysis of Factorial Experiments. Imperial Bureau of Soil Science, Harpenden, 1937.
Andreas Björklund, Petteri Kaski, and Ryan Williams
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning
We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r.
We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics.
As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.
quantum algorithms
semidefinite program
convex optimization
Theory of computation~Semidefinite programming
Theory of computation~Quantum query complexity
Theory of computation~Convex optimization
27:1-27:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1710.02581.
We thank Scott Aaronson, Joran van Apeldoorn, Andr{á}s Gilyén, Cupjin Huang, and anonymous reviewers for helpful discussions. We are also grateful to Joran van Apeldoorn and Andr{á}s Gilyén for sharing a working draft of [Apeldoorn and Gilyén, 2018] with us. FB was supported by NSF. CYL and AK were supported by the Department of Defense. TL was supported by NSF CCF-1526380. XW was supported by NSF grants CCF-1755800 and CCF-1816695, and also by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams program.
Fernando G. S. L.
Brandão
Fernando G. S. L. Brandão
Institute of Quantum Information and Matter, California Institute of Technology, USA
Amir
Kalev
Amir Kalev
Joint Center for Quantum Information and Computer Science, University of Maryland, USA
Tongyang
Li
Tongyang Li
Joint Center for Quantum Information and Computer Science, University of Maryland, USA
Cedric Yen-Yu
Lin
Cedric Yen-Yu Lin
Joint Center for Quantum Information and Computer Science, University of Maryland, USA
Krysta M.
Svore
Krysta M. Svore
Station Q, Quantum Architectures and Computation Group, Microsoft Research, USA
Xiaodi
Wu
Xiaodi Wu
Joint Center for Quantum Information and Computer Science, University of Maryland, USA
10.4230/LIPIcs.ICALP.2019.27
Scott Aaronson. The learnability of quantum states. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463(2088):3089-3114, 2007. URL: http://arxiv.org/abs/quant-ph/0608142.
http://arxiv.org/abs/quant-ph/0608142
Scott Aaronson. Shadow Tomography of Quantum States. In Proceedings of the Fiftieth Annual ACM Symposium on Theory of Computing. ACM, 2018. URL: http://arxiv.org/abs/1711.01053.
http://arxiv.org/abs/1711.01053
Scott Aaronson, Xinyi Chen, Elad Hazan, and Ashwin Nayak. Online Learning of Quantum states, 2018. URL: http://arxiv.org/abs/1802.09025.
http://arxiv.org/abs/1802.09025
Joran van Apeldoorn and András Gilyén. Improvements in Quantum SDP-solving with Applications, 2018. URL: http://arxiv.org/abs/1804.05058.
http://arxiv.org/abs/1804.05058
Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Quantum SDP-Solvers: Better upper and lower bounds. In 58th Annual IEEE Symposium on Foundations of Computer Science. IEEE, 2017. URL: http://arxiv.org/abs/1705.01843.
http://arxiv.org/abs/1705.01843
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
Sanjeev Arora and Satyen Kale. A combinatorial, primal-dual approach to semidefinite programs. In Proceedings of the Thirty-ninth Annual ACM Symposium on Theory of Computing, pages 227-236. ACM, 2007.
Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Yen-Yu Lin, Krysta M. Svore, and Xiaodi Wu. Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning, 2019. URL: http://arxiv.org/abs/1710.02581.
http://arxiv.org/abs/1710.02581
Fernando G. S. L. Brandão and Krysta Svore. Quantum speed-ups for semidefinite programming. In 58th Annual IEEE Symposium on Foundations of Computer Science. IEEE, 2017. URL: http://arxiv.org/abs/1609.05537.
http://arxiv.org/abs/1609.05537
Nai-Hui Chia, Tongyang Li, Han-Hsuan Lin, and Chunhao Wang. Quantum-inspired classical sublinear-time algorithm for solving low-rank semidefinite programming via sampling approaches, 2019. URL: http://arxiv.org/abs/1901.03254.
http://arxiv.org/abs/1901.03254
Anirban Narayan Chowdhury and Rolando D. Somma. Quantum algorithms for Gibbs sampling and hitting-time estimation. Quantum Information &Computation, 17(LA-UR-16-21218), 2017. URL: http://arxiv.org/abs/1603.02940.
http://arxiv.org/abs/1603.02940
Gus Gutoski and Xiaodi Wu. Parallel approximation of min-max problems with applications to classical and quantum zero-sum games. In Proceedings of the Twenty-seventh Annual IEEE Symposium on Computational Complexity (CCC), pages 21-31. IEEE, 2012.
Jeongwan Haah, Aram W. Harrow, Zhengfeng Ji, Xiaodi Wu, and Nengkun Yu. Sample-optimal Tomography of Quantum States. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing, pages 913-925. ACM, 2016. URL: http://arxiv.org/abs/1508.01797.
http://arxiv.org/abs/1508.01797
Aram W. Harrow, Cedric Yen-Yu Lin, and Ashley Montanaro. Sequential measurements, disturbance and property testing. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1598-1611. SIAM, 2017. URL: http://arxiv.org/abs/1607.03236.
http://arxiv.org/abs/1607.03236
Elad Hazan. Efficient algorithms for online convex optimization and their applications. PhD thesis, Princeton University, 2006.
Rahul Jain, Zhengfeng Ji, Sarvagya Upadhyay, and John Watrous. QIP=PSPACE. Journal of the ACM (JACM), 58(6):30, 2011. URL: http://arxiv.org/abs/0907.4737.
http://arxiv.org/abs/0907.4737
Edwin T. Jaynes. Information theory and statistical mechanics. Physical Review, 106(4):620, 1957.
Shelby Kimmel, Cedric Yen-Yu Lin, Guang Hao Low, Maris Ozols, and Theodore J. Yoder. Hamiltonian simulation with optimal sample complexity. npj Quantum Information, 3(1):13, 2017. URL: http://arxiv.org/abs/1608.00281.
http://arxiv.org/abs/1608.00281
James R. Lee, Prasad Raghavendra, and David Steurer. Lower Bounds on the Size of Semidefinite Programming Relaxations. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, pages 567-576, New York, NY, USA, 2015. ACM. URL: http://dx.doi.org/10.1145/2746539.2746599.
http://dx.doi.org/10.1145/2746539.2746599
Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. In Proceedings of the Fifty-sixth Annual IEEE Symposium on Foundations of Computer Science, pages 1049-1065. IEEE, 2015. URL: http://arxiv.org/abs/1508.04874.
http://arxiv.org/abs/1508.04874
Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum principal component analysis. Nature Physics, 10(9):631, 2014. URL: http://arxiv.org/abs/1307.0401.
http://arxiv.org/abs/1307.0401
Daniel Nagaj, Pawel Wocjan, and Yong Zhang. Fast amplification of QMA, 2009. URL: http://arxiv.org/abs/0904.1549.
http://arxiv.org/abs/0904.1549
Ryan O'Donnell and John Wright. Efficient Quantum Tomography. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing, pages 899-912. ACM, 2016. URL: http://arxiv.org/abs/1508.01907.
http://arxiv.org/abs/1508.01907
David Poulin and Pawel Wocjan. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. Physical Review Letters, 103(22):220502, 2009. URL: http://arxiv.org/abs/0905.2199.
http://arxiv.org/abs/0905.2199
Ewin Tang. A quantum-inspired classical algorithm for recommendation systems. In Proceedings of the 51st Annual ACM Symposium on Theory of Computing. ACM, 2019. URL: http://arxiv.org/abs/1807.04271.
http://arxiv.org/abs/1807.04271
Ronald de Wolf. Personal communication, 2017.
Xiaodi Wu. Parallelized Solution to Semidefinite Programmings in Quantum Complexity Theory, 2010. URL: http://arxiv.org/abs/1009.2211.
http://arxiv.org/abs/1009.2211
Fernando G. S. L. Brandão, Amir Kalev, Tongyang Li, Cedric Y.-Y. Lin, Krysta M. Svore, and Xiaodi Wu
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Simple Protocol for Verifiable Delegation of Quantum Computation in One Round
The importance of being able to verify quantum computation delegated to remote servers increases with recent development of quantum technologies. In some of the proposed protocols for this task, a client delegates her quantum computation to non-communicating servers in multiple rounds of communication. In this work, we propose the first protocol where the client delegates her quantum computation to two servers in one-round of communication. Another advantage of our protocol is that it is conceptually simpler than previous protocols. The parameters of our protocol also make it possible to prove security even if the servers are allowed to communicate, but respecting the plausible assumption that information cannot be propagated faster than speed of light, making it the first relativistic protocol for quantum computation.
quantum computation
quantum cryptography
delegation of quantum computation
Hardware~Quantum communication and cryptography
Theory of computation~Quantum complexity theory
28:1-28:13
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1711.09585.
I thank Iordanis Kerenidis, Damián Pitalúa-García and Thomas Vidick for useful discussions and comments in early drafts of this manuscript. I also thank anonymous reviewers helped me improving the presentation of this work. Part of this work was done when I was member of IRIF, Université Paris Diderot, Paris, France, where I was supported by ERC QCC and French Programme d’Investissement d’Avenir RISQ P141580.
Alex B.
Grilo
Alex B. Grilo
CWI, Amsterdam, The Netherlands
QuSoft, Amsterdam, The Netherlands
Supported by ERC Consolidator Grant 615307-QPROGRESS.
10.4230/LIPIcs.ICALP.2019.28
Scott Aaronson, Alexandru Cojocaru, Alexandru Gheorghiu, and Elham Kashefi. On the implausibility of classical client blind quantum computing. arXiv preprint, 2017. URL: http://arxiv.org/abs/1704.08482.
http://arxiv.org/abs/1704.08482
Dorit Aharonov, Itai Arad, and Thomas Vidick. Guest column: the quantum PCP conjecture. SIGACT News, 44(2):47-79, 2013. URL: http://dblp.uni-trier.de/db/journals/sigact/sigact44.html#AharonovAV13.
http://dblp.uni-trier.de/db/journals/sigact/sigact44.html#AharonovAV13
Dorit Aharonov, Michael Ben-Or, Elad Eban, and Urmila Mahadev. Interactive proofs for quantum computations. arXiv preprint, 2017. URL: http://arxiv.org/abs/1704.04487.
http://arxiv.org/abs/1704.04487
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http://dx.doi.org/10.1145/278298.278306
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http://arxiv.org/abs/arXiv:1603.06046
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Alex B. Grilo
Creative Commons Attribution 3.0 Unported license
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Dismantlability, Connectedness, and Mixing in Relational Structures
The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, statistical physics, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, topological properties of the solution set such as connectedness is related to the hardness of CSPs over random structures. In approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions or the free energy of spin systems. Additionally, in the decision CSPs, structural properties of the relational structures involved - like, for example, dismantlability - and their logical characterizations have been instrumental for determining the complexity and other properties of the problem.
In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that the structural property of dismantlability of the target graph, the connectedness of the set of homomorphisms, good mixing properties of the corresponding spin system, and the uniqueness of Gibbs measure are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Briceño. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif.
relational structure
constraint satisfaction problem
homomorphism
mixing properties
Gibbs measure
Mathematics of computing~Paths and connectivity problems
29:1-29:15
Track A: Algorithms, Complexity and Games
This work was done in part while the first three authors were visiting the Simons Institute for the Theory of Computing at University of California, Berkeley.
A full version of this paper is available at http://arxiv.org/abs/1901.04398.
Raimundo
Briceño
Raimundo Briceño
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
The first author was supported by ERC Starting Grants 678520 and 676970.
Andrei A.
Bulatov
Andrei A. Bulatov
School of Computing Science, Simon Fraser University, Canada
This work was supported by an NSERC Discovery grant.
Víctor
Dalmau
Víctor Dalmau
Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
The third author was supported by MICCIN grant TIN2016-76573-C2-1P and Maria de Maeztu Units of Excellence Programme MDM-2015-0502.
Benoît
Larose
Benoît Larose
LACIM, Université du Québec a Montréal, Montréal, Canada
The fourth author was supported by an NSERC Discovery grant and FRQNT.
10.4230/LIPIcs.ICALP.2019.29
Dimitris Achlioptas, Paul Beame, and Michael Molloy. Exponential bounds for DPLL below the satisfiability threshold. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, New Orleans, Louisiana, USA, January 11-14, 2004, pages 139-140, 2004.
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Raimundo Briceño. The topological strong spatial mixing property and new conditions for pressure approximation. Ergodic Theory Dynam. Systems, 38(5):1658-1696, 2018.
Raimundo Briceño and Ronnie Pavlov. Strong spatial mixing in homomorphism spaces. SIAM J. Discrete Math., 31(3):2110-2137, 2017.
Raimundo Briceño, Andrei Bulatov, Víctor Dalmau, and Benoit Larose. Long range actions, connectedness, and dismantlability in relational structures. CoRR, abs/1901.04398, 2019. URL: http://arxiv.org/abs/1901.04398.
http://arxiv.org/abs/1901.04398
Raimundo Briceño, Kevin McGoff, and Ronnie Pavlov. Factoring onto ℤ^d subshifts with the finite extension property. Proc. Amer. Math. Soc., 146(12):5129-5140, 2018.
Graham R. Brightwell and Peter Winkler. Graph homomorphisms and phase transitions. J. Combin. Theory Ser. B, 77(2):221-262, 1999.
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Graham R. Brightwell and Peter Winkler. Graph homomorphisms and long range action. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 63:29-48, 2004.
Andrei A. Bulatov. A Dichotomy Theorem for Nonuniform CSPs. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 319-330, 2017.
Andrei A. Bulatov, Víctor Dalmau, Martin Grohe, and Dániel Marx. Enumerating homomorphisms. J. Comput. Syst. Sci., 78(2):638-650, 2012.
Andrei A. Bulatov, Andrei A. Krokhin, and Benoit Larose. Dualities for Constraint Satisfaction Problems. In Complexity of Constraints - An Overview of Current Research Themes [Result of a Dagstuhl Seminar], pages 93-124, 2008.
Tullio Ceccherini-Silberstein and Michel Coornaert. On the density of periodic configurations in strongly irreducible subshifts. Nonlinearity, 25(7):2119, 2012.
Víctor Dalmau, Andrei A. Krokhin, and Benoit Larose. First-Order Definable Retraction Problems for Posets and Reflexive Graph. In 19th IEEE Symposium on Logic in Computer Science (LICS 2004), 14-17 July 2004, Turku, Finland, Proceedings, pages 232-241, 2004.
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Martin Dyer and Catherine Greenhill. The complexity of counting graph homomorphisms. Random Structures &Algorithms, 17(3-4):260-289, 2000.
Martin Dyer, Alistair Sinclair, Eric Vigoda, and Dror Weitz. Mixing in time and space for lattice spin systems: A combinatorial view. Random Structures &Algorithms, 24(4):461-479, 2004.
Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory. SIAM Journal on Computing, 28(1):57-104, 1998.
Jan Foniok, Jaroslav Nesetril, and Claude Tardif. Generalised dualities and maximal finite antichains in the homomorphism order of relational structures. Eur. J. Comb., 29(4):881-899, 2008.
David Gamarnik and Dmitriy Katz. Sequential cavity method for computing free energy and surface pressure. Journal of Statistical Physics, 137(2):205, 2009.
Hans-Otto Georgii. Gibbs Measures and Phase Transitions, volume 9 of De Gruyter Studies in Mathematics. Berlin, 2 edition, 2011.
Tatsuhiko Hatanaka, Takehiro Ito, and Xiao Zhou. Complexity of Reconfiguration Problems for Constraint Satisfaction. CoRR, abs/1812.10629, 2018.
Pavol Hell and Jaroslav Nešetřil. Graphs and homomorphisms, volume 28 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2004.
Takehiro Ito, Erik D. Demaine, Nicholas J. A. Harvey, Christos H. Papadimitriou, Martha Sideri, Ryuhei Uehara, and Yushi Uno. On the complexity of reconfiguration problems. Theor. Comput. Sci., 412(12-14):1054-1065, 2011.
David Kelly and Ivan Rival. Crowns, Fences, and Dismantlable Lattices. Canadian Journal of Mathematics, 26(5):1257-1271, 1974.
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Benoit Larose, Cynthia Loten, and Claude Tardif. A characterisation of first-order constraint satisfaction problems. Log. Methods Comput. Sci., 3(4):4:6, 22, 2007.
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Fabio Martinelli, Enzo Olivieri, and Roberto H Schonmann. For 2-D lattice spin systems weak mixing implies strong mixing. Communications in Mathematical Physics, 165(1):33-47, 1994.
Marc Mézard and Andrea Montanari. Information, Physics, and Computation. Oxford University press, 2009.
Naomi Nishimura. Introduction to Reconfiguration. Algorithms, 11(4):52, 2018.
Richard Nowakowski and Peter Winkler. Vertex-to-vertex pursuit in a graph. Discrete Mathematics, 43(2-3):235-239, 1983.
Ronnie Pavlov and Michael Schraudner. Entropies realizable by block gluing ℤ^d shifts of finite type. Journal d'Analyse Mathématique, 126(1):113-174, 2015.
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Raimundo Briceño, Andrei A. Bulatov, Víctor Dalmau, and Benoît Larose
Creative Commons Attribution 3.0 Unported license
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Sign-Rank Can Increase Under Intersection
The communication class UPP^{cc} is a communication analog of the Turing Machine complexity class PP. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds.
For a communication problem f, let f wedge f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP^{cc}(f)= O(log n), and UPP^{cc}(f wedge f) = Theta(log^2 n). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP^{cc}, the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection.
Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n^{Omega(log n)}. This matches an upper bound of (Klivans, O'Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time.
Sign rank
dimension complexity
communication complexity
learning theory
Theory of computation~Communication complexity
Theory of computation~Boolean function learning
30:1-30:14
Track A: Algorithms, Complexity and Games
https://eccc.weizmann.ac.il/report/2019/027/
Nikhil Mande and Justin Thaler were supported by NSF Grant CCF-1845125.
Mark
Bun
Mark Bun
Simons Institute for the Theory of Computing, Berkeley, CA, USA
Boston University, MA, USA
Nikhil S.
Mande
Nikhil S. Mande
Georgetown University, Washington, DC, USA
Justin
Thaler
Justin Thaler
Georgetown University, Washington, DC, USA
10.4230/LIPIcs.ICALP.2019.30
Andris Ambainis, Andrew M Childs, Ben W Reichardt, Robert Špalek, and Shengyu Zhang. Any AND-OR formula of size N can be evaluated in time N^1/2+o(1) on a quantum computer. SIAM Journal on Computing, 39(6):2513-2530, 2010.
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http://dx.doi.org/10.1109/SFCS.1986.15
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http://dx.doi.org/10.1006/jcss.1995.1017
Arnab Bhattacharyya, Suprovat Ghoshal, and Rishi Saket. Hardness of Learning Noisy Halfspaces using Polynomial Thresholds. In Conference On Learning Theory, COLT 2018, Stockholm, Sweden, 6-9 July 2018., pages 876-917, 2018. URL: http://proceedings.mlr.press/v75/bhattacharyya18a.html.
http://proceedings.mlr.press/v75/bhattacharyya18a.html
Adam Bouland, Lijie Chen, Dhiraj Holden, Justin Thaler, and Prashant Nalini Vasudevan. On the Power of Statistical Zero Knowledge. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 708-719, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.71.
http://dx.doi.org/10.1109/FOCS.2017.71
Mark Bun and Justin Thaler. Improved Bounds on the Sign-Rank of AC⁰. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 37:1-37:14, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.37.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.37
Mark Bun and Justin Thaler. The Large-Error Approximate Degree of AC⁰. Electronic Colloquium on Computational Complexity (ECCC), 25:143, 2018. URL: https://eccc.weizmann.ac.il/report/2018/143.
https://eccc.weizmann.ac.il/report/2018/143
Arkadev Chattopadhyay and Nikhil S. Mande. A Short List of Equalities Induces Large Sign Rank. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 47-58, 2018. URL: http://dx.doi.org/10.1109/FOCS.2018.00014.
http://dx.doi.org/10.1109/FOCS.2018.00014
Vitaly Feldman, Parikshit Gopalan, Subhash Khot, and Ashok Kumar Ponnuswami. New results for learning noisy parities and halfspaces. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), pages 563-574. IEEE, 2006.
Jürgen Forster. A linear lower bound on the unbounded error probabilistic communication complexity. J. Comput. Syst. Sci., 65(4):612-625, 2002. URL: http://dx.doi.org/10.1016/S0022-0000(02)00019-3.
http://dx.doi.org/10.1016/S0022-0000(02)00019-3
Mika Göös, Pritish Kamath, Toniann Pitassi, and Thomas Watson. Query-to-Communication Lifting for P^NP. In 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, pages 12:1-12:16, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.12.
http://dx.doi.org/10.4230/LIPIcs.CCC.2017.12
Mika Göös, Toniann Pitassi, and Thomas Watson. Query-to-Communication Lifting for BPP. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 132-143, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.21.
http://dx.doi.org/10.1109/FOCS.2017.21
Mika Göös, Toniann Pitassi, and Thomas Watson. The Landscape of Communication Complexity Classes. Computational Complexity, 27(2):245-304, 2018. URL: http://dx.doi.org/10.1007/s00037-018-0166-6.
http://dx.doi.org/10.1007/s00037-018-0166-6
Michael Kearns. Efficient noise-tolerant learning from statistical queries. Journal of the ACM (JACM), 45(6):983-1006, 1998.
Subhash Khot and Rishi Saket. On the hardness of learning intersections of two halfspaces. Journal of Computer and System Sciences, 77(1):129-141, 2011.
Adam R. Klivans, Ryan O'Donnell, and Rocco A. Servedio. Learning intersections and thresholds of halfspaces. J. Comput. Syst. Sci., 68(4):808-840, 2004. URL: http://dx.doi.org/10.1016/j.jcss.2003.11.002.
http://dx.doi.org/10.1016/j.jcss.2003.11.002
Adam R Klivans and Rocco A Servedio. Learning DNF in time 2^O (n^1/3). Journal of Computer and System Sciences, 68(2):303-318, 2004.
Adam R Klivans and Alexander A Sherstov. Cryptographic hardness for learning intersections of halfspaces. Journal of Computer and System Sciences, 75(1):2-12, 2009.
Marvin Minsky and Seymour Papert. Perceptrons. MIT Press, 1969.
Ryan O'Donnell and Rocco A Servedio. New degree bounds for polynomial threshold functions. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 325-334. ACM, 2003.
Ramamohan Paturi and Janos Simon. Probabilistic Communication Complexity. J. Comput. Syst. Sci., 33(1):106-123, 1986. URL: http://dx.doi.org/10.1016/0022-0000(86)90046-2.
http://dx.doi.org/10.1016/0022-0000(86)90046-2
Alexander A. Razborov and Alexander A. Sherstov. The Sign-Rank of AC⁰. SIAM J. Comput., 39(5):1833-1855, 2010. URL: http://dx.doi.org/10.1137/080744037.
http://dx.doi.org/10.1137/080744037
Alexander A. Sherstov. The Pattern Matrix Method. SIAM J. Comput., 40(6):1969-2000, 2011. URL: http://dx.doi.org/10.1137/080733644.
http://dx.doi.org/10.1137/080733644
Alexander A Sherstov. The unbounded-error communication complexity of symmetric functions. Combinatorica, 31(5):583-614, 2011.
Alexander A. Sherstov. Optimal bounds for sign-representing the intersection of two halfspaces by polynomials. Combinatorica, 33(1):73-96, 2013. URL: http://dx.doi.org/10.1007/s00493-013-2759-7.
http://dx.doi.org/10.1007/s00493-013-2759-7
Alexander A. Sherstov. The Intersection of Two Halfspaces Has High Threshold Degree. SIAM J. Comput., 42(6):2329-2374, 2013. URL: http://dx.doi.org/10.1137/100785260.
http://dx.doi.org/10.1137/100785260
Alexander A Sherstov. Breaking the Minsky-Papert Barrier for Constant-Depth Circuits. SIAM Journal on Computing, 47(5):1809-1857, 2018.
Alexander A. Sherstov and Pei Wu. Near-Optimal Lower Bounds on the Threshold Degree and Sign-Rank of AC⁰. CoRR, abs/1901.00988, 2019. To appear in STOC 2019. URL: http://arxiv.org/abs/1901.00988.
http://arxiv.org/abs/1901.00988
Mark Bun, Nikhil S. Mande, and Justin Thaler
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Covert Computation in Self-Assembled Circuits
Traditionally, computation within self-assembly models is hard to conceal because the self-assembly process generates a crystalline assembly whose computational history is inherently part of the structure itself. With no way to remove information from the computation, this computational model offers a unique problem: how can computational input and computation be hidden while still computing and reporting the final output? Designing such systems is inherently motivated by privacy concerns in biomedical computing and applications in cryptography.
In this paper we propose the problem of performing "covert computation" within tile self-assembly that seeks to design self-assembly systems that "conceal" both the input and computational history of performed computations. We achieve these results within the growth-only restricted abstract tile assembly model (aTAM) with positive and negative interactions. We show that general-case covert computation is possible by implementing a set of basic covert logic gates capable of simulating any circuit (functionally complete). To further motivate the study of covert computation, we apply our new framework to resolve an outstanding complexity question; we use our covert circuitry to show that the unique assembly verification problem within the growth-only aTAM with negative interactions is coNP-complete.
self-assembly
covert circuits
Theory of computation~Computational complexity and cryptography
31:1-31:14
Track A: Algorithms, Complexity and Games
This research was supported in part by National Science Foundation Grant CCF-1817602.
We would like to thank the anonymous reviewers for their careful review of our work and for their constructive feedback.
Angel A.
Cantu
Angel A. Cantu
Department of Computer Science, University of Texas - Rio Grande Valley, USA
Austin
Luchsinger
Austin Luchsinger
Department of Computer Science, University of Texas - Rio Grande Valley, USA
Robert
Schweller
Robert Schweller
Department of Computer Science, University of Texas - Rio Grande Valley, USA
Tim
Wylie
Tim Wylie
Department of Computer Science, University of Texas - Rio Grande Valley, USA
10.4230/LIPIcs.ICALP.2019.31
Leonard M. Adleman, Qi Cheng, Ashish Goel, Ming-Deh A. Huang, David Kempe, Pablo Moisset de Espanés, and Paul W. K. Rothemund. Combinatorial optimization problems in self-assembly. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 23-32, 2002.
Yuriy Brun. Arithmetic computation in the tile assembly model: Addition and multiplication. Theoretical Comp. Sci., 378:17-31, 2007.
Cameron Chalk, Erik D. Demiane, Martin L. Demaine, Eric Martinez, Robert Schweller, Luis Vega, and Tim Wylie. Universal Shape Replicators via Self-Assembly with Attractive and Repulsive Forces. In Proc. of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'17), 2017.
Cameron Chalk, Austin Luchsinger, Robert Schweller, and Tim Wylie. Self-Assembly of Any Shape with Constant Tile Types using High Temperature. In Proc. of the 26th Annual European Symposium on Algorithms, ESA'18, 2018.
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Angel A. Cantu, Austin Luchsinger, Robert Schweller, and Tim Wylie
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Randomness and Intractability in Kolmogorov Complexity
We introduce randomized time-bounded Kolmogorov complexity (rKt), a natural extension of Levin’s notion [Leonid A. Levin, 1984] of Kolmogorov complexity. A string w of low rKt complexity can be decompressed from a short representation via a time-bounded algorithm that outputs w with high probability.
This complexity measure gives rise to a decision problem over strings: MrKtP (The Minimum rKt Problem). We explore ideas from pseudorandomness to prove that MrKtP and its variants cannot be solved in randomized quasi-polynomial time. This exhibits a natural string compression problem that is provably intractable, even for randomized computations. Our techniques also imply that there is no n^{1 - epsilon}-approximate algorithm for MrKtP running in randomized quasi-polynomial time.
Complementing this lower bound, we observe connections between rKt, the power of randomness in computing, and circuit complexity. In particular, we present the first hardness magnification theorem for a natural problem that is unconditionally hard against a strong model of computation.
computational complexity
randomness
circuit lower bounds
Kolmogorov complexity
Theory of computation
32:1-32:14
Track A: Algorithms, Complexity and Games
I am grateful to Ján Pich, Eric Allender, Ryan Williams, Shuichi Hirahara, Michal Koucký, Rahul Santhanam, and Jan Krajíček for discussions. Part of this work was completed while the author was visiting the Simons Institute for the Theory of Computing. This work was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agreement no. 615075.
Igor Carboni
Oliveira
Igor Carboni Oliveira
Department of Computer Science, University of Oxford, UK
10.4230/LIPIcs.ICALP.2019.32
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Igor C. Oliveira
Creative Commons Attribution 3.0 Unported license
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The Power of Block-Encoded Matrix Powers: Improved Regression Techniques via Faster Hamiltonian Simulation
We apply the framework of block-encodings, introduced by Low and Chuang (under the name standard-form), to the study of quantum machine learning algorithms and derive general results that are applicable to a variety of input models, including sparse matrix oracles and matrices stored in a data structure. We develop several tools within the block-encoding framework, such as singular value estimation of a block-encoded matrix, and quantum linear system solvers using block-encodings. The presented results give new techniques for Hamiltonian simulation of non-sparse matrices, which could be relevant for certain quantum chemistry applications, and which in turn imply an exponential improvement in the dependence on precision in quantum linear systems solvers for non-sparse matrices.
In addition, we develop a technique of variable-time amplitude estimation, based on Ambainis' variable-time amplitude amplification technique, which we are also able to apply within the framework.
As applications, we design the following algorithms: (1) a quantum algorithm for the quantum weighted least squares problem, exhibiting a 6-th power improvement in the dependence on the condition number and an exponential improvement in the dependence on the precision over the previous best algorithm of Kerenidis and Prakash; (2) the first quantum algorithm for the quantum generalized least squares problem; and (3) quantum algorithms for estimating electrical-network quantities, including effective resistance and dissipated power, improving upon previous work.
Quantum algorithms
Hamiltonian simulation
Quantum machine learning
Theory of computation~Quantum computation theory
33:1-33:14
Track A: Algorithms, Complexity and Games
Full version of this submission is available at: https://arxiv.org/abs/1804.01973.
The authors are grateful for Iordanis Kerendis, Anupam Prakash and Michael Walter for useful discussions. SC is supported by the Belgian Fonds de la Recherche Scientifique - FNRS under grants no F.4515.16 (QUICTIME) and R.50.05.18.F (QuantAlgo). AG is supported by ERC Consolidator Grant QPROGRESS. SJ is supported by an NWO WISE Grant and NWO Veni Innovational Research Grant under project number 639.021.752.
Shantanav
Chakraborty
Shantanav Chakraborty
QuIC, Université libre de Bruxelles, Belgium
András
Gilyén
András Gilyén
QuSoft/CWI, The Netherlands
Stacey
Jeffery
Stacey Jeffery
QuSoft/CWI, The Netherlands
10.4230/LIPIcs.ICALP.2019.33
Dorit Aharonov and Amnon Ta-Schma. Adiabatic quantum state generation and statistical zero-knowledge. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC 2003), pages 20-29, 2003. URL: http://dx.doi.org/10.1145/780542.780546.
http://dx.doi.org/10.1145/780542.780546
Andris Ambainis. Variable time amplitude amplification and quantum algorithms for linear algebra problems. In Symposium on Theoretical Aspects of Computer Science STACS, pages 636-647, 2012. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2012.636.
http://dx.doi.org/10.4230/LIPIcs.STACS.2012.636
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http://arxiv.org/abs/1502.03450
Ryan Babbush, Dominic W Berry, Ian D Kivlichan, Annie Y Wei, Peter J Love, and Alán Aspuru-Guzik. Exponentially more precise quantum simulation of fermions in second quantization. New Journal of Physics, 18(3):033032, 2016. URL: http://arxiv.org/abs/1506.01020.
http://arxiv.org/abs/1506.01020
D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma. Exponential improvement in precision for simulating sparse Hamiltonians. In Symposium on Theory of Computing, STOC 2014, pages 283-292, 2014. URL: http://dx.doi.org/10.1145/2591796.2591854.
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Dominic W. Berry, Graeme Ahokas, Richard Cleve, and Barry C. Sanders. Efficient quantum algorithms for simulating sparse Hamiltonians. Communications in Mathematical Physics, 270(2):359-371, 2007. URL: http://dx.doi.org/10.1007/s00220-006-0150-x.
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Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. Simulating Hamiltonian Dynamics with a Truncated Taylor Series. Phys. Rev. Lett., 114:090502, 2015. URL: http://dx.doi.org/10.1103/PhysRevLett.114.090502.
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Dominic W. Berry, Andrew M. Childs, and Robin Kothari. Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters. In FOCS, pages 792-809, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.54.
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Shantanav Chakraborty, András Gilyén, and Stacey Jeffery. The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation, 2018. URL: http://arxiv.org/abs/1804.01973.
http://arxiv.org/abs/1804.01973
Andrew M. Childs. Quantum information processing in continuous time. PhD thesis, Massachusetts Institute of Technology, 2004. URL: https://dspace.mit.edu/handle/1721.1/16663.
https://dspace.mit.edu/handle/1721.1/16663
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Andrew M. Childs, Robin Kothari, and Rolando D. Somma. Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision. SIAM Journal on Computing, 46(6):1920-1950, 2017. URL: http://dx.doi.org/10.1137/16M1087072.
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Andrew M. Childs and Nathan Wiebe. Hamiltonian simulation using linear combinations of unitaries. Quantum Information and Computation, 12(11-12):901-924, 2012. URL: http://arxiv.org/abs/1202.5822.
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S. Chakraborty, A. Gilyén, and S. Jeffery
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Unlabeled Sample Compression Schemes and Corner Peelings for Ample and Maximum Classes
We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by H. Tracy Hall (2004) of a partial shelling of the cross-polytope which can not be extended. We use it to derive a maximum class of VC dimension 3 that has no corners. This refutes several previous works in machine learning from the past 11 years. In particular, it implies that the previous constructions of optimal unlabeled compression schemes for maximum classes are erroneous.
On the positive side we present a new construction of an optimal unlabeled compression scheme for maximum classes. We leave as open whether our unlabeled compression scheme extends to ample (a.k.a. lopsided or extremal) classes, which represent a natural and far-reaching generalization of maximum classes. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the 1-skeletons of associated cubical complexes.
VC-dimension
sample compression
Sauer-Shelah-Perles lemma
Sandwich lemma
maximum class
ample/extremal class
corner peeling
unique sink orientation
Mathematics of computing~Combinatorics
Theory of computation~Machine learning theory
Theory of computation~Computational geometry
34:1-34:15
Track A: Algorithms, Complexity and Games
The research of J.C. and V.C. is supported by the ANR project DISTANCIA (ANR-17-CE40-0015). The research of S.M. is supported by the Simons Foundation and the NSF; part of this project was carried while S.M. was at the Institute for Advanced Study and was supported by the National Science Foundation under agreement No. CCF-1412958. The research of M.W. is supported by the NSF grant IIS-1619271.
A full version of this paper is available at http://arxiv.org/abs/1812.02099.
The authors are grateful to Olivier Bousquet for insightful discussions and to the anonymous referees for useful remarks that helped improving the presentation of this work.
Jérémie
Chalopin
Jérémie Chalopin
CNRS, Aix-Marseille Université, Université de Toulon, LIS, Marseille, France
https://orcid.org/0000-0002-2988-8969
Victor
Chepoi
Victor Chepoi
Aix-Marseille Université, CNRS, Université de Toulon, LIS, Marseille, France
https://orcid.org/0000-0002-0481-7312
Shay
Moran
Shay Moran
Department of Computer Science, Princeton University, Princeton, USA
https://orcid.org/0000-0002-8662-2737
Manfred K.
Warmuth
Manfred K. Warmuth
Computer Science Department, University of California, Santa Cruz, USA
10.4230/LIPIcs.ICALP.2019.34
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Jérémie Chalopin, Victor Chepoi, Shay Moran, and Manfred K. Warmuth. Unlabeled Sample Compression Schemes and Corner Peelings for Ample and Maximum Classes. arXiv preprint, 1812.02099, 2018. URL: http://arxiv.org/abs/1812.02099.
http://arxiv.org/abs/1812.02099
Thorsten Doliwa, Gaojian Fan, Hans Ulrich Simon, and Sandra Zilles. Recursive Teaching Dimension, VC-dimension and Sample Compression. J. Mach. Learn. Res., 15(1):3107-3131, 2014. URL: http://jmlr.org/papers/v15/doliwa14a.html.
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H. Tracy Hall. Counterexamples in Discrete Geometry. PhD thesis, University of California, 2004.
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http://arxiv.org/abs/1211.2980
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Dömötör Pálvölgyi and Gábor Tardos. Unlabeled Compression Schemes Exceeding the VC-dimension. arXiv preprint, 1811.12471, 2018. URL: http://arxiv.org/abs/1811.12471.
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Jérémie Chalopin, Victor Chepoi, Shay Moran, and Manfred K. Warmuth
Creative Commons Attribution 3.0 Unported license
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Query-To-Communication Lifting for BPP Using Inner Product
We prove a new query-to-communication lifting for randomized protocols, with inner product as gadget. This allows us to use a much smaller gadget, leading to a more efficient lifting. Prior to this work, such a theorem was known only for deterministic protocols, due to Chattopadhyay et al. [Arkadev Chattopadhyay et al., 2017] and Wu et al. [Xiaodi Wu et al., 2017]. The only query-to-communication lifting result for randomized protocols, due to Göös, Pitassi and Watson [Mika Göös et al., 2017], used the much larger indexing gadget.
Our proof also provides a unified treatment of randomized and deterministic lifting. Most existing proofs of deterministic lifting theorems use a measure of information known as thickness. In contrast, Göös, Pitassi and Watson [Mika Göös et al., 2017] used blockwise min-entropy as a measure of information. Our proof uses the blockwise min-entropy framework to prove lifting theorems in both settings in a unified way.
lifting theorems
inner product
BPP Lifting
Deterministic Lifting
Theory of computation~Communication complexity
Theory of computation~Oracles and decision trees
35:1-35:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at [Chattopadhyay et al., 2019], https://arxiv.org/abs/1904.13056.
We thank Daniel Kane for some very enlightening conversations and suggestions. This work was done (in part) while the authors were visiting the Simons Institute for the Theory of Computing.
Arkadev
Chattopadhyay
Arkadev Chattopadhyay
School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India
http://www.tcs.tifr.res.in/~arkadev/
Yuval
Filmus
Yuval Filmus
Department of Computer Science, Technion Israel Institute of Technology, Haifa, Israel
https://filmus.net.technion.ac.il/
https://orcid.org/0000-0002-1739-0872
Taub Fellow - supported by the Taub Foundations. The research was funded by ISF grant 1337/16.
Sajin
Koroth
Sajin Koroth
Department of Computer Science, University of Haifa, Haifa, Israel
https://sites.google.com/csweb.haifa.ac.il/sajin
https://orcid.org/0000-0002-7989-1963
Supported by the Israel Science Foundation (grant No. 1445/16)
Or
Meir
Or Meir
Department of Computer Science, University of Haifa, Haifa, Israel
http://cs.haifa.ac.il/~ormeir/
https://orcid.org/0000-0001-5031-0750
Partially supported by ISF grant by the Israel Science Foundation (grant No. 1445/16).
Toniann
Pitassi
Toniann Pitassi
Department of Computer Science, University of Toronto, Canada
https://www.cs.toronto.edu/~toni/
https://orcid.org/0000-0003-0832-2760
10.4230/LIPIcs.ICALP.2019.35
Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 67-76, 2010.
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Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, and Toniann Pitassi. Query-to-communication lifting for BPP using inner product, April 2019. URL: http://arxiv.org/abs/1904.13056.
http://arxiv.org/abs/1904.13056
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Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, and Toniann Pitassi
Creative Commons Attribution 3.0 Unported license
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Estimating the Frequency of a Clustered Signal
We consider the problem of locating a signal whose frequencies are "off grid" and clustered in a narrow band. Given noisy sample access to a function g(t) with Fourier spectrum in a narrow range [f_0 - Delta, f_0 + Delta], how accurately is it possible to identify f_0? We present generic conditions on g that allow for efficient, accurate estimates of the frequency. We then show bounds on these conditions for k-Fourier-sparse signals that imply recovery of f_0 to within Delta + O~(k^3) from samples on [-1, 1]. This improves upon the best previous bound of O(Delta + O~(k^5))^{1.5}. We also show that no algorithm can do better than Delta + O~(k^2).
In the process we provide a new O~(k^3) bound on the ratio between the maximum and average value of continuous k-Fourier-sparse signals, which has independent application.
sublinear algorithms
Fourier transform
Theory of computation~Design and analysis of algorithms
Theory of computation~Streaming, sublinear and near linear time algorithms
36:1-36:13
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1904.13043.
We thank Daniel Kane and Zhao Song for many helpful discussions. We also thank the anonymous referee for the detailed feedback and comments.
Xue
Chen
Xue Chen
Northwestern University, Evanston, IL, USA
Supported by research funding from Northwestern University. Part of this work was done while the author was in the University of Texas at Austin supported by NSF Grant CCF-1526952 and a Simons Investigator Award (#409864, David Zuckerman).
Eric
Price
Eric Price
The University of Texas at Austin, USA
Supported in part by NSF Award CCF-1751040 (CAREER).
10.4230/LIPIcs.ICALP.2019.36
A. Akavia, S. Goldwasser, and S. Safra. Proving hard-core predicates using list decoding. FOCS, 44:146-159, 2003.
Haim Avron, Michael Kapralov, Cameron Musco, Christopher Musco, Ameya Velingker, and Amir Zandieh. A Universal Sampling Method for Reconstructing Signals with Simple Fourier Transforms. In Proceedings of the 51st annual ACM symposium on Theory of computing (STOC 2019), 2019. URL: http://arxiv.org/abs/1812.08723.
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http://arxiv.org/abs/1609.01361
Xue Chen and Eric Price. Active Regression via Linear-Sample Sparsification. In the 32nd Annual Conference on Learning Theory (COLT 2019), 2019.
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Anna C Gilbert, S Muthukrishnan, and Martin Strauss. Improved time bounds for near-optimal sparse Fourier representations. In Optics &Photonics 2005, pages 59141A-59141A. International Society for Optics and Photonics, 2005.
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Piotr Indyk and Michael Kapralov. Sample-Optimal Fourier Sampling in Any Constant Dimension. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 514-523. IEEE, 2014.
Y. Mansour. Randomized Interpolation and Approximation of Sparse Polynomials. ICALP, 1992.
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R Prony. Essai experimental et analytique. J. de l’Ecole Polytechnique, 1795.
Xue Chen and Eric Price
Creative Commons Attribution 3.0 Unported license
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Block Edit Errors with Transpositions: Deterministic Document Exchange Protocols and Almost Optimal Binary Codes
Document exchange and error correcting codes are two fundamental problems regarding communications. In the first problem, Alice and Bob each holds a string, and the goal is for Alice to send a short sketch to Bob, so that Bob can recover Alice’s string. In the second problem, Alice sends a message with some redundant information to Bob through a channel that can add adversarial errors, and the goal is for Bob to correctly recover the message despite the errors. In both problems, an upper bound is placed on the number of errors between the two strings or that the channel can add, and a major goal is to minimize the size of the sketch or the redundant information. In this paper we focus on deterministic document exchange protocols and binary error correcting codes.
Both problems have been studied extensively. In the case of Hamming errors (i.e., bit substitutions) and bit erasures, we have explicit constructions with asymptotically optimal parameters. However, other error types are still rather poorly understood. In a recent work [Kuan Cheng et al., 2018], the authors constructed explicit deterministic document exchange protocols and binary error correcting codes for edit errors with almost optimal parameters. Unfortunately, the constructions in [Kuan Cheng et al., 2018] do not work for other common errors such as block transpositions.
In this paper, we generalize the constructions in [Kuan Cheng et al., 2018] to handle a much larger class of errors. These include bursts of insertions and deletions, as well as block transpositions. Specifically, we consider document exchange and error correcting codes where the total number of block insertions, block deletions, and block transpositions is at most k <= alpha n/log n for some constant 0<alpha<1. In addition, the total number of bits inserted and deleted by the first two kinds of operations is at most t <= beta n for some constant 0<beta<1, where n is the length of Alice’s string or message. We construct explicit, deterministic document exchange protocols with sketch size O((k log n +t) log^2 n/{k log n + t}) and explicit binary error correcting code with O(k log n log log log n+t) redundant bits. As a comparison, the information-theoretic optimum for both problems is Theta(k log n+t). As far as we know, previously there are no known explicit deterministic document exchange protocols in this case, and the best known binary code needs Omega(n) redundant bits even to correct just one block transposition [L. J. Schulman and D. Zuckerman, 1999].
Deterministic document exchange
error correcting code
block edit error
Mathematics of computing~Coding theory
37:1-37:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1809.00725.
We thank an anonymous referee for catching an error in the previous version of this paper, and Bernhard Haeupler for very useful feedbacks.
Kuan
Cheng
Kuan Cheng
Department of Computer Science, Johns Hopkins University, USA
Zhengzhong
Jin
Zhengzhong Jin
Department of Computer Science, Johns Hopkins University, USA
Xin
Li
Xin Li
Department of Computer Science, Johns Hopkins University, USA
Ke
Wu
Ke Wu
Department of Computer Science, Johns Hopkins University, USA
10.4230/LIPIcs.ICALP.2019.37
Djamal Belazzougui. Efficient Deterministic Single Round Document Exchange for Edit Distance. CoRR, abs/1511.09229, 2015. URL: http://arxiv.org/abs/1511.09229.
http://arxiv.org/abs/1511.09229
Djamal Belazzougui and Qin Zhang. Edit Distance: Sketching, Streaming, and Document Exchange. In Proceedings of the 57th IEEE Annual Symposium on Foundations of Computer Science, pages 51-60. IEEE, 2016.
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http://dx.doi.org/10.1109/TIT.2017.2746566
Boris Bukh and Venkatesan Guruswami. An improved bound on the fraction of correctable deletions. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms, pages 1893-1901. ACM, 2016.
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K. Cheng, B. Haeupler, X. Li, A. Shahrasbi, and K. Wu. Synchronization Strings: Efficient and Fast Deterministic Constructions over Small Alphabets. ArXiv e-prints, March 2018. URL: http://arxiv.org/abs/1803.03530.
http://arxiv.org/abs/1803.03530
Kuan Cheng, Zhengzhong Jin, Xin Li, and Ke Wu. Deterministic Document Exchange Protocols, and Almost Optimal Binary Codes for Edit Errors. In Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS). IEEE, 2018.
Graham Cormode and S. Muthukrishnan. The string edit distance matching problem with moves. ACM Transactions on Algorithms, 3(1), 2007.
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V. Guruswami and R. Li. Efficiently decodable insertion/deletion codes for high-noise and high-rate regimes. In 2016 IEEE International Symposium on Information Theory (ISIT), pages 620-624, July 2016. URL: http://dx.doi.org/10.1109/ISIT.2016.7541373.
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V. Guruswami and C. Wang. Deletion Codes in the High-Noise and High-Rate Regimes. IEEE Transactions on Information Theory, 63(4):1961-1970, April 2017. URL: http://dx.doi.org/10.1109/TIT.2017.2659765.
http://dx.doi.org/10.1109/TIT.2017.2659765
Bernhard Haeupler. Optimal Document Exchange and New Codes for Small Number of Insertions and Deletions. arXiv preprint, 2018. URL: http://arxiv.org/abs/1804.03604.
http://arxiv.org/abs/1804.03604
Bernhard Haeupler and Amirbehshad Shahrasbi. Synchronization strings: codes for insertions and deletions approaching the Singleton bound. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 33-46. ACM, 2017.
Bernhard Haeupler and Amirbehshad Shahrasbi. Synchronization Strings: Explicit Constructions, Local Decoding, and Applications. In Proceedings of the 50th Annual ACM Symposium on Theory of Computing, 2018.
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Tom Høholdt, Jacobus H Van Lint, and Ruud Pellikaan. Algebraic geometry codes. Handbook of coding theory, 1(Part 1):871-961, 1998.
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Hossein Jowhari. Efficient Communication Protocols for Deciding Edit Distance. In ESA, 2012.
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http://dx.doi.org/10.1109/SURV.2010.020110.00079
A. Orlitsky. Interactive communication: balanced distributions, correlated files, and average-case complexity. In [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science, pages 228-238, October 1991. URL: http://dx.doi.org/10.1109/SFCS.1991.185373.
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http://dx.doi.org/10.1109/18.796406
D Shapira and J. A. Storer. Edit distance with move operations. In Proceedings of the 13th Symposium on Combinatorial Pattern Matching, pages 85-98, 2002.
Kuan Cheng, Zhengzhong Jin, Xin Li, and Ke Wu
Creative Commons Attribution 3.0 Unported license
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Restricted Max-Min Allocation: Approximation and Integrality Gap
Asadpour, Feige, and Saberi proved that the integrality gap of the configuration LP for the restricted max-min allocation problem is at most 4. However, their proof does not give a polynomial-time approximation algorithm. A lot of efforts have been devoted to designing an efficient algorithm whose approximation ratio can match this upper bound for the integrality gap. In ICALP 2018, we present a (6 + delta)-approximation algorithm where delta can be any positive constant, and there is still a gap of roughly 2. In this paper, we narrow the gap significantly by proposing a (4+delta)-approximation algorithm where delta can be any positive constant. The approximation ratio is with respect to the optimal value of the configuration LP, and the running time is poly(m,n)* n^{poly(1/(delta))} where n is the number of players and m is the number of resources. We also improve the upper bound for the integrality gap of the configuration LP to 3 + 21/26 =~ 3.808.
fair allocation
configuration LP
approximation
integrality gap
Theory of computation~Scheduling algorithms
38:1-38:13
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1905.06084.
Siu-Wing
Cheng
Siu-Wing Cheng
Department of Computer Science and Engineering, HKUST, Hong Kong
https://orcid.org/0000-0002-3557-9935
Yuchen
Mao
Yuchen Mao
Department of Computer Science and Engineering, HKUST, Hong Kong
https://orcid.org/0000-0002-1075-344X
10.4230/LIPIcs.ICALP.2019.38
C. Annamalai, C. Kalaitzis, and O. Svensson. Combinatorial Algorithm for Restricted Max-Min Fair Allocation. ACM Transactions on Algorithms, 13(3):37:1-37:28, 2017.
A. Asadpour, U. Feige, and A. Saberi. Santa Claus meets hypergraph matchings. ACM Transactions on Algorithms, 8(3):24:1-24:9, 2012.
A. Asadpour and A. Saberi. An approximation algorithm for max-min fair allocation of indivisible goods. In Proceedings of the 39th ACM Symposium on Theory of Computing, pages 114-121, 2007.
N. Bansal and M. Sviridenko. The Santa Claus problem. In Proceedings of the 38th ACM Symposium on Theory of Computing, pages 31-40, 2006.
M. Bateni, M. Charikar, and V. Guruswami. Max-Min Allocation via Degree Lower-bounded Arborescences. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pages 543-552, 2009.
I. Bezáková and Varsha Dani. Allocating indivisible goods. SIGecom Exchanges, 5(3):11-18, 2005.
D. Chakrabarty, J. Chuzhoy, and S. Khanna. On Allocating Goods to Maximize Fairness. In Proceedings of the 50th IEEE Symposium on Foundations of Computer Science, pages 107-116, 2009.
S.-W. Cheng and Y. Mao. Integrality Gap of the Configuration LP for the Restricted Max-Min Fair Allocation. CoRR, abs/1807.04152, 2018. URL: http://arxiv.org/abs/1807.04152.
http://arxiv.org/abs/1807.04152
S.-W. Cheng and Y. Mao. Restricted Max-Min Fair Allocation. In Proceedings of the 45th International Colloquium on Automata, Languages, and Programming, pages 37:1-37:13, 2018.
S. Davies, T. Rothvoss, and Y. Zhang. A Tale of Santa Claus, Hypergraphs and Matroids. CoRR, abs/1807.07189, 2018. URL: http://arxiv.org/abs/1807.07189.
http://arxiv.org/abs/1807.07189
U. Feige. On allocations that maximize fairness. In Proceedings of the 19th ACM-SIAM Symposium on Discrete Algorithms, pages 287-293, 2008.
B. Haeupler, B. Saha, and A. Srinivasan. New Constructive Aspects of the Lovász Local Lemma. Journal of the ACM, 58(6):28:1-28:28, 2011.
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K. Jansen and L. Rohwedder. A note on the integrality gap of the configuration LP for restricted Santa Claus. CoRR, abs/1807.03626, 2018. URL: http://arxiv.org/abs/1807.03626.
http://arxiv.org/abs/1807.03626
J.K. Lenstra, D.B. Shmoys, and É. Tardos. Approximation Algorithms for Scheduling Unrelated Parallel Machines. In Proceedings of the 28th IEEE Symposium on Foundations of Computer Science, pages 217-224, 1987.
L. Polacek and O. Svensson. Quasi-polynomial Local Search for Restricted Max-Min Fair Allocation. In 39th International Colloquium on Automata, Languages, and Programming, pages 726-737, 2012.
B. Saha and A. Srinivasan. A New Approximation Technique for Resource-Allocation Problems. In Proceedings of the 1st Symposium on Innovations in Computer Science, pages 342-357, 2010.
Siu-Wing Cheng and Yuchen Mao
Creative Commons Attribution 3.0 Unported license
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Circuit Lower Bounds for MCSP from Local Pseudorandom Generators
The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function f can be computed by a Boolean circuit of size at most theta, for a given parameter theta. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a PRG is called local if its output bit strings, when viewed as the truth table of a Boolean function, can be computed by a Boolean circuit of small size. We get new and improved lower bounds for MCSP that almost match the best-known lower bounds against several circuit models. Specifically, we show that computing MCSP, on functions with a truth table of length N, requires
- N^{3-o(1)}-size de Morgan formulas, improving the recent N^{2-o(1)} lower bound by Hirahara and Santhanam (CCC, 2017),
- N^{2-o(1)}-size formulas over an arbitrary basis or general branching programs (no non-trivial lower bound was known for MCSP against these models), and
- 2^{Omega (N^{1/(d+2.01)})}-size depth-d AC^0 circuits, improving the superpolynomial lower bound by Allender et al. (SICOMP, 2006).
The AC^0 lower bound stated above matches the best-known AC^0 lower bound (for PARITY) up to a small additive constant in the depth. Also, for the special case of depth-2 circuits (i.e., CNFs or DNFs), we get an almost optimal lower bound of 2^{N^{1-o(1)}} for MCSP.
minimum circuit size problem (MCSP)
circuit lower bounds
pseudorandom generators (PRGs)
local PRGs
de Morgan formulas
branching programs
constant depth circuits
Theory of computation~Circuit complexity
Theory of computation~Pseudorandomness and derandomization
39:1-39:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://eccc.weizmann.ac.il/report/2019/022/.
We thank the anonymous ICALP'19 reviewers for their excellent comments.
Mahdi
Cheraghchi
Mahdi Cheraghchi
Department of Computing, Imperial College London, London, UK
http://mahdi.ch
https://orcid.org/0000-0001-8957-0306
Valentine
Kabanets
Valentine Kabanets
School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
https://www.cs.sfu.ca/~kabanets/
Zhenjian
Lu
Zhenjian Lu
School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
Dimitrios
Myrisiotis
Dimitrios Myrisiotis
Department of Computing, Imperial College London, London, UK
10.4230/LIPIcs.ICALP.2019.39
Miklós Ajtai and Avi Wigderson. Deterministic Simulation of Probabilistic Constant Depth Circuits. Advances in Computing Research, 5:199-222, 1989. URL: http://dx.doi.org/10.1109/SFCS.1985.19.
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https://eccc.weizmann.ac.il/report/2018/138/
Shuichi Hirahara and Rahul Santhanam. On the Average-Case Complexity of MCSP and Its Variants. In CCC, pages 7:1-7:20, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.7.
http://dx.doi.org/10.4230/LIPIcs.CCC.2017.7
Russell Impagliazzo, Raghu Meka, and David Zuckerman. Pseudorandomness from Shrinkage. In FOCS, pages 111-119, 2012. URL: https://eccc.weizmann.ac.il/report/2012/057/.
https://eccc.weizmann.ac.il/report/2012/057/
Valentine Kabanets and Jin-yi Cai. Circuit minimization problem. In STOC, pages 73-79, 2000. URL: https://eccc.weizmann.ac.il/report/1999/045/.
https://eccc.weizmann.ac.il/report/1999/045/
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http://dx.doi.org/10.1006/jcss.1996.0004
Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam. Hardness magnification near state-of-the-art lower bounds. Electronic Colloquium on Computational Complexity (ECCC), 25:158, 2018. URL: https://eccc.weizmann.ac.il/report/2018/158/.
https://eccc.weizmann.ac.il/report/2018/158/
Igor Carboni Oliveira and Rahul Santhanam. Conspiracies Between Learning Algorithms, Circuit Lower Bounds, and Pseudorandomness. In CCC, pages 18:1-18:49, 2017. URL: https://eccc.weizmann.ac.il/report/2016/197/.
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Igor Carboni Oliveira and Rahul Santhanam. Hardness Magnification for Natural Problems. In FOCS, pages 65-76, 2018. URL: https://eccc.weizmann.ac.il/report/2018/139/.
https://eccc.weizmann.ac.il/report/2018/139/
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Avishay Tal. Formula lower bounds via the quantum method. In STOC, pages 1256-1268, 2017. URL: http://dx.doi.org/10.1145/3055399.3055472.
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Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, and Dimitrios Myrisiotis
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Norms of Graph Spanners
A t-spanner of a graph G is a subgraph H in which all distances are preserved up to a multiplicative t factor. A classical result of Althöfer et al. is that for every integer k and every graph G, there is a (2k-1)-spanner of G with at most O(n^{1+1/k}) edges. But for some settings the more interesting notion is not the number of edges, but the degrees of the nodes. This spurred interest in and study of spanners with small maximum degree. However, this is not necessarily a robust enough objective: we would like spanners that not only have small maximum degree, but also have "few" nodes of "large" degree. To interpolate between these two extremes, in this paper we initiate the study of graph spanners with respect to the l_p-norm of their degree vector, thus simultaneously modeling the number of edges (the l_1-norm) and the maximum degree (the l_{infty}-norm). We give precise upper bounds for all ranges of p and stretch t: we prove that the greedy (2k-1)-spanner has l_p norm of at most max(O(n), O(n^{{k+p}/{kp}})), and that this bound is tight (assuming the Erdős girth conjecture). We also study universal lower bounds, allowing us to give "generic" guarantees on the approximation ratio of the greedy algorithm which generalize and interpolate between the known approximations for the l_1 and l_{infty} norm. Finally, we show that at least in some situations, the l_p norm behaves fundamentally differently from l_1 or l_{infty}: there are regimes (p=2 and stretch 3 in particular) where the greedy spanner has a provably superior approximation to the generic guarantee.
spanners
approximations
Theory of computation~Sparsification and spanners
40:1-40:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1903.07418.
Eden
Chlamtáč
Eden Chlamtáč
Ben Gurion University of the Negev, Beersheva, Israel
Supported in part by ISF grant 1002/14.
Michael
Dinitz
Michael Dinitz
Johns Hopkins University, Baltimore, MD, USA
Supported in part by NSF awards CCF-1464239 and CCF-1535887.
Thomas
Robinson
Thomas Robinson
Ben Gurion University of the Negev, Beersheva, Israel
Supported in part by ISF grant 1002/14.
10.4230/LIPIcs.ICALP.2019.40
Noga Alon, Yossi Azar, Gerhard J. Woeginger, and Tal Yadid. Approximation Schemes for Scheduling. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '97, 1997.
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http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.80
Eden Chlamtac, Michael Dinitz, and Robert Krauthgamer. Everywhere-Sparse Spanners via Dense Subgraphs. In Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, FOCS '12, pages 758-767, Washington, DC, USA, 2012. IEEE Computer Society. URL: http://dx.doi.org/10.1109/FOCS.2012.61.
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Eden Chlamtác, Michael Dinitz, and Yury Makarychev. Minimizing the Union: Tight Approximations for Small Set Bipartite Vertex Expansion. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 881-899. SIAM, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.56.
http://dx.doi.org/10.1137/1.9781611974782.56
Eden Chlamtác, Michael Dinitz, and Thomas Robinson. The Norms of Graph Spanners. CoRR, abs/1903.07418, 2019. URL: http://arxiv.org/abs/1903.07418.
http://arxiv.org/abs/1903.07418
Eden Chlamtác and Pasin Manurangsi. Sherali-Adams Integrality Gaps Matching the Log-Density Threshold. In Eric Blais, Klaus Jansen, José D. P. Rolim, and David Steurer, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2018, August 20-22, 2018 - Princeton, NJ, USA, volume 116 of LIPIcs, pages 10:1-10:19. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.10.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.10
Eden Chlamtác, Pasin Manurangsi, Dana Moshkovitz, and Aravindan Vijayaraghavan. Approximation Algorithms for Label Cover and The Log-Density Threshold. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 900-919. SIAM, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.57.
http://dx.doi.org/10.1137/1.9781611974782.57
Michael Dinitz, Guy Kortsarz, and Ran Raz. Label Cover Instances with Large Girth and the Hardness of Approximating Basic k-Spanner. ACM Trans. Algorithms, 12(2):25:1-25:16, December 2015. URL: http://dx.doi.org/10.1145/2818375.
http://dx.doi.org/10.1145/2818375
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http://dx.doi.org/10.1145/1993636.1993680
Michael Dinitz and Robert Krauthgamer. Fault-tolerant Spanners: Better and Simpler. In Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, PODC '11, pages 169-178, New York, NY, USA, 2011. ACM. URL: http://dx.doi.org/10.1145/1993806.1993830.
http://dx.doi.org/10.1145/1993806.1993830
Michael Dinitz and Zeyu Zhang. Approximating Low-stretch Spanners. In Proceedings of the Twenty-seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 821-840, Philadelphia, PA, USA, 2016. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2884435.2884494.
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Arnold Filtser and Shay Solomon. The Greedy Spanner is Existentially Optimal. In Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC '16, pages 9-17, 2016.
Daniel Golovin, Anupam Gupta, Amit Kumar, and Kanat Tangwongsan. All-Norms and All-L_p-Norms Approximation Algorithms. In FSTTCS, volume 2 of LIPIcs, pages 199-210. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2008.
Tapas Kanungo, David M. Mount, Nathan S. Netanyahu, Christine D. Piatko, Ruth Silverman, and Angela Y. Wu. A local search approximation algorithm for k-means clustering. Computational Geometry, 28(2):89-112, 2004. Special Issue on the 18th Annual Symposium on Computational Geometry - SoCG2002. URL: http://dx.doi.org/10.1016/j.comgeo.2004.03.003.
http://dx.doi.org/10.1016/j.comgeo.2004.03.003
Guy Kortsarz and David Peleg. Generating Low-Degree 2-Spanners. SIAM J. Comput., 27(5):1438-1456, 1998. URL: http://dx.doi.org/10.1137/S0097539794268753.
http://dx.doi.org/10.1137/S0097539794268753
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Eden Chlamtáč, Michael Dinitz, and Thomas Robinson
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On the Fixed-Parameter Tractability of Capacitated Clustering
We study the complexity of the classic capacitated k-median and k-means problems parameterized by the number of centers, k. These problems are notoriously difficult since the best known approximation bound for high dimensional Euclidean space and general metric space is Theta(log k) and it remains a major open problem whether a constant factor exists.
We show that there exists a (3+epsilon)-approximation algorithm for the capacitated k-median and a (9+epsilon)-approximation algorithm for the capacitated k-means problem in general metric spaces whose running times are f(epsilon,k) n^{O(1)}. For Euclidean inputs of arbitrary dimension, we give a (1+epsilon)-approximation algorithm for both problems with a similar running time. This is a significant improvement over the (7+epsilon)-approximation of Adamczyk et al. for k-median in general metric spaces and the (69+epsilon)-approximation of Xu et al. for Euclidean k-means.
approximation algorithms
fixed-parameter tractability
capacitated
k-median
k-means
clustering
core-sets
Euclidean
Theory of computation~Facility location and clustering
Theory of computation~Fixed parameter tractability
Mathematics of computing~Probabilistic algorithms
Mathematics of computing~Dimensionality reduction
41:1-41:14
Track A: Algorithms, Complexity and Games
Vincent
Cohen-Addad
Vincent Cohen-Addad
CNRS & Sorbonne Université, Paris, France
Ce projet a bénéficié d'une aide de l'Etat gérée par l'Agence Nationale de la Recherche au titre du Programme Appel à projets générique JCJC 2018 portant la référence suivante: ANR-18-CE40-0004-01.
Jason
Li
Jason Li
Carnegie Mellon University, Pittsburgh, PA, USA
Supported in part by NSF awards CCF-1536002, CCF-1540541, and CCF-1617790.
10.4230/LIPIcs.ICALP.2019.41
M. Adamczyk, J. Byrka, J. Marcinkowski, S. M. Meesum, and M. Włodarczyk. Constant factor FPT approximation for capacitated k-median. ArXiv e-prints, September 2018. URL: http://arxiv.org/abs/1809.05791.
http://arxiv.org/abs/1809.05791
Sanjeev Arora, Prabhakar Raghavan, and Satish Rao. Approximation Schemes for Euclidean k-Medians and Related Problems. In Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 106-113, 1998. URL: http://dx.doi.org/10.1145/276698.276718.
http://dx.doi.org/10.1145/276698.276718
Jarosław Byrka, Krzysztof Fleszar, Bartosz Rybicki, and Joachim Spoerhase. Bi-factor approximation algorithms for hard capacitated k-median problems. In Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms, pages 722-736. SIAM, 2014.
Jarosław Byrka, Bartosz Rybicki, and Sumedha Uniyal. An Approximation Algorithm for Uniform Capacitated k-Median Problem with 1+ε Capacity Violation. In International Conference on Integer Programming and Combinatorial Optimization, pages 262-274. Springer, 2016.
Moses Charikar, Chandra Chekuri, Ashish Goel, and Sudipto Guha. Rounding via Trees: Deterministic Approximation Algorithms for Group Steiner Trees and k-Median. In STOC, volume 98, pages 114-123. Citeseer, 1998.
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.
K. Chen. On Coresets for k-Median and k-Means Clustering in Metric and ELuclidean Spaces and Their Applications. SIAM Journal on Computing, 39(3):923-947, 2009.
Julia Chuzhoy and Yuval Rabani. Approximating K-median with Non-uniform Capacities. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '05, pages 952-958, Philadelphia, PA, USA, 2005. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=1070432.1070569.
http://dl.acm.org/citation.cfm?id=1070432.1070569
Vincent Cohen-Addad. Approximation Schemes for Capacitated Clustering in Doubling Metrics. CoRR, abs/1812.07721, 2018. URL: http://arxiv.org/abs/1812.07721.
http://arxiv.org/abs/1812.07721
Vincent Cohen-Addad, Arnaud de Mesmay, Eva Rotenberg, and Alan Roytman. The Bane of Low-Dimensionality Clustering. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 441-456, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.30.
http://dx.doi.org/10.1137/1.9781611975031.30
Vincent Cohen-Addad, Anupam Gupta, Amit Kumar, Euiwoong Lee, and Jason Li. Tight FPT Approximations for k-Median and k-Means. In ICALP 2019, 2019.
Wenceslas Fernandez de la Vega, Marek Karpinski, Claire Kenyon, and Yuval Rabani. Approximation schemes for clustering problems. In Lawrence L. Larmore and Michel X. Goemans, editors, Proceedings of the 35th Annual ACM Symposium on Theory of Computing, June 9-11, 2003, San Diego, CA, USA, pages 50-58. ACM, 2003. URL: http://dx.doi.org/10.1145/780542.780550.
http://dx.doi.org/10.1145/780542.780550
H. Gökalp Demirci and Shi Li. Constant Approximation for Capacitated k-Median with (1+epsilon)-Capacity Violation. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 73:1-73:14, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.73.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.73
G. Frahling and C. Sohler. Coresets in dynamic geometric data streams. In STOC, pages 209-217, 2005.
Anupam Gupta, Euiwoong Lee, and Jason Li. An FPT Algorithm Beating 2-approximation for K-cut. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '18, pages 2821-2837, Philadelphia, PA, USA, 2018. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=3174304.3175483.
http://dl.acm.org/citation.cfm?id=3174304.3175483
Anupam Gupta, Euiwoong Lee, Jason Li, Pasin Manurangsi, and Michal Wlodarczyk. Losing Treewidth by Separating Subsets. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1731-1749, 2019. URL: http://dx.doi.org/10.1137/1.9781611975482.104.
http://dx.doi.org/10.1137/1.9781611975482.104
Sariel Har-Peled and Akash Kushal. Smaller Coresets for k-Median and k-Means Clustering. Discrete & Computational Geometry, 37(1):3-19, 2007. URL: http://dx.doi.org/10.1007/s00454-006-1271-x.
http://dx.doi.org/10.1007/s00454-006-1271-x
Sariel Har-Peled and Soham Mazumdar. On coresets for k-means and k-median clustering. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 291-300, 2004. URL: http://dx.doi.org/10.1145/1007352.1007400.
http://dx.doi.org/10.1145/1007352.1007400
Amit Kumar, Yogish Sabharwal, and Sandeep Sen. A Simple Linear Time (1+ ") -Approximation Algorithm for k-Means Clustering in Any Dimensions. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS '04, pages 454-462, Washington, DC, USA, 2004. IEEE Computer Society. URL: http://dx.doi.org/10.1109/FOCS.2004.7.
http://dx.doi.org/10.1109/FOCS.2004.7
Amit Kumar, Yogish Sabharwal, and Sandeep Sen. Linear-time approximation schemes for clustering problems in any dimensions. J. ACM, 57(2), 2010. URL: http://dx.doi.org/10.1145/1667053.1667054.
http://dx.doi.org/10.1145/1667053.1667054
Euiwoong Lee. Partitioning a graph into small pieces with applications to path transversal. Mathematical Programming, March 2018. URL: http://dx.doi.org/10.1007/s10107-018-1255-7.
http://dx.doi.org/10.1007/s10107-018-1255-7
Shi Li. On Uniform Capacitated k-Median Beyond the Natural LP Relaxation. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 696-707, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.47.
http://dx.doi.org/10.1137/1.9781611973730.47
Shi Li. Approximating capacitated k-median with (1 + k open facilities. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 786-796, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch56.
http://dx.doi.org/10.1137/1.9781611974331.ch56
Shi Li. On Uniform Capacitated k-Median Beyond the Natural LP Relaxation. ACM Trans. Algorithms, 13(2):22:1-22:18, 2017. URL: http://dx.doi.org/10.1145/2983633.
http://dx.doi.org/10.1145/2983633
Sepideh Mahabadi, Konstantin Makarychev, Yury Makarychev, and Ilya Razenshteyn. Nonlinear Dimension Reduction via Outer Bi-Lipschitz Extensions. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 1088-1101, New York, NY, USA, 2018. ACM. URL: http://dx.doi.org/10.1145/3188745.3188828.
http://dx.doi.org/10.1145/3188745.3188828
Dániel Marx and Michal Pilipczuk. Optimal Parameterized Algorithms for Planar Facility Location Problems Using Voronoi Diagrams. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 865-877, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_72.
http://dx.doi.org/10.1007/978-3-662-48350-3_72
Shyam Narayanan and Jelani Nelson. Optimal terminal dimensionality reduction in Euclidean space. CoRR - To appear in the proceedings of STOC'19, abs/1810.09250, 2018. URL: http://arxiv.org/abs/1810.09250.
http://arxiv.org/abs/1810.09250
Yicheng Xu, Yong Zhang, and Yifei Zou. A constant parameterized approximation for hard-capacitated k-means. CoRR, abs/1901.04628, 2019. URL: http://arxiv.org/abs/1901.04628.
http://arxiv.org/abs/1901.04628
Vincent Cohen-Addad and Jason Li
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Tight FPT Approximations for k-Median and k-Means
We investigate the fine-grained complexity of approximating the classical k-Median/k-Means clustering problems in general metric spaces. We show how to improve the approximation factors to (1+2/e+epsilon) and (1+8/e+epsilon) respectively, using algorithms that run in fixed-parameter time. Moreover, we show that we cannot do better in FPT time, modulo recent complexity-theoretic conjectures.
approximation algorithms
fixed-parameter tractability
k-median
k-means
clustering
core-sets
Theory of computation~Facility location and clustering
Theory of computation~Fixed parameter tractability
Theory of computation~Submodular optimization and polymatroids
42:1-42:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1904.12334.
We thank Deeparnab Chakrabarty, Ola Svensson, and Pasin Manurangsi for useful discussions. This research was partially conducted when A. Kumar was visiting A. Gupta and Carnegie Mellon University as part of the Joint Indo-US Virtual Center for Algorithms under Uncertainty.
Vincent
Cohen-Addad
Vincent Cohen-Addad
CNRS & Sorbonne Université, Paris, France
Ce projet a bénéficié d'une aide de l'Etat gérée par l'Agence Nationale de la Recherche au titre du Programme Appel à projets générique JCJC 2018 portant la référence suivante: ANR-18-CE40-0004-01.
Anupam
Gupta
Anupam Gupta
Carnegie Mellon University, Pittsburgh, PA, USA
Supported in part by NSF awards CCF-1536002, CCF-1540541, and CCF-1617790.
Amit
Kumar
Amit Kumar
IIT Delhi, India
Euiwoong
Lee
Euiwoong Lee
New York University, NY, USA
Supported in part by the Simons Collaboration on Algorithms and Geometry.
Jason
Li
Jason Li
Carnegie Mellon University, Pittsburgh, PA, USA
Supported in part by NSF awards CCF-1536002, CCF-1540541, and CCF-1617790.
10.4230/LIPIcs.ICALP.2019.42
Sara Ahmadian, Ashkan Norouzi-Fard, Ola Svensson, and Justin Ward. Better Guarantees for k-Means and Euclidean k-Median by Primal-Dual Algorithms. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 61-72, 2017.
Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The Hardness of Approximation of Euclidean k-Means. In 31st International Symposium on Computational Geometry, SoCG 2015, June 22-25, 2015, Eindhoven, The Netherlands, pages 754-767, 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 twenty-sixth annual ACM-SIAM symposium on Discrete algorithms, pages 737-756. SIAM, 2014.
Gruia Calinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput., 40(6):1740-1766, 2011. URL: http://dx.doi.org/10.1137/080733991.
http://dx.doi.org/10.1137/080733991
Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From gap-ETH to FPT-inapproximability: Clique, dominating set, and more. In Foundations of Computer Science (FOCS), 2017 IEEE 58th Annual Symposium on, pages 743-754. IEEE, 2017.
Moses Charikar, Sudipto Guha, Éva Tardos, and David B. Shmoys. A Constant-Factor Approximation Algorithm for the k-Median Problem. J. Comput. Syst. Sci., 65(1):129-149, 2002. URL: http://dx.doi.org/10.1006/jcss.2002.1882.
http://dx.doi.org/10.1006/jcss.2002.1882
Ke Chen. On k-Median clustering in high dimensions. In SODA, 2006.
Ke Chen. On coresets for k-median and k-means clustering in metric and Euclidean spaces and their applications. SIAM Journal on Computing, 39(3):923-947, 2009.
Yijia Chen and Bingkai Lin. The constant inapproximability of the parameterized dominating set problem. In Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, pages 505-514. IEEE, 2016.
Karthik C.S., Bundit Laekhanukit, and Pasin Manurangsi. On the parameterized complexity of approximating dominating set. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 1283-1296. ACM, 2018.
Artur Czumaj and Christian Sohler. Small Space Representations for Metric Min-sum k-Clustering and Their Applications. Theory Comput. Syst., 46(3):416-442, 2010. URL: http://dx.doi.org/10.1007/s00224-009-9235-1.
http://dx.doi.org/10.1007/s00224-009-9235-1
Irit Dinur. Mildly exponential reduction from gap 3SAT to polynomial-gap label cover. Electronic Colloquium on Computational Complexity (ECCC), pages TR 16-128, 2016.
Uriel Feige. A Threshold of Ln N for Approximating Set Cover. J. ACM, 45(4), July 1998.
Dan Feldman and Michael Langberg. A Unified Framework for Approximating and Clustering Data. CoRR, abs/1106.1379, 2011. (An extended abstract appeared at STOC 11). URL: http://arxiv.org/abs/1106.1379.
http://arxiv.org/abs/1106.1379
Dan Feldman, Melanie Schmidt, and Christian Sohler. Turning big data into tiny data: Constant-size coresets for k-means, PCA and projective clustering. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pages 1434-1453. Society for Industrial and Applied Mathematics, 2013.
Sudipto Guha and Samir Khuller. Greedy strikes back: improved facility location algorithms. J. Algorithms, 31(1):228-248, 1999. URL: http://dx.doi.org/10.1006/jagm.1998.0993.
http://dx.doi.org/10.1006/jagm.1998.0993
Tapas Kanungo, David M. Mount, Nathan S. Netanyahu, Christine D. Piatko, Ruth Silverman, and Angela Y. Wu. A local search approximation algorithm for k-means clustering. Comput. Geom., 28(2-3):89-112, 2004. URL: http://dx.doi.org/10.1016/j.comgeo.2004.03.003.
http://dx.doi.org/10.1016/j.comgeo.2004.03.003
Ravishankar Krishnaswamy, Amit Kumar, Viswanath Nagarajan, Yogish Sabharwal, and Barna Saha. Facility Location with Matroid or Knapsack Constraints. Math. Oper. Res., 40(2):446-459, 2015. URL: http://dx.doi.org/10.1287/moor.2014.0678.
http://dx.doi.org/10.1287/moor.2014.0678
Amit Kumar, Yogish Sabharwal, and Sandeep Sen. Linear-time approximation schemes for clustering problems in any dimensions. J. ACM, 57(2):5:1-5:32, 2010. URL: http://dx.doi.org/10.1145/1667053.1667054.
http://dx.doi.org/10.1145/1667053.1667054
Euiwoong Lee, Melanie Schmidt, and John Wright. Improved and simplified inapproximability for k-means. Inf. Process. Lett., 120:40-43, 2017. URL: http://dx.doi.org/10.1016/j.ipl.2016.11.009.
http://dx.doi.org/10.1016/j.ipl.2016.11.009
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. SIAM J. Comput., 45(2):530-547, 2016. URL: http://dx.doi.org/10.1137/130938645.
http://dx.doi.org/10.1137/130938645
Bingkai Lin. The parameterized complexity of k-biclique. In Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms, pages 605-615. SIAM, 2015.
Pasin Manurangsi and Prasad Raghavendra. A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), pages 78:1-78:15, 2017.
Ran Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763-803, 1998.
Chaitanya Swamy. Improved Approximation Algorithms for Matroid and Knapsack Median Problems and Applications. ACM Trans. Algorithms, 12(4):49:1-49:22, 2016. URL: http://dx.doi.org/10.1145/2963170.
http://dx.doi.org/10.1145/2963170
Vincent Cohen-Addad, Anupam Gupta, Amit Kumar, Euiwoong Lee, and Jason Li
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Information-Theoretic and Algorithmic Thresholds for Group Testing
In the group testing problem we aim to identify a small number of infected individuals within a large population. We avail ourselves to a procedure that can test a group of multiple individuals, with the test result coming out positive iff at least one individual in the group is infected. With all tests conducted in parallel, what is the least number of tests required to identify the status of all individuals? In a recent test design [Aldridge et al. 2016] the individuals are assigned to test groups randomly, with every individual joining an equal number of groups. We pinpoint the sharp threshold for the number of tests required in this randomised design so that it is information-theoretically possible to infer the infection status of every individual. Moreover, we analyse two efficient inference algorithms. These results settle conjectures from [Aldridge et al. 2014, Johnson et al. 2019].
Group testing problem
phase transitions
information theory
efficient algorithms
sharp threshold
Bayesian inference
Theory of computation~Theory and algorithms for application domains
Theory of computation~Bayesian analysis
Theory of computation~Machine learning theory
43:1-43:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/pdf/1902.02202.pdf.
We thank Arya Mazumdar for bringing the group testing problem to our attention.
Amin
Coja-Oghlan
Amin Coja-Oghlan
Goethe University, Frankfurt, Germany
Supported of DFG 646/3.
Oliver
Gebhard
Oliver Gebhard
Goethe University, Frankfurt, Germany
Max
Hahn-Klimroth
Max Hahn-Klimroth
Goethe University, Frankfurt, Germany
Supported by Stiftung Polytechnische Gesellschaft.
Philipp
Loick
Philipp Loick
Goethe University, Frankfurt, Germany
Supported of DFG 646/3.
10.4230/LIPIcs.ICALP.2019.43
E. Abbe. Community Detection and Stochastic Block Models: Recent Developments. Journal of Machine Learning Research, 18:1-86, 2018.
D. Achlioptas and A. Coja-Oghlan. Algorithmic barriers from phase transitions. Proc. 49th FOCS, pages 793-802, 2008.
D. Achlioptas, A. Coja-Oghlan, and F. Ricci-Tersenghi. On the solution space geometry of random formulas. Random Structures and Algorithms, 38:251-268, 2011.
D. Achlioptas and C. Moore. Random k-SAT: two moments suffice to cross a sharp threshold. SIAM Journal on Computing, 36:740-762, 2006.
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A. Coja-Oghlan, F. Krzakala, W. Perkins, and L. Zdeborová. Information-theoretic thresholds from the cavity method. Advances in Mathematics, 333:694-795, 2018.
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A. Decelle, F. Krzakala, C. Moore, and L. Zdeborová. Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Phys. Rev. E, 84:066106, 2011.
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Amin Coja-Oghlan, Oliver Gebhard, Max Hahn-Klimroth, and Philipp Loick
Creative Commons Attribution 3.0 Unported license
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On Reachability Problems for Low-Dimensional Matrix Semigroups
We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group over rational numbers. Furthermore, we prove two decidability results for the Half-Space Reachability Problem. Namely, we show that this problem is decidable for sub-semigroups of GL(2,Z) and of the Heisenberg group over rational numbers.
membership problem
half-space reachability problem
matrix semigroups
Heisenberg group
general linear group
Theory of computation~Formal languages and automata theory
Computing methodologies~Symbolic and algebraic algorithms
44:1-44:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1902.09597, [Thomas Colcombet et al., 2019].
Thomas
Colcombet
Thomas Colcombet
IRIF, CNRS, Université Paris Diderot, France
https://orcid.org/0000-0001-6529-6963
Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.670624), and by the DeLTA ANR project (ANR-16-CE40-0007).
Joël
Ouaknine
Joël Ouaknine
The Max Planck Institute for Software Systems, Saarbrücken, Germany
Department of Computer Science, University of Oxford, United Kingdom
https://orcid.org/0000-0003-0031-9356
Supported by ERC grant AVS-ISS (648701) and by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).
Pavel
Semukhin
Pavel Semukhin
Department of Computer Science, University of Oxford, United Kingdom
https://orcid.org/0000-0002-7547-6391
Supported by ERC grant AVS-ISS (648701).
James
Worrell
James Worrell
Department of Computer Science, University of Oxford, United Kingdom
https://orcid.org/0000-0001-8151-2443
Supported by EPSRC Fellowship EP/N008197/1.
10.4230/LIPIcs.ICALP.2019.44
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http://arxiv.org/abs/1404.0644
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Thomas Colcombet, Joël Ouaknine, Pavel Semukhin, and James Worrell. On reachability problems for low dimensional matrix semigroups. CoRR, abs/1902.09597, 2019. URL: http://arxiv.org/abs/1902.09597.
http://arxiv.org/abs/1902.09597
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Thomas Colcombet, Joël Ouaknine, Pavel Semukhin, and James Worrell
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Independent Sets in Vertex-Arrival Streams
We consider the maximal and maximum independent set problems in three models of graph streams:
- In the edge model we see a stream of edges which collectively define a graph; this model is well-studied for a variety of problems. We show that the space complexity for a one-pass streaming algorithm to find a maximal independent set is quadratic (i.e. we must store all edges). We further show that it is not much easier if we only require approximate maximality. This contrasts strongly with the other two vertex-based models, where one can greedily find an exact solution in only the space needed to store the independent set.
- In the "explicit" vertex model, the input stream is a sequence of vertices making up the graph. Every vertex arrives along with its incident edges that connect to previously arrived vertices. Various graph problems require substantially less space to solve in this setting than in edge-arrival streams. We show that every one-pass c-approximation streaming algorithm for maximum independent set (MIS) on explicit vertex streams requires Omega({n^2}/{c^6}) bits of space, where n is the number of vertices of the input graph. It is already known that Theta~({n^2}/{c^2}) bits of space are necessary and sufficient in the edge arrival model (Halldórsson et al. 2012), thus the MIS problem is not significantly easier to solve under the explicit vertex arrival order assumption. Our result is proved via a reduction from a new multi-party communication problem closely related to pointer jumping.
- In the "implicit" vertex model, the input stream consists of a sequence of objects, one per vertex. The algorithm is equipped with a function that maps pairs of objects to the presence or absence of edges, thus defining the graph. This model captures, for example, geometric intersection graphs such as unit disc graphs. Our final set of results consists of several improved upper and lower bounds for interval and square intersection graphs, in both explicit and implicit streams. In particular, we show a gap between the hardness of the explicit and implicit vertex models for interval graphs.
streaming algorithms
independent set size
lower bounds
Theory of computation~Lower bounds and information complexity
Theory of computation~Streaming models
45:1-45:14
Track A: Algorithms, Complexity and Games
The full version of this paper is available at: https://arxiv.org/abs/1807.08331.
Graham
Cormode
Graham Cormode
University of Warwick, UK
https://orcid.org/0000-0002-0698-0922
Supported by European Research Council grant ERC-2014-CoG 647557.
Jacques
Dark
Jacques Dark
University of Warwick, UK
Supported by an EMEA Microsoft Research scholarship. Part of the work was done while J.D. was at the Alan Turing Institute, under EPSRC grant EP/N510129/1.
Christian
Konrad
Christian Konrad
University of Bristol, UK
https://orcid.org/0000-0003-1802-4011
C.K. carried out most work on this paper while being at the University of Warwick. He was supported by the Centre for Discrete Mathematics and its Applications (DIMAP) at Warwick University and by EPSRC award EP/N011163/1.
10.4230/LIPIcs.ICALP.2019.45
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 International Conference on Machine Learning - Volume 37, ICML'15, pages 2237-2246. JMLR.org, 2015. URL: http://dl.acm.org/citation.cfm?id=3045118.3045356.
http://dl.acm.org/citation.cfm?id=3045118.3045356
Noga Alon, Ankur Moitra, and Benny Sudakov. Nearly Complete Graphs Decomposable into Large Induced Matchings and Their Applications. In Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC '12, pages 1079-1090, New York, NY, USA, 2012. ACM. URL: http://dx.doi.org/10.1145/2213977.2214074.
http://dx.doi.org/10.1145/2213977.2214074
Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Sublinear Algorithms for (Δ + 1) Vertex Coloring. In SODA, 2019.
Sepehr 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, Philadelphia, PA, USA, 2016. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2884435.2884528.
http://dl.acm.org/citation.cfm?id=2884435.2884528
Béla Bollobás. The chromatic number of random graphs. Combinatorica, 8(1):49-55, 1988.
Vladimir Braverman, Zaoxing Liu, Tejasvam Singh, N. V. Vinodchandran, and Lin F. Yang. New Bounds for the CLIQUE-GAP Problem Using Graph Decomposition Theory. Algorithmica, 80(2):652-667, February 2018. URL: http://dx.doi.org/10.1007/s00453-017-0277-5.
http://dx.doi.org/10.1007/s00453-017-0277-5
Sergio Cabello and Pablo Pérez-Lantero. Interval selection in the streaming model. Theoretical Computer Science, 702:77-96, 2017.
Amit Chakrabarti. Lower bounds for multi-player pointer jumping. In Computational Complexity, 2007. CCC'07. Twenty-Second Annual IEEE Conference on, pages 33-45. IEEE, 2007.
Graham Cormode, Jacques Dark, and Christian Konrad. Approximating the Caro-Wei Bound for Independent Sets in Graph Streams. In Jon Lee, Giovanni Rinaldi, and A. Ridha Mahjoub, editors, Combinatorial Optimization, pages 101-114, Cham, 2018. Springer International Publishing.
Graham Cormode, Jacques Dark, and Christian Konrad. Independent Sets in Vertex-Arrival Streams. CoRR, abs/1807.08331, 2018. URL: http://arxiv.org/abs/1807.08331.
http://arxiv.org/abs/1807.08331
Yuval Emek, Magnús M Halldórsson, and Adi Rosén. Space-constrained interval selection. ACM Transactions on Algorithms (TALG), 12(4):51, 2016.
Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 468-485, 2012.
Bjarni V Halldórsson, Magnús M Halldórsson, Elena Losievskaja, and Mario Szegedy. Streaming algorithms for independent sets. In International Colloquium on Automata, Languages, and Programming, pages 641-652. Springer, 2010.
Magnús M. Halldórsson and Christian Konrad. Computing Large Independent Sets in a Single Round. Distrib. Comput., 31(1):69-82, February 2018. URL: http://dx.doi.org/10.1007/s00446-017-0298-y.
http://dx.doi.org/10.1007/s00446-017-0298-y
Magnús M Halldórsson, Xiaoming Sun, Mario Szegedy, and Chengu Wang. Streaming and communication complexity of clique approximation. In International Colloquium on Automata, Languages, and Programming, pages 449-460. Springer, 2012.
R. M. Karp. Reducibility among combinatorial problems. In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, pages 85-103. Plenum Press, 1972.
Christian Konrad. Maximum Matching in Turnstile Streams. In Nikhil Bansal and Irene Finocchi, editors, Algorithms - ESA 2015, pages 840-852, Berlin, Heidelberg, 2015. Springer Berlin Heidelberg.
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http://dx.doi.org/10.1145/2627692.2627694
David Zuckerman. Linear Degree Extractors and the Inapproximability of Max Clique and Chromatic Number. Theory of Computing, 3(1):103-128, 2007.
Graham Cormode, Jacques Dark, and Christian Konrad
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Approximation Algorithms for Min-Distance Problems
We study fundamental graph parameters such as the Diameter and Radius in directed graphs, when distances are measured using a somewhat unorthodox but natural measure: the distance between u and v is the minimum of the shortest path distances from u to v and from v to u. The center node in a graph under this measure can for instance represent the optimal location for a hospital to ensure the fastest medical care for everyone, as one can either go to the hospital, or a doctor can be sent to help.
By computing All-Pairs Shortest Paths, all pairwise distances and thus the parameters we study can be computed exactly in O~(mn) time for directed graphs on n vertices, m edges and nonnegative edge weights. Furthermore, this time bound is tight under the Strong Exponential Time Hypothesis [Roditty-Vassilevska W. STOC 2013] so it is natural to study how well these parameters can be approximated in O(mn^{1-epsilon}) time for constant epsilon>0. Abboud, Vassilevska Williams, and Wang [SODA 2016] gave a polynomial factor approximation for Diameter and Radius, as well as a constant factor approximation for both problems in the special case where the graph is a DAG. We greatly improve upon these bounds by providing the first constant factor approximations for Diameter, Radius and the related Eccentricities problem in general graphs. Additionally, we provide a hierarchy of algorithms for Diameter that gives a time/accuracy trade-off.
fine-grained complexity
graph algorithms
diameter
radius
eccentricities
Mathematics of computing~Graph algorithms
46:1-46:14
Track A: Algorithms, Complexity and Games
https://arxiv.org/abs/1904.11606
The authors would like to thank the members of the MIT course 6.S078 open problem sessions, especially Thuy-Duong Vuong, Robin Hui, and Ali Vakilian. These sessions were organized by Erik Demaine, Ryan Williams, and Virginia Vassilevska Williams.
Mina
Dalirrooyfard
Mina Dalirrooyfard
MIT, Cambridge, MA, USA
Virginia Vassilevska
Williams
Virginia Vassilevska Williams
MIT, Cambridge, MA, USA
Nikhil
Vyas
Nikhil Vyas
MIT, Cambridge, MA, USA
Nicole
Wein
Nicole Wein
MIT, Cambridge, MA, USA
Yinzhan
Xu
Yinzhan Xu
MIT, Cambridge, MA, USA
Yuancheng
Yu
Yuancheng Yu
MIT, Cambridge, MA, USA
10.4230/LIPIcs.ICALP.2019.46
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, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1681-1697, 2015.
Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 377-391, 2016.
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Shiri Chechik, Daniel H. Larkin, Liam Roditty, Grant Schoenebeck, Robert Endre Tarjan, and Virginia Vassilevska Williams. Better Approximation Algorithms for the Graph Diameter. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1041-1052, 2014.
V. Chepoi, F. Dragan, and Y. Vaxès. Center and diameter problems in plane triangulations and quadrangulations. In Proc. SODA, pages 346-355, 2002.
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D.G. Corneil, F.F. Dragan, M. Habib, and C. Paul. Diameter determination on restricted graph families. Discr. Appl. Math., 113:143-166, 2001.
L. Cowen and C. Wagner. Compact roundtrip routing for digraphs. In SODA, pages 885-886, 1999.
Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, Nicole Wein, Yinzhan Xu, and Yuancheng Yu. Approximation Algorithms for Min-Distance Problems, 2019. URL: http://arxiv.org/abs/1904.11606.
http://arxiv.org/abs/1904.11606
D. Dvir and G. Handler. The absolute center of a network. Networks, 43:109-118, 2004.
D. Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms and Applications, 3(3):1-27, 1999.
Silvio Frischknecht, Stephan Holzer, and Roger Wattenhofer. Networks cannot compute their diameter in sublinear time. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 1150-1162. SIAM, 2012.
S.L. Hakimi. Optimum location of switching centers and absolute centers and medians of a graph. Oper. Res., 12:450-459, 1964.
Seth Pettie. A faster all-pairs shortest path algorithm for real-weighted sparse graphs. In International Colloquium on Automata, Languages, and Programming, pages 85-97. Springer, 2002.
Seth Pettie and Vijaya Ramachandran. A Shortest Path Algorithm for Real-Weighted Undirected Graphs. SIAM J. Comput., 34(6):1398-1431, 2005. URL: http://dx.doi.org/10.1137/S0097539702419650.
http://dx.doi.org/10.1137/S0097539702419650
Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, STOC '13, pages 515-524, New York, NY, USA, 2013. ACM. URL: http://dx.doi.org/10.1145/2488608.2488673.
http://dx.doi.org/10.1145/2488608.2488673
O. Weimann and R. Yuster. Approximating the Diameter of Planar Graphs in Near Linear Time. In Proc. ICALP, 2013.
Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 664-673, 2014.
C. Wulff-Nilsen. Wiener index, diameter, and stretch factor of a weighted planar graph in subquadratic time. Technical report, University of Copenhagen, 2008.
Raphael Yuster. Computing the diameter polynomially faster than APSP. arXiv preprint, 2010. URL: http://arxiv.org/abs/1011.6181.
http://arxiv.org/abs/1011.6181
Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, Nicole Wein, Yinzhan Xu, and Yuancheng Yu
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems
Some of the most fundamental and well-studied graph parameters are the Diameter (the largest shortest paths distance) and Radius (the smallest distance for which a "center" node can reach all other nodes). The natural and important ST-variant considers two subsets S and T of the vertex set and lets the ST-diameter be the maximum distance between a node in S and a node in T, and the ST-radius be the minimum distance for a node of S to reach all nodes of T. The bichromatic variant is the special case in which S and T partition the vertex set.
In this paper we present a comprehensive study of the approximability of ST and Bichromatic Diameter, Radius, and Eccentricities, and variants, in graphs with and without directions and weights. We give the first nontrivial approximation algorithms for most of these problems, including time/accuracy trade-off upper and lower bounds. We show that nearly all of our obtained bounds are tight under the Strong Exponential Time Hypothesis (SETH), or the related Hitting Set Hypothesis.
For instance, for Bichromatic Diameter in undirected weighted graphs with m edges, we present an O~(m^{3/2}) time 5/3-approximation algorithm, and show that under SETH, neither the running time, nor the approximation factor can be significantly improved while keeping the other unchanged.
approximation algorithms
fine-grained complexity
diameter
radius
eccentricities
Theory of computation~Graph algorithms analysis
Theory of computation~Approximation algorithms analysis
Theory of computation~Problems, reductions and completeness
Theory of computation~Shortest paths
47:1-47:15
Track A: Algorithms, Complexity and Games
https://arxiv.org/abs/1904.11601
The authors would like to thank Arturs Backurs for discussions during the early stages of this work.
Mina
Dalirrooyfard
Mina Dalirrooyfard
MIT, Cambridge, MA, USA
Supported by an Akamai Presidential Fellowship and NSF Grant CCF-6936484.
Virginia Vassilevska
Williams
Virginia Vassilevska Williams
MIT, Cambridge, MA, USA
Supported by an NSF CAREER Award, NSF Grants CCF-1417238, CCF-1528078 and CCF-1514339, a BSF Grant BSF:2012338 and a Sloan Research Fellowship.
Nikhil
Vyas
Nikhil Vyas
MIT, Cambridge, MA, USA
Supported by an Akamai Presidential Fellowship and NSF Grant CCF-1552651.
Nicole
Wein
Nicole Wein
MIT, Cambridge, MA, USA
Supported by an NSF Graduate Fellowship and NSF Grant CCF-1514339.
10.4230/LIPIcs.ICALP.2019.47
Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 377-391, 2016.
Pankaj K. Agarwal, Herbert Edelsbrunner, Otfried Schwarzkopf, and Emo Welzl. Euclidean Minimum Spanning Trees and Bichromatic Closest Pairs. Discrete Comput. Geom., 6(1):407-422, December 1991.
D. Aingworth, C. Chekuri, P. Indyk, and R. Motwani. Fast Estimation of Diameter and Shortest Paths (Without Matrix Multiplication). SIAM Journal on Computing, 28(4):1167-1181, 1999.
Arturs Backurs, Liam Roditty, Gilad Segal, Virginia Vassilevska Williams, and Nicole Wein. Towards Tight Approximation Bounds for Graph Diameter and Eccentricities. In Proceedings of STOC'18, page to appear, 2018.
Massimo Cairo, Roberto Grossi, and Romeo Rizzi. New Bounds for Approximating Extremal Distances in Undirected Graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 363-376, 2016.
T. M. Chan. More Algorithms for All-Pairs Shortest Paths in Weighted Graphs. In Proc. STOC, pages 590-598, 2007.
Bernard Chazelle, Herbert Edelsbrunner, Leonidas J Guibas, and Micha Sharir. Algorithms for bichromatic line-segment problems and polyhedral terrains. Algorithmica, 11(2):116-132, 1994.
Shiri Chechik, Daniel H. Larkin, Liam Roditty, Grant Schoenebeck, Robert Endre Tarjan, and Virginia Vassilevska Williams. Better Approximation Algorithms for the Graph Diameter. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1041-1052, 2014.
Don Coppersmith and Michael Elkin. Sparse sourcewise and pairwise distance preservers. SIAM Journal on Discrete Mathematics, 20(2):463-501, 2006.
Pierluigi Crescenzi, Roberto Grossi, Leonardo Lanzi, and Andrea Marino. On Computing the Diameter of Real-World Directed (Weighted) Graphs. In Ralf Klasing, editor, Experimental Algorithms: 11th International Symposium, SEA 2012, Bordeaux, France, June 7-9, 2012. Proceedings, pages 99-110, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg.
Marek Cygan, Fabrizio Grandoni, and Telikepalli Kavitha. On pairwise spanners. arXiv preprint, 2013. URL: http://arxiv.org/abs/1301.1999.
http://arxiv.org/abs/1301.1999
Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, and Nicole Wein. Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems, 2019. URL: http://arxiv.org/abs/1904.11601.
http://arxiv.org/abs/1904.11601
Adrian Dumitrescu and Sumanta Guha. Extreme Distances in Multicolored Point Sets. J. Graph Algorithms and Applications, 8(1):27-38, 2004.
Jiawei Gao, Russell Impagliazzo, Antonina Kolokolova, and R. Ryan Williams. Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2162-2181, 2017.
R. Impagliazzo and R. Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001.
Piotr Indyk. A near linear time constant factor approximation for Euclidean bichromatic matching (cost). In SODA, volume 7, pages 39-42, 2007.
Naoki Katoh and Kazuo Iwano. Finding K Farthest Pairs and K Closest/Farthest Bichromatic Pairs for Points in the Plane. In Proceedings of the Eighth Annual Symposium on Computational Geometry, SCG '92, pages 320-329, 1992.
Telikepalli Kavitha. New pairwise spanners. Theory of Computing Systems, 61(4):1011-1036, 2017.
Philip N Klein. A subset spanner for planar graphs, with application to subset TSP. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 749-756. ACM, 2006.
Clémence Magnien, Matthieu Latapy, and Michel Habib. Fast Computation of Empirically Tight Bounds for the Diameter of Massive Graphs. J. Exp. Algorithmics, 13:10:1.10-10:1.9, February 2009.
David Peleg, Liam Roditty, and Elad Tal. Distributed Algorithms for Network Diameter and Girth. In Automata, Languages, and Programming: 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part II, pages 660-672, 2012.
S. Pettie. A new approach to all-pairs shortest paths on real-weighted graphs. Theor. Comput. Sci., 312(1):47-74, 2004.
Seth Pettie and Vijaya Ramachandran. A Shortest Path Algorithm for Real-Weighted Undirected Graphs. SIAM J. Comput., 34(6):1398-1431, 2005.
Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, STOC '13, pages 515-524, New York, NY, USA, 2013. ACM. URL: http://dx.doi.org/10.1145/2488608.2488673.
http://dx.doi.org/10.1145/2488608.2488673
Frank W. Takes and Walter A. Kosters. Determining the Diameter of Small World Networks. In Proceedings of the 20th ACM International Conference on Information and Knowledge Management, CIKM '11, pages 1191-1196, 2011.
Virginia Vassilevska Williams. Hardness of Easy Problems: Basing Hardness on Popular Conjectures such as the Strong Exponential Time Hypothesis (Invited Talk). In 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, September 16-18, 2015, Patras, Greece, pages 17-29, 2015.
R. Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005.
Ryan Williams. On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-linear vs Barely-subquadratic Complexity. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '18, pages 1207-1215, 2018.
A. Yao. On Constructing Minimum Spanning Trees in k-Dimensional Spaces and Related Problems. SIAM Journal on Computing, 11(4):721-736, 1982.
Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, and Nicole Wein
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Faster Algorithms for All Pairs Non-Decreasing Paths Problem
In this paper, we present an improved algorithm for the All Pairs Non-decreasing Paths (APNP) problem on weighted simple digraphs, which has running time O~(n^{{3 + omega}/{2}}) = O~(n^{2.686}). Here n is the number of vertices, and omega < 2.373 is the exponent of time complexity of fast matrix multiplication [Williams 2012, Le Gall 2014]. This matches the current best upper bound for (max, min)-matrix product [Duan, Pettie 2009] which is reducible to APNP. Thus, further improvement for APNP will imply a faster algorithm for (max, min)-matrix product. The previous best upper bound for APNP on weighted digraphs was O~(n^{1/2(3 + {3 - omega}/{omega + 1} + omega)}) = O~(n^{2.78}) [Duan, Gu, Zhang 2018]. We also show an O~(n^2) time algorithm for APNP in undirected simple graphs which also reaches optimal within logarithmic factors.
graph optimization
matrix multiplication
non-decreasing paths
Mathematics of computing~Graph algorithms
48:1-48:13
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1904.10701.
Ran
Duan
Ran Duan
Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
Ce
Jin
Ce Jin
Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
Hongxun
Wu
Hongxun Wu
Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
10.4230/LIPIcs.ICALP.2019.48
Alfred V. Aho and John E. Hopcroft. The Design and Analysis of Computer Algorithms. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1st edition, 1974.
D. Coppersmith and S. Winograd. Matrix Multiplication via Arithmetic Progressions. In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, STOC '87, pages 1-6, New York, NY, USA, 1987. ACM. URL: http://dx.doi.org/10.1145/28395.28396.
http://dx.doi.org/10.1145/28395.28396
Artur Czumaj, Mirosław Kowaluk, and Andrzej Lingas. Faster algorithms for finding lowest common ancestors in directed acyclic graphs. Theoretical Computer Science, 380(1-2):37-46, 2007.
Edsger W. Dijkstra. A note on two problems in connexion with graphs. Numerische mathematik, 1(1):269-271, 1959.
Ran Duan, Yong Gu, and Le Zhang. Improved Time Bounds for All Pairs Non-decreasing Paths in General Digraphs. In 45th International Colloquium on Automata, Languages, and Programming, pages 44:1-44:14, 2018.
Ran Duan and Seth Pettie. Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths. In Proceedings of the 20th annual ACM-SIAM Symposium on Discrete Algorithms, pages 384-391. SIAM, 2009.
François Le Gall and Florent Urrutia. Improved rectangular matrix multiplication using powers of the Coppersmith-Winograd tensor. In Proceedings of the 29th annual ACM-SIAM Symposium on Discrete Algorithms, pages 1029-1046. SIAM, 2018.
François Le Gall. Powers of tensors and fast matrix multiplication. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, pages 296-303. ACM, 2014.
François Le Gall and Harumichi Nishimura. Quantum algorithms for matrix products over semirings. In Scandinavian Workshop on Algorithm Theory, pages 331-343. Springer, 2014.
Kurt Mehlhorn, Rajamani Sundar, and Christian Uhrig. Maintaining dynamic sequences under equality tests in polylogarithmic time. Algorithmica, 17(2):183-198, 1997.
George J. Minty. A Variant on the Shortest-Route Problem. Operations Research, 6(6):882-883, 1958.
Virginia Vassilevska, Ryan Williams, and Raphael Yuster. All Pairs Bottleneck Paths and Max-min Matrix Products in Truly Subcubic Time. Theory of Computing, 5(9):173-189, 2009.
Virginia Vassilevska Williams. Nondecreasing Paths in a Weighted Graph or: How to Optimally Read a Train Schedule. ACM Trans. Algorithms, 6(4):70:1-70:24, September 2010.
Virginia Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In Proceedings of the 44th annual ACM Symposium on Theory of Computing, pages 887-898. ACM, 2012.
Uri Zwick. All Pairs Shortest Paths Using Bridging Sets and Rectangular Matrix Multiplication. J. ACM, 49(3):289-317, May 2002.
Ran Duan, Ce Jin, and Hongxun Wu
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Faster Approximation Algorithms for Computing Shortest Cycles on Weighted Graphs
Given an n-vertex m-edge graph G with non-negative edge-weights, a shortest cycle of G is one minimizing the sum of the weights on its edges. The girth of G is the weight of such a shortest cycle. We obtain several new approximation algorithms for computing the girth of weighted graphs:
- For any graph G with polynomially bounded integer weights, we present a deterministic algorithm that computes, in O~(n^{5/3}+m)-time, a cycle of weight at most twice the girth of G. This matches both the approximation factor and - almost - the running time of the best known subquadratic-time approximation algorithm for the girth of unweighted graphs.
- Then, we turn our algorithm into a deterministic (2+epsilon)-approximation for graphs with arbitrary non-negative edge-weights, at the price of a slightly worse running-time in O~(n^{5/3}polylog(1/epsilon)+m). For that, we introduce a generic method in order to obtain a polynomial-factor approximation of the girth in subquadratic time, that may be of independent interest.
- Finally, if we assume that the adjacency lists are sorted then we can get rid off the dependency in the number m of edges. Namely, we can transform our algorithms into an O~(n^{5/3})-time randomized 4-approximation for graphs with non-negative edge-weights. This can be derandomized, thereby leading to an O~(n^{5/3})-time deterministic 4-approximation for graphs with polynomially bounded integer weights, and an O~(n^{5/3}polylog(1/epsilon))-time deterministic (4+epsilon)-approximation for graphs with non-negative edge-weights.
To the best of our knowledge, these are the first known subquadratic-time approximation algorithms for computing the girth of weighted graphs.
girth
weighted graphs
approximation algorithms
Theory of computation~Design and analysis of algorithms
Theory of computation~Graph algorithms analysis
Theory of computation~Approximation algorithms analysis
49:1-49:13
Track A: Algorithms, Complexity and Games
This work was supported by a grant of Romanian Ministry of Research and Innovation CCCDI-UEFISCDI. project no. 17PCCDI/2018.
A full version of the paper is available at https://arxiv.org/abs/1810.10229.
Guillaume
Ducoffe
Guillaume Ducoffe
National Institute for Research and Development in Informatics, Romania
The Research Institute of the University of Bucharest ICUB, Romania
University of Bucharest, Romania
10.4230/LIPIcs.ICALP.2019.49
D. Aingworth, C. Chekuri, P. Indyk, and R. Motwani. Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM Journal on Computing, 28(4):1167-1181, 1999.
S. Baswana and S. Sen. Approximate distance oracles for unweighted graphs in expected O (n 2) time. ACM Transactions on Algorithms (TALG), 2(4):557-577, 2006.
J. A. Bondy and U. S. R. Murty. Graph theory. Grad. Texts in Math., 2008.
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S. Chechik. Approximate distance oracles with constant query time. In ACM STOC'14, pages 654-663, 2014.
S. Chechik, D. Larkin, L. Roditty, G. Schoenebeck, R. Tarjan, and V. Vassilevska Williams. Better approximation algorithms for the graph diameter. In SODA, pages 1041-1052. SIAM, 2014.
S. Dahlgaard, M. Knudsen, and M. Stöckel. New Subquadratic Approximation Algorithms for the Girth. Technical report, arXiv, 2017. URL: http://arxiv.org/abs/1704.02178.
http://arxiv.org/abs/1704.02178
G. Ducoffe. Faster approximation algorithms for computing shortest cycles on weighted graphs. Technical report, arXiv, 2018. URL: http://arxiv.org/abs/1810.10229.
http://arxiv.org/abs/1810.10229
A. Itai and M. Rodeh. Finding a minimum circuit in a graph. SIAM Journal on Computing, 7(4):413-423, 1978.
A. Lingas and E. Lundell. Efficient approximation algorithms for shortest cycles in undirected graphs. Information Processing Letters, 109(10):493-498, 2009.
J. Pachocki, L. Roditty, A. Sidford, R. Tov, and V. Vassilevska Williams. Approximating cycles in directed graphs: Fast algorithms for girth and roundtrip spanners. In SODA, pages 1374-1392. SIAM, 2018.
M. Patrascu and L. Roditty. Distance Oracles beyond the Thorup-Zwick Bound. SIAM Journal on Computing, 43(1):300-311, 2014.
S. Robinson. Toward an optimal algorithm for matrix multiplication. SIAM news, 38(9):1-3, 2005.
L. Roditty, M. Thorup, and U. Zwick. Deterministic constructions of approximate distance oracles and spanners. In ICALP, pages 261-272. Springer, 2005.
L. Roditty and R. Tov. Approximating the girth. ACM Transactions on Algorithms (TALG), 9(2):15, 2013.
L. Roditty and V. Vassilevska Williams. Minimum weight cycles and triangles: Equivalences and algorithms. In FOCS, pages 180-189. IEEE, 2011.
L. Roditty and V. Vassilevska Williams. Subquadratic time approximation algorithms for the girth. In SODA, pages 833-845. SIAM, 2012.
M. Thorup and U. Zwick. Approximate distance oracles. Journal of the ACM (JACM), 52(1):1-24, 2005.
V. Vassilevska Williams and R. Williams. Subcubic equivalences between path, matrix and triangle problems. In FOCS, pages 645-654. IEEE, 2010.
C. Wulff-Nilsen. Approximate distance oracles with improved preprocessing time. In ACM/SIAM SODA'12, pages 202-208, 2012.
R. Yuster and U. Zwick. Finding even cycles even faster. SIAM Journal on Discrete Mathematics, 10(2):209-222, 1997.
Guillaume Ducoffe
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Algorithmically Efficient Syntactic Characterization of Possibility Domains
We call domain any arbitrary subset of a Cartesian power of the set {0,1} when we think of it as reflecting abstract rationality restrictions on vectors of two-valued judgments on a number of issues. In Computational Social Choice Theory, and in particular in the theory of judgment aggregation, a domain is called a possibility domain if it admits a non-dictatorial aggregator, i.e. if for some k there exists a unanimous (idempotent) function F:D^k - > D which is not a projection function. We prove that a domain is a possibility domain if and only if there is a propositional formula of a certain syntactic form, sometimes called an integrity constraint, whose set of satisfying truth assignments, or models, comprise the domain. We call possibility integrity constraints the formulas of the specific syntactic type we define. Given a possibility domain D, we show how to construct a possibility integrity constraint for D efficiently, i.e, in polynomial time in the size of the domain. We also show how to distinguish formulas that are possibility integrity constraints in linear time in the size of the input formula. Finally, we prove the analogous results for local possibility domains, i.e. domains that admit an aggregator which is not a projection function, even when restricted to any given issue. Our result falls in the realm of classical results that give syntactic characterizations of logical relations that have certain closure properties, like e.g. the result that logical relations component-wise closed under logical AND are precisely the models of Horn formulas. However, our techniques draw from results in judgment aggregation theory as well from results about propositional formulas and logical relations.
collective decision making
computational social choice
judgment aggregation
logical relations
algorithm complexity
Theory of computation~Theory and algorithms for application domains
50:1-50:13
Track A: Algorithms, Complexity and Games
A full version of the paper is available at [Díaz et al., 2019], https://arxiv.org/abs/1901.00138.
We are grateful to Bruno Zanuttini for his comments that improved the presentation and simplified several proofs. Lefteris Kirousis is grateful to Phokion Kolaitis for initiating him to the area of Computational Social Choice Theory. We thank Eirini Georgoulaki for her valuable help in the final stages of writing this paper.
Josep
Díaz
Josep Díaz
Computer Science Department, Universitat Politècnica de Catalunya, Barcelona
Research partially supported by TIN2017-86727-C2-1-R, GRAMM.
Lefteris
Kirousis
Lefteris Kirousis
Department of Mathematics, National and Kapodistrian University of Athens
Computer Science Department, Universitat Politècnica de Catalunya, Barcelona
https://orcid.org/0000-0002-4912-8959
Research carried out while visiting the Computer Science Department of the Universitat Politècnica de Catalunya and supported by TIN2017-86727-C2-1-R, GRAMM.
Sofia
Kokonezi
Sofia Kokonezi
Department of Mathematics, National and Kapodistrian University of Athens
https://orcid.org/0000-0002-4580-6150
John
Livieratos
John Livieratos
Department of Mathematics, National and Kapodistrian University of Athens
https://orcid.org/0000-0001-6409-4286
10.4230/LIPIcs.ICALP.2019.50
Rina Dechter and Judea Pearl. Structure identification in relational data. Artificial Intelligence, 58(1-3):237-270, 1992.
Alvaro del Val. On 2-SAT and renamable Horn. In Proceedings of the National Conference on Artificial Intelligence, pages 279-284. Menlo Park, CA; Cambridge, MA; London; AAAI Press; MIT Press; 1999, 2000.
Josep Díaz, Lefteris Kirousis, Sofia Kokonezi, and John Livieratos. Algorithmically Efficient Syntactic Characterization of Possibility Domains. arXiv preprint, 2019. URL: http://arxiv.org/abs/1901.00138.
http://arxiv.org/abs/1901.00138
Franz Dietrich. A generalised model of judgment aggregation. Social Choice and Welfare, 28(4):529-565, 2007.
Elad Dokow and Ron Holzman. Aggregation of binary evaluations for truth-functional agendas. Social Choice and Welfare, 32(2):221-241, 2009.
Elad Dokow and Ron Holzman. Aggregation of binary evaluations. Journal of Economic Theory, 145(2):495-511, 2010.
Ramez Elmasri and Sham Navathe. Fundamentals of database systems. Pearson London, 2016.
Herbert Enderton and Herbert B Enderton. A mathematical introduction to logic. Elsevier, 2001.
Ulle Endriss and Ronald de Haan. Complexity of the winner determination problem in judgment aggregation: Kemeny, Slater, Tideman, Young. In Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems, pages 117-125. International Foundation for Autonomous Agents and Multiagent Systems, 2015.
Umberto Grandi and Ulle Endriss. Binary aggregation with integrity constraints. In IJCAI Proceedings-International Joint Conference on Artificial Intelligence, volume 22, page 204, 2011.
Umberto Grandi and Ulle Endriss. Lifting integrity constraints in binary aggregation. Artificial Intelligence, 199:45-66, 2013.
Lefteris Kirousis, Phokion G Kolaitis, and John Livieratos. Aggregation of votes with multiple positions on each issue. In Proceedings 16th International Conference on Relational and Algebraic Methods in Computer Science, pages 209-225. Springer, 2017. Expanded version to appear in ACM Transactions on Economics and Computation.
Harry R Lewis. Renaming a set of clauses as a Horn set. Journal of the ACM (JACM), 25(1):134-135, 1978.
Christian List. The theory of judgment aggregation: An introductory review. Synthese, 187(1):179-207, 2012.
Klaus Nehring and Clemens Puppe. Abstract arrowian aggregation. Journal of Economic Theory, 145(2):467-494, 2010.
Gabriella Pigozzi. Belief merging and the discursive dilemma: an argument-based account to paradoxes of judgment aggregation. Synthese, 152(2):285-298, 2006.
Willard V Quine. On cores and prime implicants of truth functions. The American Mathematical Monthly, 66(9):755-760, 1959.
Thomas J. Schaefer. The complexity of satisfiability problems. In Proc. of the 10th Annual ACM Symp. on Theory of Computing, pages 216-226, 1978.
Robert Wilson. On the theory of aggregation. Journal of Economic Theory, 10(1):89-99, 1975.
Susumu Yamasaki and Shuji Doshita. The satisfiabilty problem for a class consisting of Horn sentences and some non-Horn sentences in proportional logic. Information and Control, 59(1-3):1-12, 1983.
Bruno Zanuttini and Jean-Jacques Hébrard. A unified framework for structure identification. Information Processing Letters, 81(6):335-339, 2002.
Joseph Díaz, Lefteris Kirousis, Sofia Kokonezi, and John Livieratos
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions
Geometric Complexity Theory as initiated by Mulmuley and Sohoni in two papers (SIAM J Comput 2001, 2008) aims to separate algebraic complexity classes via representation theoretic multiplicities in coordinate rings of specific group varieties. We provide the first toy setting in which a separation can be achieved for a family of polynomials via these multiplicities.
Mulmuley and Sohoni’s papers also conjecture that the vanishing behavior of multiplicities would be sufficient to separate complexity classes (so-called occurrence obstructions). The existence of such strong occurrence obstructions has been recently disproven in 2016 in two successive papers, Ikenmeyer-Panova (Adv. Math.) and Bürgisser-Ikenmeyer-Panova (J. AMS). This raises the question whether separating group varieties via representation theoretic multiplicities is stronger than separating them via occurrences. We provide first finite settings where a separation via multiplicities can be achieved, while the separation via occurrences is provably impossible. These settings are surprisingly simple and natural: We study the variety of products of homogeneous linear forms (the so-called Chow variety) and the variety of polynomials of bounded border Waring rank (i.e. a higher secant variety of the Veronese variety).
As a side result we prove a slight generalization of Hermite’s reciprocity theorem, which proves Foulkes' conjecture for a new infinite family of cases.
Algebraic complexity theory
geometric complexity theory
Waring rank
plethysm coefficients
occurrence obstructions
multiplicity obstructions
Theory of computation~Algebraic complexity theory
51:1-51:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1901.04576.
This work was done in part while CI and GP were visiting the Simons Institute for the Theory of Computing.
Julian
Dörfler
Julian Dörfler
Saarland University, Saarbrücken, Germany
Partially supported by DFG grant IK 116/2-1.
Christian
Ikenmeyer
Christian Ikenmeyer
Max Planck Institute for Software Systems, Saarbrücken, Germany
Partially supported by DFG grant IK 116/2-1.
Greta
Panova
Greta Panova
University of Southern Californa, Los Angeles, CA, USA
University of Pennsylvania, Philadelphia, PA, USA
Partially funded by the NSF.
10.4230/LIPIcs.ICALP.2019.51
Markus Bläser and Christian Ikenmeyer. Introduction to geometric complexity theory. lecture notes, summer 2017 at Saarland University, version July 25, 2018, http://people.mpi-inf.mpg.de/~cikenmey/teaching/summer17/introtogct/gct.pdf, 2018.
http://people.mpi-inf.mpg.de/~cikenmey/teaching/summer17/introtogct/gct.pdf
Emmanuel Briand, Rosa Orellana, and Mercedes Rosas. Reduced Kronecker coefficients and counter-examples to Mulmuley’s strong saturation conjecture SH. Computational Complexity, 18:577-600, 2009.
Peter Bürgisser, Michael Clausen, and M. Amin Shokrollahi. Algebraic complexity theory, volume 315 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1997. With the collaboration of Thomas Lickteig.
Peter Bürgisser, Jesko Hüttenhain, and Christian Ikenmeyer. Permanent versus determinant: not via saturations. Proceedings of the American Mathematical Society, 145(3):1247-1258, 2017.
Peter Bürgisser and Christian Ikenmeyer. The complexity of computing Kronecker coefficients. In FPSAC 2008, Valparaiso-Viña del Mar, Chile, DMTCS proc. AJ, pages 357-368, 2008.
Peter Bürgisser and Christian Ikenmeyer. Geometric Complexity Theory and Tensor Rank. Proceedings 43rd Annual ACM Symposium on Theory of Computing 2011, pages 509-518, 2011.
Peter Bürgisser and Christian Ikenmeyer. Explicit Lower Bounds via Geometric Complexity Theory. Proceedings 45th Annual ACM Symposium on Theory of Computing 2013, pages 141-150, 2013.
Peter Bürgisser, Christian Ikenmeyer, and Greta Panova. No occurrence obstructions in geometric complexity theory. Journal of the AMS, 32:163-193, 2019. An earlier version was presented at the IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) 2016 in New Brunswick, New Jersey.
Man-Wai Cheung, Christian Ikenmeyer, and Sevak Mkrtchyan. Symmetrizing tableaux and the 5th case of the Foulkes conjecture. Journal of Symbolic Computation, 80(3):833-843, 2016. URL: http://dx.doi.org/10.1016/j.jsc.2016.09.002.
http://dx.doi.org/10.1016/j.jsc.2016.09.002
W. Fulton. Young tableaux, volume 35 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997.
Roe Goodman and Nolan R. Wallach. Symmetry, representations, and invariants, volume 255 of Graduate Texts in Mathematics. Springer, Dordrecht, 2009.
Yonghui Guan. Brill’s equations as a GL(V)-module. Linear Algebra and its Applications, 548:273-292, 2018. URL: http://dx.doi.org/10.1016/j.laa.2018.02.026.
http://dx.doi.org/10.1016/j.laa.2018.02.026
Charles Hermite. Sur la théorie des fonctions homogenes à deux indéterminées. Cambridge and Dublin Mathematical Journal, 9:172-217, 1854.
Christian Ikenmeyer. Geometric Complexity Theory, Tensor Rank, and Littlewood-Richardson Coefficients. PhD thesis, Institute of Mathematics, University of Paderborn, 2012. URL: http://nbn-resolving.de/urn:nbn:de:hbz:466:2-10472.
http://nbn-resolving.de/urn:nbn:de:hbz:466:2-10472
Christian Ikenmeyer. GCT and symmetries. unpublished lecture notes, version from January 29, 2018.
Christian Ikenmeyer and Greta Panova. Rectangular Kronecker Coefficients and Plethysms in Geometric Complexity Theory. Advances in Mathematics, 319:40-66, 2017. An earlier version was presented at the IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) 2016 in New Brunswick, New Jersey.
Harlan Kadish and J. M. Landsberg. Padded Polynomials, Their Cousins, and Geometric Complexity Theory. Communications in Algebra, 42(5):2171-2180, 2014. URL: http://dx.doi.org/10.1080/00927872.2012.758268.
http://dx.doi.org/10.1080/00927872.2012.758268
Hanspeter Kraft. Geometrische Methoden in der Invariantentheorie. Friedr. Vieweg und Sohn Verlagsgesellschaft, Braunschweig, 1985.
J. M. Landsberg. Geometry and Complexity Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2017. URL: http://dx.doi.org/10.1017/9781108183192.
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Joseph Landsberg. Tensors: Geometry and Applications, volume 128 of Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island, 2011.
I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications.
K.D. Mulmuley and M. Sohoni. Geometric Complexity Theory. I. An approach to the P vs. NP and related problems. SIAM J. Comput., 31(2):496-526 (electronic), 2001.
K.D. Mulmuley and M. Sohoni. Geometric Complexity Theory. II. Towards explicit obstructions for embeddings among class varieties. SIAM J. Comput., 38(3):1175-1206, 2008.
D. Mumford. Algebraic geometry. I: Complex projective varieties. Classics in mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1976 edition in Grundlehren der mathematischen Wissenschaften, vol. 221.
Hariharan Narayanan. On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients. J. Algebraic Combin., 24(3):347-354, 2006.
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http://dx.doi.org/10.1016/j.crma.2013.06.008
Claudio Procesi. Lie groups. Universitext. Springer, New York, 2007. An approach through invariants and representations.
Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity. , version 3.1.4, 2017. URL: https://github.com/dasarpmar/lowerbounds-survey.
https://github.com/dasarpmar/lowerbounds-survey
Richard P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.
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Richard P. Stanley. Enumerative combinatorics. Vol. 1. Cambridge University Press, Cambridge, 2011. second edition.
Bernd Sturmfels. Algorithms in Invariant Theory. Texts & Monographs in Symbolic Computation. Springer, 2008.
Julian Dörfler, Christian Ikenmeyer, and Greta Panova
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Arboricity Captures the Complexity of Sampling Edges
In this paper, we revisit the problem of sampling edges in an unknown graph G = (V, E) from a distribution that is (pointwise) almost uniform over E. We consider the case where there is some a priori upper bound on the arboriciy of G. Given query access to a graph G over n vertices and of average degree {d} and arboricity at most alpha, we design an algorithm that performs O(alpha/d * {log^3 n}/epsilon) queries in expectation and returns an edge in the graph such that every edge e in E is sampled with probability (1 +/- epsilon)/m. The algorithm performs two types of queries: degree queries and neighbor queries. We show that the upper bound is tight (up to poly-logarithmic factors and the dependence in epsilon), as Omega(alpha/d) queries are necessary for the easier task of sampling edges from any distribution over E that is close to uniform in total variational distance. We also prove that even if G is a tree (i.e., alpha = 1 so that alpha/d = Theta(1)), Omega({log n}/{loglog n}) queries are necessary to sample an edge from any distribution that is pointwise close to uniform, thus establishing that a poly(log n) factor is necessary for constant alpha. Finally we show how our algorithm can be applied to obtain a new result on approximately counting subgraphs, based on the recent work of Assadi, Kapralov, and Khanna (ITCS, 2019).
sampling
graph algorithms
arboricity
sublinear-time algorithms
Theory of computation~Graph algorithms analysis
Theory of computation~Approximation algorithms analysis
Theory of computation~Streaming, sublinear and near linear time algorithms
Theory of computation~Sketching and sampling
52:1-52:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1902.08086.
Talya
Eden
Talya Eden
Tel Aviv University, Tel Aviv, Israel
Supported by the Azrieli fellowship program for graduate students, by the Sephora Scholarship and by the Weinstein Graduate Studies Prize.
Dana
Ron
Dana Ron
Tel Aviv University, Tel Aviv, Israel
Partially supported by the Israel Science Foundation grant No. 1146/18.
Will
Rosenbaum
Will Rosenbaum
Max Planck Institute for Informatics, Saarbrücken, Germany
10.4230/LIPIcs.ICALP.2019.52
Maryam Aliakbarpour, Amartya Shankha Biswas, Themis Gouleakis, John Peebles, Ronitt Rubinfeld, and Anak Yodpinyanee. Sublinear-Time Algorithms for Counting Star Subgraphs via Edge Sampling. Algorithmica, pages 1-30, 2017. URL: http://dx.doi.org/10.1007/s00453-017-0287-3.
http://dx.doi.org/10.1007/s00453-017-0287-3
Sepehr Assadi, Michael Kapralov, and Sanjeev Khanna. A Simple Sublinear-Time Algorithm for Counting Arbitrary Subgraphs via Edge Sampling. In ITCS, volume 124 of LIPIcs, pages 6:1-6:20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019.
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Leonid Barenboim and Michael Elkin. Sublogarithmic distributed MIS algorithm for sparse graphs using Nash-Williams decomposition. Distributed Computing, 22(5-6):363-379, 2010. URL: http://dx.doi.org/10.1007/s00446-009-0088-2.
http://dx.doi.org/10.1007/s00446-009-0088-2
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http://dx.doi.org/10.1137/1.9781611975031.136
Talya Eden, Dana Ron, and Will Rosenbaum. The Arboricity Captures the Complexity of Sampling Edges, 2019. URL: http://arxiv.org/abs/1902.08086.
http://arxiv.org/abs/1902.08086
Talya Eden, Dana Ron, and C. Seshadhri. Sublinear Time Estimation of Degree Distribution Moments: The Degeneracy Connection. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1-7:13, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.7.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.7
Talya Eden, Dana Ron, and C. Seshadhri. Faster sublinear approximations of k-cliques for low arboricity graphs. CoRR, abs/1811.04425, 2018. URL: http://arxiv.org/abs/1811.04425.
http://arxiv.org/abs/1811.04425
Talya Eden, Dana Ron, and C. Seshadhri. On approximating the number of k-cliques in sublinear time. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 722-734, 2018. URL: http://dx.doi.org/10.1145/3188745.3188810.
http://dx.doi.org/10.1145/3188745.3188810
Talya Eden and Will Rosenbaum. Lower Bounds for Approximating Graph Parameters via Communication Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2018, August 20-22, 2018 - Princeton, NJ, USA, pages 11:1-11:18, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.11.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.11
Talya Eden and Will Rosenbaum. On Sampling Edges Almost Uniformly. In Raimund Seidel, editor, 1st Symposium on Simplicity in Algorithms (SOSA 2018), volume 61 of OpenAccess Series in Informatics (OASIcs), pages 7:1-7:9, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://dx.doi.org/10.4230/OASIcs.SOSA.2018.7.
http://dx.doi.org/10.4230/OASIcs.SOSA.2018.7
David Eppstein and Darren Strash. Listing all maximal cliques in large sparse real-world graphs. In International Symposium on Experimental Algorithms, pages 364-375. Springer, 2011.
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Gaurav Goel and Jens Gustedt. Bounded arboricity to determine the local structure of sparse graphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 159-167. Springer, 2006.
Oded Goldreich and Dana Ron. Approximating average parameters of graphs. Random Struct. Algorithms, 32(4):473-493, 2008.
Mira Gonen, Dana Ron, and Yuval Shavitt. Counting stars and other small subgraphs in sublinear-time. SIAM Journal on Discrete Mathematics, 25(3):1365-1411, 2011.
C. St. JA. Nash-Williams. Edge-disjoint spanning trees of finite graphs. Journal of the London Mathematical Society, 1(1):445-450, 1961.
Kijung Shin, Tina Eliassi-Rad, and Christos Faloutsos. Patterns and anomalies in k-cores of real-world graphs with applications. Knowledge and Information Systems, 54(3):677-710, 2018.
Talya Eden, Dana Ron, and Will Rosenbaum
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Nearly-Linear Time Algorithm for Submodular Maximization with a Knapsack Constraint
We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint. Our main contribution is an algorithm that achieves a nearly-optimal, 1 - 1/e - epsilon approximation, using (1/epsilon)^{O(1/epsilon^4)} n log^2{n} function evaluations and arithmetic operations. Our algorithm is impractical but theoretically interesting, since it overcomes a fundamental running time bottleneck of the multilinear extension relaxation framework. This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to Omega(n^2) running times, since evaluating the multilinear extension is expensive. Our algorithm maintains a fractional solution with only a constant number of entries that are strictly fractional, which allows us to overcome this obstacle.
submodular maximization
knapsack constraint
fast algorithms
Theory of computation~Submodular optimization and polymatroids
53:1-53:12
Track A: Algorithms, Complexity and Games
https://arxiv.org/abs/1709.09767
This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing.
Alina
Ene
Alina Ene
Department of Computer Science, Boston University, MA, USA
Partially supported by NSF CAREER grant CCF-1750333 and NSF grant CCF-1718342.
Huy L.
Nguyen
Huy L. Nguyen
College of Computer and Information Science, Northeastern University, Boston, MA, USA
Partially supported by NSF CAREER grant CCF-1750716.
10.4230/LIPIcs.ICALP.2019.53
Ashwinkumar Badanidiyuru and Jan Vondrák. Fast algorithms for maximizing submodular functions. In ACM-SIAM Symposium on Discrete Algorithms (SODA), 2014.
Shaddin Dughmi, Tim Roughgarden, and Mukund Sundararajan. Revenue Submodularity. Theory of Computing, 8(1):95-119, 2012.
Ryan Gomes and Andreas Krause. Budgeted Nonparametric Learning from Data Streams. In International Conference on Machine Learning (ICML), pages 391-398, 2010.
Stefanie Jegelka and Jeff A. Bilmes. Submodularity beyond submodular energies: Coupling edges in graph cuts. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2011.
David Kempe, Jon M. Kleinberg, and Éva Tardos. Maximizing the spread of influence through a social network. In ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), pages 137-146, 2003.
Samir Khuller, Anna Moss, and Joseph Seffi Naor. The budgeted maximum coverage problem. Information processing letters, 70(1):39-45, 1999.
Andreas Krause, Ajit Paul Singh, and Carlos Guestrin. Near-Optimal Sensor Placements in Gaussian Processes: Theory, Efficient Algorithms and Empirical Studies. Journal of Machine Learning Research, 9:235-284, 2008.
Ariel Kulik, Hadas Shachnai, and Tami Tamir. Approximations for monotone and nonmonotone submodular maximization with knapsack constraints. Mathematics of Operations Research, 38(4):729-739, 2013.
Hui Lin and Jeff A. Bilmes. Multi-document Summarization via Budgeted Maximization of Submodular Functions. In Human Language Technologies: Conference of the North American Chapter of the Association of Computational Linguistics, pages 912-920, 2010.
Maxim Sviridenko. A note on maximizing a submodular set function subject to a knapsack constraint. Operations Research Letters, 32(1):41-43, 2004.
Laurence A Wolsey. Maximising real-valued submodular functions: Primal and dual heuristics for location problems. Mathematics of Operations Research, 7(3):410-425, 1982.
Yuichi Yoshida. Maximizing a Monotone Submodular Function with a Bounded Curvature under a Knapsack Constraint. CoRR, abs/1607.04527, 2016. URL: http://arxiv.org/abs/1607.04527.
http://arxiv.org/abs/1607.04527
Alina Ene and Huy L. Nguyen
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Towards Nearly-Linear Time Algorithms for Submodular Maximization with a Matroid Constraint
We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a general matroid constraint with a 1 - 1/e - epsilon approximation that achieves a fast running time provided we have a fast data structure for maintaining an approximately maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the first algorithms for these classes of matroids that achieve a nearly-optimal, 1 - 1/e - epsilon approximation, using a nearly-linear number of function evaluations and arithmetic operations.
submodular maximization
matroid constraints
fast running times
Theory of computation~Submodular optimization and polymatroids
54:1-54:14
Track A: Algorithms, Complexity and Games
https://arxiv.org/abs/1811.07464
This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing.
Alina
Ene
Alina Ene
Department of Computer Science, Boston University, MA, USA
Partially supported by NSF CAREER grant CCF-1750333 and NSF grant CCF-1718342.
Huy L.
Nguyen
Huy L. Nguyen
College of Computer and Information Science, Northeastern University, Boston, MA, USA
Partially supported by NSF CAREER grant CCF-1750716.
10.4230/LIPIcs.ICALP.2019.54
Alexander Ageev and Maxim Sviridenko. Pipage Rounding: A New Method of Constructing Algorithms with Proven Performance Guarantee. Journal of Combinatorial Optimization, 8(3):307-328, 2004.
Yossi Azar and Iftah Gamzu. Efficient Submodular Function Maximization under Linear Packing Constraints. In International Colloquium on Automata, Languages and Programming (ICALP), 2012.
Ashwinkumar Badanidiyuru and Jan Vondrák. Fast algorithms for maximizing submodular functions. In ACM-SIAM Symposium on Discrete Algorithms (SODA), 2014.
Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. Submodular Maximization with Cardinality Constraints. In ACM-SIAM Symposium on Discrete Algorithms (SODA), 2014.
Niv Buchbinder, Moran Feldman, and Roy Schwartz. Comparing Apples and Oranges: Query Trade-off in Submodular Maximization. Math. Oper. Res., 42(2):308-329, 2017.
Gruia Calinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a Submodular Set Function Subject to a Matroid Constraint. SIAM Journal on Computing, 40(6):1740-1766, 2011.
Chandra Chekuri, T. S. Jayram, and Jan Vondrák. On Multiplicative Weight Updates for Concave and Submodular Function Maximization. In Conference on Innovations in Theoretical Computer Science (ITCS), 2015. URL: http://dx.doi.org/10.1145/2688073.2688086.
http://dx.doi.org/10.1145/2688073.2688086
Chandra Chekuri, Jan Vondrák, and Rico Zenklusen. Dependent Randomized Rounding via Exchange Properties of Combinatorial Structures. In IEEE Foundations of Computer Science (FOCS), pages 575-584. IEEE Computer Society, 2010.
Shaddin Dughmi, Tim Roughgarden, and Mukund Sundararajan. Revenue Submodularity. Theory of Computing, 8(1):95-119, 2012.
Uriel Feige. A threshold of ln n for approximating set cover. jacm, 45:634-652, 1998.
Yuval Filmus and Justin Ward. Monotone Submodular Maximization over a Matroid via Non-Oblivious Local Search. SIAM Journal on Computing, 43(2):514-542, 2014.
M L Fisher, G L Nemhauser, and L A Wolsey. An analysis of approximations for maximizing submodular set functions - II. Mathematical Programming Studies, 8:73-87, 1978.
Bernard A Galler and Michael J Fisher. An improved equivalence algorithm. Communications of the ACM, 7(5):301-303, 1964.
Ryan Gomes and Andreas Krause. Budgeted Nonparametric Learning from Data Streams. In International Conference on Machine Learning (ICML), pages 391-398, 2010.
Jacob Holm, Kristian De Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. Journal of the ACM (JACM), 48(4):723-760, 2001.
David Kempe, Jon M. Kleinberg, and Éva Tardos. Maximizing the spread of influence through a social network. In ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), pages 137-146, 2003.
Andreas Krause, Ajit Paul Singh, and Carlos Guestrin. Near-Optimal Sensor Placements in Gaussian Processes: Theory, Efficient Algorithms and Empirical Studies. Journal of Machine Learning Research, 9:235-284, 2008.
Hui Lin and Jeff A. Bilmes. Multi-document Summarization via Budgeted Maximization of Submodular Functions. In Human Language Technologies: Conference of the North American Chapter of the Association of Computational Linguistics, pages 912-920, 2010.
Baharan Mirzasoleiman, Ashwinkumar Badanidiyuru, Amin Karbasi, Jan Vondrák, and Andreas Krause. Lazier Than Lazy Greedy. In AAAI Conference on Artificial Intelligence (AAAI), 2015.
G L Nemhauser and L A Wolsey. Best Algorithms for Approximating the Maximum of a Submodular Set Function. Mathematics of Operations Research, 3(3):177-188, 1978.
G L Nemhauser, L A Wolsey, and M L Fisher. An analysis of approximations for maximizing submodular set functions - I. Mathematical Programming, 14(1):265-294, 1978.
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Robert Endre Tarjan. Efficiency of a good but not linear set union algorithm. Journal of the ACM (JACM), 22(2):215-225, 1975.
Jan Vondrák. Optimal approximation for the submodular welfare problem in the value oracle model. In ACM Symposium on Theory of Computing (STOC), 2008.
Alina Ene and Huy L. Nguyen
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On the Complexity of String Matching for Graphs
Exact string matching in labeled graphs is the problem of searching paths of a graph G=(V,E) such that the concatenation of their node labels is equal to the given pattern string P[1..m]. This basic problem can be found at the heart of more complex operations on variation graphs in computational biology, of query operations in graph databases, and of analysis operations in heterogeneous networks.
We prove a conditional lower bound stating that, for any constant epsilon>0, an O(|E|^{1 - epsilon} m)-time, or an O(|E| m^{1 - epsilon})-time algorithm for exact string matching in graphs, with node labels and patterns drawn from a binary alphabet, cannot be achieved unless the Strong Exponential Time Hypothesis (SETH) is false. This holds even if restricted to undirected graphs with maximum node degree two, i.e. to zig-zag matching in bidirectional strings, or to deterministic directed acyclic graphs whose nodes have maximum sum of indegree and outdegree three. These restricted cases make the lower bound stricter than what can be directly derived from related bounds on regular expression matching (Backurs and Indyk, FOCS'16). In fact, our bounds are tight in the sense that lowering the degree or the alphabet size yields linear-time solvable problems.
An interesting corollary is that exact and approximate matching are equally hard (quadratic time) in graphs under SETH. In comparison, the same problems restricted to strings have linear-time vs quadratic-time solutions, respectively (approximate pattern matching having also a matching SETH lower bound (Backurs and Indyk, STOC'15)).
exact pattern matching
graph query
graph search
labeled graphs
string matching
string search
strong exponential time hypothesis
heterogeneous networks
variation graphs
Theory of computation~Pattern matching
55:1-55:15
Track A: Algorithms, Complexity and Games
This work has been partially supported by Academy of Finland (grant 309048).
This paper is based on our two technical reports [{Equi} et al., 2019; {Equi} et al., 2019] available at https://arxiv.org/abs/1901.05264 and https://arxiv.org/abs/1902.03560.
Massimo
Equi
Massimo Equi
Department of Computer Science, University of Helsinki, Finland
Roberto
Grossi
Roberto Grossi
Dipartimento di Informatica, Università di Pisa, Italy
Veli
Mäkinen
Veli Mäkinen
Department of Computer Science, University of Helsinki, Finland
Alexandru I.
Tomescu
Alexandru I. Tomescu
Department of Computer Science, University of Helsinki, Finland
10.4230/LIPIcs.ICALP.2019.55
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Massimo Equi, Roberto Grossi, Veli Mäkinen, and Alexandru I. Tomescu
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Unique End of Potential Line
The complexity class CLS was proposed by Daskalakis and Papadimitriou in 2011 to understand the complexity of important NP search problems that admit both path following and potential optimizing algorithms. Here we identify a subclass of CLS - called UniqueEOPL - that applies a more specific combinatorial principle that guarantees unique solutions. We show that UniqueEOPL contains several important problems such as the P-matrix Linear Complementarity Problem, finding Fixed Point of Contraction Maps, and solving Unique Sink Orientations (USOs). UniqueEOPL seems to a proper subclass of CLS and looks more likely to be the right class for the problems of interest. We identify a problem - closely related to solving contraction maps and USOs - that is complete for UniqueEOPL. Our results also give the fastest randomised algorithm for P-matrix LCP.
P-matrix linear complementarity problem
unique sink orientation
contraction map
TFNP
total search problems
continuous local search
Theory of computation~Problems, reductions and completeness
56:1-56:15
Track A: Algorithms, Complexity and Games
This paper substantially revises and extends the work described in our previous preprint "End of Potential Line" arXiv:1804.03450 [John Fearnley et al., 2018]. The full version of the paper is available at https://arxiv.org/abs/1811.03841.
We thank Aviad Rubinstein and Kousha Etessami for alerting us to the work in [Qi Qi, 2012], and we thank Rasmus Ibsen Jensen for helpful comments on a preprint of this paper.
John
Fearnley
John Fearnley
University of Liverpool, UK
Research supported by EPSRC grant EP/P020909/1.
Spencer
Gordon
Spencer Gordon
California Institute of Technology, Pasadena, CA, USA
Ruta
Mehta
Ruta Mehta
University of Illinois at Urbana-Champaign, IL, USA
Research supported by NSF grant CCF-1750436.
Rahul
Savani
Rahul Savani
University of Liverpool, UK
10.4230/LIPIcs.ICALP.2019.56
Ilan Adler and Sushil Verma. The Linear Complementarity Problem, Lemke Algorithm, Perturbation, and the Complexity Class PPAD. Technical report, Manuscript, Depatrment of IEOR, University of California, Berkeley, CA 94720, 2011.
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John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Dichotomy for Symmetric Boolean PCSPs
In one of the most actively studied version of Constraint Satisfaction Problem, a CSP is defined by a relational structure called a template. In the decision version of the problem the goal is to determine whether a structure given on input admits a homomorphism into this template. Two recent independent results of Bulatov [FOCS'17] and Zhuk [FOCS'17] state that each finite template defines CSP which is tractable or NP-complete.
In a recent paper Brakensiek and Guruswami [SODA'18] proposed an extension of the CSP framework. This extension, called Promise Constraint Satisfaction Problem, includes many naturally occurring computational questions, e.g. approximate coloring, that cannot be cast as CSPs. A PCSP is a combination of two CSPs defined by two similar templates; the computational question is to distinguish a YES instance of the first one from a NO instance of the second.
The computational complexity of many PCSPs remains unknown. Even the case of Boolean templates (solved for CSP by Schaefer [STOC'78]) remains wide open. The main result of Brakensiek and Guruswami [SODA'18] shows that Boolean PCSPs exhibit a dichotomy (PTIME vs. NPC) when "all the clauses are symmetric and allow for negation of variables". In this paper we remove the "allow for negation of variables" assumption from the theorem. The "symmetric" assumption means that changing the order of variables in a constraint does not change its satisfiability. The "negation of variables" means that both of the templates share a relation which can be used to effectively negate Boolean variables.
The main result of this paper establishes dichotomy for all the symmetric boolean templates. The tractability case of our theorem and the theorem of Brakensiek and Guruswami are almost identical. The main difference, and the main contribution of this work, is the new reason for hardness and the reasoning proving the split.
promise constraint satisfaction problem
PCSP
algebraic approach
Theory of computation~Complexity theory and logic
Theory of computation~Constraint and logic programming
57:1-57:12
Track A: Algorithms, Complexity and Games
Research was partially supported by National Science Centre, Poland grant no. 2014/2013/B/ST6/01812.
A full version of the paper is avaialable at https://arxiv.org/abs/1904.12424, [Miron Ficak et al., 2019] .
Miron
Ficak
Miron Ficak
Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
https://orcid.org/0000-0003-3104-6354
Marcin
Kozik
Marcin Kozik
Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
https://orcid.org/0000-0002-1839-4824
Miroslav
Olšák
Miroslav Olšák
Department of Algebra, Charles University, Prague, Czech Republic
Szymon
Stankiewicz
Szymon Stankiewicz
Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
https://orcid.org/0000-0003-2235-4849
10.4230/LIPIcs.ICALP.2019.57
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http://dx.doi.org/10.4230/LIPIcs.CCC.2016.14
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http://arxiv.org/abs/1807.05194
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http://arxiv.org/abs/1904.12424
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Vladimir Kolmogorov, Andrei Krokhin, and Michal Rolinek. The Complexity of General-Valued CSPs. In Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), FOCS '15, pages 1246-1258, Washington, DC, USA, 2015. IEEE Computer Society. URL: http://dx.doi.org/10.1109/FOCS.2015.80.
http://dx.doi.org/10.1109/FOCS.2015.80
Thomas J. Schaefer. The complexity of satisfiability problems. In Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, Calif., 1978), pages 216-226. ACM, New York, 1978.
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http://dx.doi.org/10.1109/FOCS.2017.38
Miron Ficak, Marcin Kozik, Miroslav Olšák, and Szymon Stankiewicz
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions
The seminal result of Kahn, Kalai and Linial shows that a coalition of O(n/(log n)) players can bias the outcome of any Boolean function {0,1}^n -> {0,1} with respect to the uniform measure. We extend their result to arbitrary product measures on {0,1}^n, by combining their argument with a completely different argument that handles very biased input bits.
We view this result as a step towards proving a conjecture of Friedgut, which states that Boolean functions on the continuous cube [0,1]^n (or, equivalently, on {1,...,n}^n) can be biased using coalitions of o(n) players. This is the first step taken in this direction since Friedgut proposed the conjecture in 2004.
Russell, Saks and Zuckerman extended the result of Kahn, Kalai and Linial to multi-round protocols, showing that when the number of rounds is o(log^* n), a coalition of o(n) players can bias the outcome with respect to the uniform measure. We extend this result as well to arbitrary product measures on {0,1}^n.
The argument of Russell et al. relies on the fact that a coalition of o(n) players can boost the expectation of any Boolean function from epsilon to 1-epsilon with respect to the uniform measure. This fails for general product distributions, as the example of the AND function with respect to mu_{1-1/n} shows. Instead, we use a novel boosting argument alongside a generalization of our first result to arbitrary finite ranges.
Boolean function analysis
coin flipping
Theory of computation
58:1-58:13
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://eccc.weizmann.ac.il/report/2019/029/.
Part of the work on this paper was done while the first three authors were at the Simons Institute for the Theory of Computing at Berkeley, CA, USA.
Yuval
Filmus
Yuval Filmus
Computer Science Department, Technion, Haifa, Israel
http://www.cs.toronto.edu/~yuvalf/
https://orcid.org/0000-0002-1739-0872
Taub Fellow - supported by the Taub Foundations. The research was funded by ISF grant 1337/16.
Lianna
Hambardzumyan
Lianna Hambardzumyan
School of Computer Science, McGill University, Montreal, QC, Canada
Hamed
Hatami
Hamed Hatami
School of Computer Science, McGill University, Montreal, QC, Canada
https://www.cs.mcgill.ca/~hatami/
https://orcid.org/0000-0002-4732-434X
Supported by an NSERC grant.
Pooya
Hatami
Pooya Hatami
Department of Computer Science, UT Austin, Austin, TX, USA
https://pooyahatami.org/
https://orcid.org/0000-0001-7928-8008
Supported by a Simons Investigator Award (#409864, David Zuckerman).
David
Zuckerman
David Zuckerman
Department of Computer Science, UT Austin, Austin, TX, USA
http://www.cs.utexas.edu/~diz/
Supported by NSF Grant CCF-1705028 and a Simons Investigator Award (#409864).
10.4230/LIPIcs.ICALP.2019.58
Miklós Ajtai and Nathan Linial. The influence of large coalitions. Combinatorica, 13(2):129-145, 1993.
Michael Ben-Or and Nathan Linial. Collective coin flipping. Advances in Computing Research, 5:91-115, 1989.
Michael Ben-Or, Nathan Linial, and Michael Saks. Collective coin flipping and other models of imperfect randomness. IBM Thomas J. Watson Research Division, 1989.
Jean Bourgain, Jeff Kahn, Gil Kalai, Yitzhak Katznelson, and Nathan Linial. The influence of variables in product spaces. Israel Journal of Mathematics, 77(1-2):55-64, 1992.
Eshan Chattopadhyay and David Zuckerman. Explicit two-source extractors and resilient functions. Annals of Mathematics, to appear, 2016. Preliminary version in STOC 2016.
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Yevgeniy Dodis. Fault-tolerant leader election and collective coin-flipping in the full information model. Survey, 2006.
Uriel Feige. Noncryptographic Selection Protocols. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, page 142. IEEE Computer Society, 1999.
Ehud Friedgut. Influences in Product Spaces: KKL and BKKKL Revisited. Combinatorics, Probability and Computing, 13(1):17-29, 2004.
Jeff Kahn, Gil Kalai, and Nathan Linial. The influence of variables on Boolean functions. In Proceedings of the 29th annual FOCS, pages 68-80, 1988.
Raghu Meka. Explicit resilient functions matching Ajtai-Linial. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1132-1148. SIAM, 2017.
Alexander Russell, Michael Saks, and David Zuckerman. Lower bounds for leader election and collective coin-flipping in the perfect information model. SIAM Journal on Computing, 31(6):1645-1662, 2002.
Alexander Russell and David Zuckerman. Perfect information leader election in log* n+ O (1) rounds. Journal of Computer and System Sciences, 63(4):612-626, 2001.
Terence Tao. Soft analysis, hard analysis, and the finite convergence principle. URL: https://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/, 2007. Accessed 10 Feb 2019.
https://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/
Yuval Filmus, Lianna Hambardzumyan, Hamed Hatami, Pooya Hatami, and David Zuckerman
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Covering Vectors by Spaces in Perturbed Graphic Matroids and Their Duals
Perturbed graphic matroids are binary matroids that can be obtained from a graphic matroid by adding a noise of small rank. More precisely, an r-rank perturbed graphic matroid M is a binary matroid that can be represented in the form I +P, where I is the incidence matrix of some graph and P is a binary matrix of rank at most r. Such matroids naturally appear in a number of theoretical and applied settings. The main motivation behind our work is an attempt to understand which parameterized algorithms for various problems on graphs could be lifted to perturbed graphic matroids.
We study the parameterized complexity of a natural generalization (for matroids) of the following fundamental problems on graphs: Steiner Tree and Multiway Cut. In this generalization, called the Space Cover problem, we are given a binary matroid M with a ground set E, a set of terminals T subseteq E, and a non-negative integer k. The task is to decide whether T can be spanned by a subset of E \ T of size at most k.
We prove that on graphic matroid perturbations, for every fixed r, Space Cover is fixed-parameter tractable parameterized by k. On the other hand, the problem becomes W[1]-hard when parameterized by r+k+|T| and it is NP-complete for r <= 2 and |T|<= 2.
On cographic matroids, that are the duals of graphic matroids, Space Cover generalizes another fundamental and well-studied problem, namely Multiway Cut. We show that on the duals of perturbed graphic matroids the Space Cover problem is fixed-parameter tractable parameterized by r+k.
Binary matroids
perturbed graphic matroids
spanning set
parameterized complexity
Mathematics of computing~Combinatorial algorithms
Theory of computation~Parameterized complexity and exact algorithms
59:1-59:13
Track A: Algorithms, Complexity and Games
The research leading to these results has received funding from the Research Council of Norway via the projects "CLASSIS" and "MULTIVAL".
A full version of the paper is available at http://arxiv.org/abs/1902.06957.
We thank Jim Geelen for valuable insights regarding matroid minors.
Fedor V.
Fomin
Fedor V. Fomin
Department of Informatics, University of Bergen, Norway
Petr A.
Golovach
Petr A. Golovach
Department of Informatics, University of Bergen, Norway
Daniel
Lokshtanov
Daniel Lokshtanov
Department of Computer Science, University of California Santa Barbara, USA
Saket
Saurabh
Saket Saurabh
The Institute of Mathematical Sciences, HBNI, Chennai, India
Meirav
Zehavi
Meirav Zehavi
Ben-Gurion University, Israel
10.4230/LIPIcs.ICALP.2019.59
Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. J. ACM, 42(4):844-856, 1995.
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Édouard Bonnet, László Egri, and Dániel Marx. Fixed-parameter Approximability of Boolean MinCSPs. CoRR, abs/1601.04935, 2016. URL: http://arxiv.org/abs/1601.04935.
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http://dx.doi.org/10.1137/090761793
Rajesh Chitnis, Marek Cygan, MohammadTaghi Hajiaghayi, Marcin Pilipczuk, and Michal Pilipczuk. Designing FPT Algorithms for Cut Problems Using Randomized Contractions. SIAM J. Comput., 45(4):1171-1229, 2016. URL: http://dx.doi.org/10.1137/15M1032077.
http://dx.doi.org/10.1137/15M1032077
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
Elias Dahlhaus, David S. Johnson, Christos H. Papadimitriou, Paul D. Seymour, and Mihalis Yannakakis. The Complexity of Multiterminal Cuts. SIAM J. Comput., 23(4):864-894, 1994. URL: http://dx.doi.org/10.1137/S0097539792225297.
http://dx.doi.org/10.1137/S0097539792225297
Reinhard Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 3rd edition, 2005.
Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013.
Rodney G. Downey, Michael R. Fellows, Alexander Vardy, and Geoff Whittle. The Parametrized Complexity of Some Fundamental Problems in Coding Theory. SIAM J. Comput., 29(2):545-570, 1999. URL: http://dx.doi.org/10.1137/S0097539797323571.
http://dx.doi.org/10.1137/S0097539797323571
Stuart E. Dreyfus and Robert 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
Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Covering Vectors by Spaces: Regular Matroids. SIAM J. Discrete Math., 32(4):2512-2565, 2018.
Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Covering Vectors by Spaces in Perturbed Graphic Matroids and Their Duals. CoRR, abs/1902.06957, 2019. URL: http://arxiv.org/abs/1902.06957.
http://arxiv.org/abs/1902.06957
Jim Geelen, Bert Gerards, and Geoff Whittle. Solving Rota’s conjecture. Notices Amer. Math. Soc., 61(7):736-743, 2014.
Jim Geelen, Bert Gerards, and Geoff Whittle. The Highly Connected Matroids in Minor-Closed Classes. Ann. Comb., 19(1):107-123, 2015.
Jim Geelen and Rohan Kapadia. Computing Girth and Cogirth in Perturbed Graphic Matroids. Combinatorica, 38(1):167-191, 2018. URL: http://dx.doi.org/10.1007/s00493-016-3445-3.
http://dx.doi.org/10.1007/s00493-016-3445-3
Jiong Guo, Jens Gramm, Falk Hüffner, Rolf Niedermeier, and Sebastian Wernicke. Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. J. Computer and System Sciences, 72(8):1386-1396, 2006. URL: http://dx.doi.org/10.1016/j.jcss.2006.02.001.
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http://dx.doi.org/10.1109/CDC.2018.8619666
Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Decomposition of Map Graphs with Applications
Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a sqrt{k} x sqrt{k}-grid as a minor, or its treewidth is O(sqrt{k}). However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs like unit disk or map graphs. This is mainly due to the presence of large cliques in these classes of graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new decomposition lemma for map graphs, the intersection graphs of finitely many simply-connected and interior-disjoint regions of the Euclidean plane. Informally, our lemma states the following. For any map graph G, there exists a collection (U_1,...,U_t) of cliques of G with the following property: G either contains a sqrt{k} x sqrt{k}-grid as a minor, or it admits a tree decomposition where every bag is the union of O(sqrt{k}) cliques in the above collection.
The new lemma appears to be a handy tool in the design of subexponential parameterized algorithms on map graphs. We demonstrate its usability by designing algorithms on map graphs with running time 2^{O({sqrt{k}log{k}})} * n^{O(1)} for Connected Planar F-Deletion (that encompasses problems such as Feedback Vertex Set and Vertex Cover). Obtaining subexponential algorithms for Longest Cycle/Path and Cycle Packing is more challenging. We have to construct tree decompositions with more powerful properties and to prove sublinear bounds on the number of ways an optimum solution could "cross" bags in these decompositions.
For Longest Cycle/Path, these are the first subexponential-time parameterized algorithm on map graphs. For Feedback Vertex Set and Cycle Packing, we improve upon known 2^{O({k^{0.75}log{k}})} * n^{O(1)}-time algorithms on map graphs.
Longest Cycle
Cycle Packing
Feedback Vertex Set
Map Graphs
FPT
Theory of computation~Parameterized complexity and exact algorithms
Theory of computation~Computational geometry
60:1-60:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at http://arxiv.org/abs/1903.01291.
This work is supported by the European Research Council (ERC) via grant LOPPRE, reference 819416, the Norwegian Research Council via project MULTIVAL, and Israel Science Foundation individual research grant no. 1176/18.
Fedor V.
Fomin
Fedor V. Fomin
University of Bergen, Norway
Daniel
Lokshtanov
Daniel Lokshtanov
University of California, Santa Barbara, USA
Fahad
Panolan
Fahad Panolan
University of Bergen, Norway
Saket
Saurabh
Saket Saurabh
The Institute of Mathematical Sciences, HBNI, Chennai, India
Meirav
Zehavi
Meirav Zehavi
Ben-Gurion University of the Negev, Beer-Sheva, Israel
10.4230/LIPIcs.ICALP.2019.60
J. Alber, H. L. Bodlaender, H. Fernau, T. Kloks, and R. Niedermeier. Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica, 33(4):461-493, 2002.
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Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
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 Proceedings of the 52nd Annual Symposium on Foundations of Computer Science (FOCS), pages 150-159. IEEE, 2011.
Erik D. Demaine, Fedor V. Fomin, Mohammadtaghi Hajiaghayi, and Dimitrios M. Thilikos. Fixed-parameter algorithms for (k,r)-center in planar graphs and map graphs. ACM Trans. Algorithms, 1(1):33-47, 2005. URL: http://dx.doi.org/10.1145/1077464.1077468.
http://dx.doi.org/10.1145/1077464.1077468
Erik D. Demaine, Fedor V. Fomin, Mohammadtaghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential Parameterized Algorithms on Graphs of Bounded Genus and H-minor-free Graphs. J. ACM, 52(6):866-893, 2005.
Erik D. Demaine and MohammadTaghi Hajiaghayi. Bidimensionality: new connections between FPT algorithms and PTASs. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), pages 590-601, New York, 2005. ACM-SIAM.
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http://dx.doi.org/10.1093/comjnl/bxm033
Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
Frederic Dorn, Eelko Penninkx, Hans L. Bodlaender, and Fedor V. Fomin. Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions. Algorithmica, 58(3):790-810, 2010. URL: http://dx.doi.org/10.1007/s00453-009-9296-1.
http://dx.doi.org/10.1007/s00453-009-9296-1
Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms. In Proceedings of the 53rd Annual Symposium on Foundations of Computer Science (FOCS), pages 470-479. IEEE, 2012.
Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Efficient Computation of Representative Families with Applications in Parameterized and Exact Algorithms. J. ACM, 63(4):29, 2016. URL: http://dx.doi.org/10.1145/2886094.
http://dx.doi.org/10.1145/2886094
Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs. Discrete & Computational Geometry, January 2019. URL: http://dx.doi.org/10.1007/s00454-018-00054-x.
http://dx.doi.org/10.1007/s00454-018-00054-x
Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Bidimensionality and geometric graphs. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1563-1575. SIAM, 2012.
Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Excluded grid minors and efficient polynomial-time approximation schemes. J. ACM, 65(2):Art. 10, 44, 2018. URL: http://dx.doi.org/10.1145/3154833.
http://dx.doi.org/10.1145/3154833
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Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Satisfiability Threshold for Non-Uniform Random 2-SAT
Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. Its worst-case hardness lies at the core of computational complexity theory, for example in the form of NP-hardness and the (Strong) Exponential Time Hypothesis. In practice however, SAT instances can often be solved efficiently. This contradicting behavior has spawned interest in the average-case analysis of SAT and has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures.
Despite a long line of research and substantial progress, most theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a non-uniform distribution of the variables, which can result in distributions closer to industrial SAT instances.
We study satisfiability thresholds of non-uniform random 2-SAT with n variables and m clauses and with an arbitrary probability distribution (p_i)_{i in[n]} with p_1 >=slant p_2 >=slant ... >=slant p_n>0 over the n variables. We show for p_{1}^2=Theta (sum_{i=1}^n p_i^2) that the asymptotic satisfiability threshold is at {m=Theta ((1-{sum_{i=1}^n p_i^2})/(p_1 * (sum_{i=2}^n p_i^2)^{1/2}))} and that it is coarse. For p_{1}^2=o (sum_{i=1}^n p_i^2) we show that there is a sharp satisfiability threshold at m=(sum_{i=1}^n p_i^2)^{-1}. This result generalizes the seminal works by Chvatal and Reed [FOCS 1992] and by Goerdt [JCSS 1996].
random SAT
satisfiability threshold
sharpness
non-uniform distribution
2-SAT
Theory of computation~Generating random combinatorial structures
61:1-61:14
Track A: Algorithms, Complexity and Games
This paper is partially funded by the project Skalenfreie Erfüllbarkeit (project no. 416061626) of the German Research Foundation (DFG).
A full version of the paper is available at https://arxiv.org/abs/1904.02027.
Tobias
Friedrich
Tobias Friedrich
Algorithm Engineering Group, Hasso Plattner Institute, University of Potsdam, Germany
https://orcid.org/0000-0003-0076-6308
Ralf
Rothenberger
Ralf Rothenberger
Algorithm Engineering Group, Hasso Plattner Institute, University of Potsdam, Germany
https://orcid.org/0000-0002-4133-2437
10.4230/LIPIcs.ICALP.2019.61
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Tobias Friedrich and Ralf Rothenberger
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Determinant Equivalence Test over Finite Fields and over Q
The determinant polynomial Det_n(x) of degree n is the determinant of a n x n matrix of formal variables. A polynomial f is equivalent to Det_n(x) over a field F if there exists a A in GL(n^2,F) such that f = Det_n(A * x). Determinant equivalence test over F is the following algorithmic task: Given black-box access to a f in F[x], check if f is equivalent to Det_n(x) over F, and if so then output a transformation matrix A in GL(n^2,F). In (Kayal, STOC 2012), a randomized polynomial time determinant equivalence test was given over F = C. But, to our knowledge, the complexity of the problem over finite fields and over Q was not well understood.
In this work, we give a randomized poly(n,log |F|) time determinant equivalence test over finite fields F (under mild restrictions on the characteristic and size of F). Over Q, we give an efficient randomized reduction from factoring square-free integers to determinant equivalence test for quadratic forms (i.e. the n=2 case), assuming GRH. This shows that designing a polynomial-time determinant equivalence test over Q is a challenging task. Nevertheless, we show that determinant equivalence test over Q is decidable: For bounded n, there is a randomized polynomial-time determinant equivalence test over Q with access to an oracle for integer factoring. Moreover, for any n, there is a randomized polynomial-time algorithm that takes input black-box access to a f in Q[x] and if f is equivalent to Det_n over Q then it returns a A in GL(n^2,L) such that f = Det_n(A * x), where L is an extension field of Q and [L : Q] <= n.
The above algorithms over finite fields and over Q are obtained by giving a polynomial-time randomized reduction from determinant equivalence test to another problem, namely the full matrix algebra isomorphism problem. We also show a reduction in the converse direction which is efficient if n is bounded. These reductions, which hold over any F (under mild restrictions on the characteristic and size of F), establish a close connection between the complexity of the two problems. This then leads to our results via applications of known results on the full algebra isomorphism problem over finite fields (Rónyai, STOC 1987 and Rónyai, J. Symb. Comput. 1990) and over Q (Ivanyos {et al}., Journal of Algebra 2012 and Babai {et al}., Mathematics of Computation 1990).
Determinant equivalence test
full matrix algebra isomorphism
Lie algebra
Theory of computation~Algebraic complexity theory
62:1-62:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://eccc.weizmann.ac.il/report/2019/042/.
We would like to thank Youming Qiao for pointing us to the module decomposition algorithm in [Alexander L. Chistov et al., 1997]. NG would like to thank Vineet Nair for discussions on the structure of the Lie algebra of Det. We thank him for sharing his proof of Theorem 10. We also thank anonymous reviewers for their comments.
Ankit
Garg
Ankit Garg
Microsoft Research India, Bangalore, India
Nikhil
Gupta
Nikhil Gupta
Department of Computer Science and Automation, Indian Institute of Science, India
Neeraj
Kayal
Neeraj Kayal
Microsoft Research India, Bangalore, India
Chandan
Saha
Chandan Saha
Department of Computer Science and Automation, Indian Institute of Science, India
10.4230/LIPIcs.ICALP.2019.62
Manindra Agrawal and Nitin Saxena. Automorphisms of Finite Rings and Applications to Complexity of Problems. In STACS 2005, 22nd Annual Symposium on Theoretical Aspects of Computer Science, Stuttgart, Germany, February 24-26, 2005, Proceedings, pages 1-17, 2005.
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Ankit Garg, Nikhil Gupta, Neeraj Kayal, and Chandan Saha. Determinant equivalence test over finite fields and over dollarbackslashmathbfQdollar. Electronic Colloquium on Computational Complexity (ECCC), 26:42, 2019. URL: https://eccc.weizmann.ac.il/report/2019/042.
https://eccc.weizmann.ac.il/report/2019/042
Joshua A. Grochow. Matrix Lie algebra isomorphism. In IEEE Conference on Computational Complexity (CCC12), pages 203-213, 2012.
Joshua Abraham Grochow. Symmetry and equivalence relations in classical and geometric complexity theory. PhD thesis, Department of Computer Science, The University of Chicago, Chicago, Illinois, 2012.
Gábor Ivanyos, Lajos Rónyai, and Josef Schicho. Splitting full matrix algebras over algebraic number fields. Journal of Algebra, 354:211-223, 2012. URL: http://arxiv.org/abs/1106.6191.
http://arxiv.org/abs/1106.6191
Neeraj Kayal. Efficient algorithms for some special cases of the polynomial equivalence problem. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 1409-1421, 2011.
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Ankit Garg, Nikhil Gupta, Neeraj Kayal, and Chandan Saha
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Non-Clairvoyant Precedence Constrained Scheduling
We consider the online problem of scheduling jobs on identical machines, where jobs have precedence constraints. We are interested in the demanding setting where the jobs sizes are not known up-front, but are revealed only upon completion (the non-clairvoyant setting). Such precedence-constrained scheduling problems routinely arise in map-reduce and large-scale optimization. For minimizing the total weighted completion time, we give a constant-competitive algorithm. And for total weighted flow-time, we give an O(1/epsilon^2)-competitive algorithm under (1+epsilon)-speed augmentation and a natural "no-surprises" assumption on release dates of jobs (which we show is necessary in this context).
Our algorithm proceeds by assigning virtual rates to all waiting jobs, including the ones which are dependent on other uncompleted jobs. We then use these virtual rates to decide on the actual rates of minimal jobs (i.e., jobs which do not have dependencies and hence are eligible to run). Interestingly, the virtual rates are obtained by allocating time in a fair manner, using a Eisenberg-Gale-type convex program (which we can solve optimally using a primal-dual scheme). The optimality condition of this convex program allows us to show dual-fitting proofs more easily, without having to guess and hand-craft the duals. This idea of using fair virtual rates may have broader applicability in scheduling problems.
Online algorithms
Scheduling
Primal-Dual analysis
Nash welfare
Theory of computation~Online algorithms
Theory of computation~Scheduling algorithms
63:1-63:14
Track A: Algorithms, Complexity and Games
This research was supported in part by NSF awards CCF-1536002, CCF-1540541, and CCF-1617790, and the Indo-US Joint Center for Algorithms Under Uncertainty. Sahil Singla was supported in part by the Schmidt Foundation.
https://arxiv.org/pdf/1905.02133.pdf
Naveen
Garg
Naveen Garg
Computer Science and Engineering Department, Indian Institute of Technology, Delhi, India
Anupam
Gupta
Anupam Gupta
Computer Science Department, Carnegie Mellon University, USA
Amit
Kumar
Amit Kumar
Computer Science and Engineering Department, Indian Institute of Technology, Delhi, India
Sahil
Singla
Sahil Singla
Princeton University and Institute for Advanced Study, USA
10.4230/LIPIcs.ICALP.2019.63
Kunal Agrawal, Jing Li, Kefu Lu, and Benjamin Moseley. Scheduling Parallel DAG Jobs Online to Minimize Average Flow Time. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 176-189, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch14.
http://dx.doi.org/10.1137/1.9781611974331.ch14
S. Anand, Naveen Garg, and Amit Kumar. Resource augmentation for weighted flow-time explained by dual fitting. In SODA'12, pages 1228-1241. ACM, New York, 2012.
Abbas Bazzi and Ashkan Norouzi-Fard. Towards Tight Lower Bounds for Scheduling Problems. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 118-129, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_11.
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http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.16
Shi Li. Scheduling to minimize total weighted completion time via time-indexed linear programming relaxations. In 58th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2017, pages 283-294. IEEE Computer Soc., Los Alamitos, CA, 2017.
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http://dl.acm.org/citation.cfm?id=1347082.1347136
Naveen Garg, Anupam Gupta, Amit Kumar, and Sahil Singla
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity
For any relation f subseteq {0,1}^n x S and any partial Boolean function g:{0,1}^m -> {0,1,*}, we show that R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * sqrt{R_{1/3}(g)}) , where R_epsilon(*) stands for the bounded-error randomized query complexity with error at most epsilon, and f o g^n subseteq ({0,1}^m)^n x S denotes the composition of f with n instances of g.
The new composition theorem is optimal, at least, for the general case of relational problems: A relation f_0 and a partial Boolean function g_0 are constructed, such that R_{4/9}(f_0) in Theta(sqrt n), R_{1/3}(g_0)in Theta(n) and R_{1/3}(f_0 o g_0^n) in Theta(n).
The theorem is proved via introducing a new complexity measure, max-conflict complexity, denoted by bar{chi}(*). Its investigation shows that bar{chi}(g) in Omega(sqrt{R_{1/3}(g)}) for any partial Boolean function g and R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * bar{chi}(g)) for any relation f, which readily implies the composition statement. It is further shown that bar{chi}(g) is always at least as large as the sabotage complexity of g.
query complexity
lower bounds
Theory of computation~Oracles and decision trees
64:1-64:13
Track A: Algorithms, Complexity and Games
Part of this work was conducted while T.L. and S.S. were at the Nanyang Technological University and the Centre for Quantum Technologies, supported by the Singapore National Research Foundation under NRF RF Award No. NRF-NRFF2013-13. This work was additionally supported by the Singapore National Research Foundation, the Prime Minister’s Office, Singapore and the Ministry of Education, Singapore under the Research Centres of Excellence programme under research grant R 710-000-012-135. This research was supported in part by the QuantERA ERA-NET Cofund project QuantAlgo. D.G. is partially funded by the grant 19-27871X of GA ČR.
https://arxiv.org/abs/1811.10752 [Dmitry Gavinsky et al., 2018]
We thank Rahul Jain for useful discussions. We thank Srijita Kundu and Jevgēnijs Vihrovs for their helpful comments on the manuscript. We thank Yuval Filmus for suggesting to look at the min-max version of conflict complexity, which led to the development of max-conflict complexity. We thank the anonymous reviewers for their helpful comments.
Dmitry
Gavinsky
Dmitry Gavinsky
Institute of Mathematics, Czech Academy of Sciences, 115 67 Žitna 25, Praha 1, Czech Republic
Troy
Lee
Troy Lee
Centre for Quantum Software and Information, Faculty of Engineering and Information Technology, University of Technology Sydney, Australia
Miklos
Santha
Miklos Santha
CNRS, IRIF, Université de Paris, 75205 Paris, France
Centre for Quantum Technologies, National University of Singapore, Singapore 117543
MajuLab, UMI 3654, Singapore
Swagato
Sanyal
Swagato Sanyal
Indian Institute of Technology Kharagpur, India
10.4230/LIPIcs.ICALP.2019.64
Scott Aaronson. Quantum certificate complexity. Journal of Computer and System Sciences, 74(3):313-322, 2008.
Anurag Anshu, Dmitry Gavinsky, Rahul Jain, Srijita Kundu, Troy Lee, Priyanka Mukhopadhyay, Miklos Santha, and Swagato Sanyal. A Composition Theorem for Randomized Query Complexity. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017, December 11-15, 2017, Kanpur, India, pages 10:1-10:13, 2017.
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http://dx.doi.org/10.1137/100811969
Shalev Ben-David and Robin Kothari. Randomized Query Complexity of Sabotaged and Composed Functions. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 60:1-60:14, 2016.
Dmitry Gavinsky, Troy Lee, and Miklos Santha. On the randomised query complexity of composition. Technical report, arXiv, 2018. URL: http://arxiv.org/abs/1801.02226.
http://arxiv.org/abs/1801.02226
Dmitry Gavinsky, Troy Lee, Miklos Santha, and Swagato Sanyal. A composition theorem for randomized query complexity via max conflict complexity. CoRR, abs/1811.10752, 2018. URL: http://arxiv.org/abs/1811.10752.
http://arxiv.org/abs/1811.10752
Justin Gilmer, Michael Saks, and Srikanth Srinivasan. Composition limits and separating examples for some Boolean function complexity measures. Combinatorica, 36(3):265-311, 2016.
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Yaqiao Li. Conflict complexity is lower bounded by block sensitivity. Technical report, arXiv, 2018. URL: http://arxiv.org/abs/1810.08873.
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http://arxiv.org/abs/1801.03285
Avishay Tal. Properties and applications of boolean function composition. In Innovations in Theoretical Computer Science, ITCS '13, pages 441-454, 2013.
John von Neumann. Zur Theorie der Gessellschaftsspiele. Math. Ann., 100:295-320, 1928.
Dmitry Gavinsky, Troy Lee, Miklos Santha, and Swagato Sanyal
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Hairy Ball Problem is PPAD-Complete
The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of computing an approximate zero is PPAD-complete. We also give a FIXP-hardness result for the general exact computation problem.
In order to show that this problem lies in PPAD, we provide new results on multiple-source variants of End-of-Line, the canonical PPAD-complete problem. In particular, finding an approximate zero of a Hairy Ball vector field on an even-dimensional sphere reduces to a 2-source End-of-Line problem. If the domain is changed to be the torus of genus g >= 2 instead (where the Hairy Ball Theorem also holds), then the problem reduces to a 2(g-1)-source End-of-Line problem.
These multiple-source End-of-Line results are of independent interest and provide new tools for showing membership in PPAD. In particular, we use them to provide the first full proof of PPAD-completeness for the Imbalance problem defined by Beame et al. in 1998.
Computational Complexity
TFNP
PPAD
End-of-Line
Theory of computation~Problems, reductions and completeness
65:1-65:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1902.07657.
Paul W.
Goldberg
Paul W. Goldberg
Department of Computer Science, University of Oxford, United Kingdom
https://orcid.org/0000-0002-5436-7890
Alexandros
Hollender
Alexandros Hollender
Department of Computer Science, University of Oxford, United Kingdom
https://orcid.org/0000-0001-5255-9349
Supported by an EPSRC doctoral studentship (Reference 1892947).
10.4230/LIPIcs.ICALP.2019.65
James Aisenberg, Maria Luisa Bonet, and Sam Buss. 2-D Tucker is PPA complete. Technical Report TR15-163, ECCC, 2015.
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Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195-259, 2009.
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Murray Eisenberg and Robert Guy. A Proof of the Hairy Ball Theorem. The American Mathematical Monthly, 86(7):571-574, 1979.
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http://arxiv.org/abs/1804.03450
Aris Filos-Ratsikas and Paul W. Goldberg. Consensus Halving is PPA-complete. In 50th STOC, pages 51-64. ACM, 2018.
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http://arxiv.org/abs/1805.12559
Paul W. Goldberg and Christos H. Papadimitriou. Towards a unified complexity theory of total functions. Journal of Computer and System Sciences, 94:167-192, 2018.
Victor Guillemin and Alan Pollack. Differential topology. Prentice-Hall, 1974.
Alexandros Hollender and Paul W. Goldberg. The Complexity of Multi-source Variants of the End-of-Line Problem, and the Concise Mutilated Chessboard. Technical Report TR18-120, ECCC, 2018.
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Peter McGrath. An extremely short proof of the hairy ball theorem. The American Mathematical Monthly, 123(5):502-503, 2016.
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Paul W. Goldberg and Alexandros Hollender
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
AC^0[p] Lower Bounds Against MCSP via the Coin Problem
Minimum Circuit Size Problem (MCSP) asks to decide if a given truth table of an n-variate boolean function has circuit complexity less than a given parameter s. We prove that MCSP is hard for constant-depth circuits with mod p gates, for any prime p >= 2 (the circuit class AC^0[p]). Namely, we show that MCSP requires d-depth AC^0[p] circuits of size at least exp(N^{0.49/d}), where N=2^n is the size of an input truth table of an n-variate boolean function. Our circuit lower bound proof shows that MCSP can solve the coin problem: distinguish uniformly random N-bit strings from those generated using independent samples from a biased random coin which is 1 with probability 1/2+N^{-0.49}, and 0 otherwise. Solving the coin problem with such parameters is known to require exponentially large AC^0[p] circuits. Moreover, this also implies that MAJORITY is computable by a non-uniform AC^0 circuit of polynomial size that also has MCSP-oracle gates. The latter has a few other consequences for the complexity of MCSP, e.g., we get that any boolean function in NC^1 (i.e., computable by a polynomial-size formula) can also be computed by a non-uniform polynomial-size AC^0 circuit with MCSP-oracle gates.
Minimum Circuit Size Problem (MCSP)
circuit lower bounds
AC0[p]
coin problem
hybrid argument
MKTP
biased random boolean functions
Theory of computation~Circuit complexity
Theory of computation~Problems, reductions and completeness
66:1-66:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://eccc.weizmann.ac.il/report/2019/018/.
This work was partly carried out while many of the authors were visiting the Simons Institute for the Theory of Computing in association with the DIMACS/Simons Collaboration on Lower Bounds in Computational Complexity, which is conducted with support from the National Science Foundation. We also thank Chris Umans and Ronen Shaltiel for helpful discussions in the early stages of this project (during the Dagstuhl 2018 workshop on "Algebraic Methods in Complexity"). We thank Eric Allender and Shuichi Hirahara for their comments, and special thanks to Eric for pointing us to the paper of Dančik [Vladimir Dančik, 1996] and the discussion of various circuit and formula complexity measures for constant-depth circuit models. We are grateful to our anonymous reviewers for helpful comments on this paper.
Alexander
Golovnev
Alexander Golovnev
Harvard University, Cambridge, USA
Supported by a Rabin Postdoctoral Fellowship.
Rahul
Ilango
Rahul Ilango
Rutgers University, New Brunswick, USA
Russell
Impagliazzo
Russell Impagliazzo
University of California San Diego, USA
Work supported by a Simons Investigator Award from the Simons Foundation.
Valentine
Kabanets
Valentine Kabanets
Simon Fraser University, Burnaby, Canada
Supported in part by an NSERC Discovery grant.
Antonina
Kolokolova
Antonina Kolokolova
Memorial University of Newfoundland, St. John’s, Canada
Supported in part by an NSERC Discovery grant.
Avishay
Tal
Avishay Tal
Stanford University, USA
Supported by a Motwani Postdoctoral Fellowship and by NSF grant CCF-1763299. Part of this work was done while the last four authors were visiting Simons Institute for the Theory of Computing.
10.4230/LIPIcs.ICALP.2019.66
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Alexander Golovnev, Rahul Ilango, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, and Avishay Tal
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Stochastic Online Metric Matching
We study the minimum-cost metric perfect matching problem under online i.i.d arrivals. We are given a fixed metric with a server at each of the points, and then requests arrive online, each drawn independently from a known probability distribution over the points. Each request has to be matched to a free server, with cost equal to the distance. The goal is to minimize the expected total cost of the matching.
Such stochastic arrival models have been widely studied for the maximization variants of the online matching problem; however, the only known result for the minimization problem is a tight O(log n)-competitiveness for the random-order arrival model. This is in contrast with the adversarial model, where an optimal competitive ratio of O(log n) has long been conjectured and remains a tantalizing open question.
In this paper, we show that the i.i.d model admits substantially better algorithms: our main result is an O((log log log n)^2)-competitive algorithm in this model, implying a strict separation between the i.i.d model and the adversarial and random order models. Along the way we give a 9-competitive algorithm for the line and tree metrics - the first O(1)-competitive algorithm for any non-trivial arrival model for these much-studied metrics.
stochastic
online
online matching
metric matching
Theory of computation~Online algorithms
Mathematics of computing~Matchings and factors
67:1-67:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1904.09284.
Anupam
Gupta
Anupam Gupta
Carnegie Mellon University, Pittsburgh, PA, USA
http://www.cs.cmu.edu/~anupamg/
Supported in part by NSF awards CCF-1536002, CCF-1540541, and CCF-1617790, and the Indo-US Joint Center for Algorithms Under Uncertainty.
Guru
Guruganesh
Guru Guruganesh
Google Research, United States
Binghui
Peng
Binghui Peng
Tsinghua University, China
David
Wajc
David Wajc
Carnegie Mellon University, Pittsburgh, PA, USA
http://www.cs.cmu.edu/~dwajc/
Supported in part by NSF grants CCF-1618280, CCF-1814603, CCF-1527110, NSF CAREER award CCF-1750808 and a Sloan Research Fellowship.
10.4230/LIPIcs.ICALP.2019.67
Gagan Aggarwal, Gagan Goel, Chinmay Karande, and Aranyak Mehta. Online vertex-weighted bipartite matching and single-bid budgeted allocations. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1253-1264, 2011.
Antonios Antoniadis, Neal Barcelo, Michael Nugent, Kirk Pruhs, and Michele Scquizzato. A o (n)-Competitive Deterministic Algorithm for Online Matching on a Line. In Proceedings of the 12th Workshop on Approximation and Online Algorithms (WAOA), pages 11-22, 2014.
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Anupam Gupta, Guru Guruganesh, Binghui Peng, and David Wajc
Creative Commons Attribution 3.0 Unported license
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Constructions of Maximally Recoverable Local Reconstruction Codes via Function Fields
Local Reconstruction Codes (LRCs) allow for recovery from a small number of erasures in a local manner based on just a few other codeword symbols. They have emerged as the codes of choice for large scale distributed storage systems due to the very efficient repair of failed storage nodes in the typical scenario of a single or few nodes failing, while also offering fault tolerance against worst-case scenarios with more erasures. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups. In an (n,r,h,a)-LRC, the n codeword symbols are partitioned into r disjoint groups each of which include a local parity checks capable of locally correcting a erasures. The codeword symbols further obey h heavy (global) parity checks. Such a code is maximally recoverable if it can correct all patterns of a erasures per local group plus up to h additional erasures anywhere in the codeword. This property amounts to linear independence of all such subsets of columns of the parity check matrix.
MR LRCs have received much attention recently, with many explicit constructions covering different regimes of parameters. Unfortunately, all known constructions require a large field size that is exponential in h or a, and it is of interest to obtain MR LRCs of minimal possible field size. In this work, we develop an approach based on function fields to construct MR LRCs. Our method recovers, and in most parameter regimes improves, the field size of previous approaches. For instance, for the case of small r << epsilon log n and large h >=slant Omega(n^{1-epsilon}), we improve the field size from roughly n^h to n^{epsilon h}. For the case of a=1 (one local parity check), we improve the field size quadratically from r^{h(h+1)} to r^{h floor[(h+1)/2]} for some range of r. The improvements are modest, but more importantly are obtained in a unified manner via a promising new idea.
Erasure codes
Algebraic constructions
Linear algebra
Locally Repairable Codes
Explicit constructions
Mathematics of computing~Coding theory
68:1-68:14
Track A: Algorithms, Complexity and Games
A full version of this paper is posted at https://arxiv.org/abs/1808.04539.
Venkatesan
Guruswami
Venkatesan Guruswami
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
This research is supported in part by NSF grants CCF-1422045, CCF-1563742 and CCF-1814603.
Lingfei
Jin
Lingfei Jin
Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai, China
Shanghai Institute of Intelligent Electronics & Systems, Shanghai, China
Shanghai Bolckchain Engineering Research Center, Fudan University, Shanghai 200433, China
https://orcid.org/0000-0002-1523-880X
This research is supported by the National Natural Science Foundation of China under Grant 11871154.
Chaoping
Xing
Chaoping Xing
School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore
This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Strategic Capability Research Centres Funding Initiative; and the Singapore MoE Tier 1 grants RG25/16 and RG21/18.
10.4230/LIPIcs.ICALP.2019.68
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Venkatesan Guruswami, Chaoping Xing, and Chen Yuan. How long can optimal locally repairable codes be? In Proceedings of RANDOM 2018, pages 41:1-41:11, 2018.
Guangda Hu and Sergey Yekhanin. New constructions of SD and MR codes over small finite fields. In 2016 IEEE International Symposium on Information Theory (ISIT), pages 1591-1595. IEEE, 2016.
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Alessandro Neri and Anna-Lena Horlemann-Trautmann. Random Construction of Partial MDS Codes. arXiv preprint arXiv:1801.05848, 2018.
Dimitris S Papailiopoulos and Alexandros G Dimakis. Locally repairable codes. IEEE Transactions on Information Theory, 60(10):5843-5855, 2014.
Itzhak Tamo and Alexander Barg. A family of optimal locally recoverable codes. IEEE Transactions on Information Theory, 60(8):4661-4676, 2014.
Itzhak Tamo, Dimitris S Papailiopoulos, and Alexandros G Dimakis. Optimal locally repairable codes and connections to matroid theory. IEEE Transactions on Information Theory, 62(12):6661-6671, 2016.
Venkatesan Guruswami, Lingfei Jin, and Chaoping Xing
Creative Commons Attribution 3.0 Unported license
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Quantum Chebyshev’s Inequality and Applications
In this paper we provide new quantum algorithms with polynomial speed-up for a range of problems for which no such results were known, or we improve previous algorithms. First, we consider the approximation of the frequency moments F_k of order k >= 3 in the multi-pass streaming model with updates (turnstile model). We design a P-pass quantum streaming algorithm with memory M satisfying a tradeoff of P^2 M = O~(n^{1-2/k}), whereas the best classical algorithm requires P M = Theta(n^{1-2/k}). Then, we study the problem of estimating the number m of edges and the number t of triangles given query access to an n-vertex graph. We describe optimal quantum algorithms that perform O~(sqrt{n}/m^{1/4}) and O~(sqrt{n}/t^{1/6} + m^{3/4}/sqrt{t}) queries respectively. This is a quadratic speed-up compared to the classical complexity of these problems.
For this purpose we develop a new quantum paradigm that we call Quantum Chebyshev’s inequality. Namely we demonstrate that, in a certain model of quantum sampling, one can approximate with relative error the mean of any random variable with a number of quantum samples that is linear in the ratio of the square root of the variance to the mean. Classically the dependence is quadratic. Our algorithm subsumes a previous result of Montanaro [Montanaro, 2015]. This new paradigm is based on a refinement of the Amplitude Estimation algorithm of Brassard et al. [Brassard et al., 2002] and of previous quantum algorithms for the mean estimation problem. We show that this speed-up is optimal, and we identify another common model of quantum sampling where it cannot be obtained. Finally, we develop a new technique called "variable-time amplitude estimation" that reduces the dependence of our algorithm on the sample preparation time.
Quantum algorithms
approximation algorithms
sublinear-time algorithms
Monte Carlo method
streaming algorithms
subgraph counting
Theory of computation~Quantum computation theory
69:1-69:16
Track A: Algorithms, Complexity and Games
This research was supported by the French ANR project ANR-18-CE47-0010 (QUDATA) and the QuantERA ERA-NET Cofund project QuantAlgo.
A full version of the paper is available at https://arxiv.org/abs/1807.06456.
The authors want to thank the anonymous referees for their valuable comments and suggestions which helped to improve this paper.
Yassine
Hamoudi
Yassine Hamoudi
Université de Paris, IRIF, CNRS, F-75013 Paris, France
https://orcid.org/0000-0002-3762-0612
Frédéric
Magniez
Frédéric Magniez
Université de Paris, IRIF, CNRS, F-75013 Paris, France
https://orcid.org/0000-0003-2384-9026
10.4230/LIPIcs.ICALP.2019.69
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Yassine Hamoudi and Frédéric Magniez
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Retracting Graphs to Cycles
We initiate the algorithmic study of retracting a graph into a cycle in the graph, which seeks a mapping of the graph vertices to the cycle vertices so as to minimize the maximum stretch of any edge, subject to the constraint that the restriction of the mapping to the cycle is the identity map. This problem has its roots in the rich theory of retraction of topological spaces, and has strong ties to well-studied metric embedding problems such as minimum bandwidth and 0-extension. Our first result is an O(min{k, sqrt{n}})-approximation for retracting any graph on n nodes to a cycle with k nodes. We also show a surprising connection to Sperner’s Lemma that rules out the possibility of improving this result using certain natural convex relaxations of the problem. Nevertheless, if the problem is restricted to planar graphs, we show that we can overcome these integrality gaps by giving an optimal combinatorial algorithm, which is the technical centerpiece of the paper. Building on our planar graph algorithm, we also obtain a constant-factor approximation algorithm for retraction of points in the Euclidean plane to a uniform cycle.
Graph algorithms
Graph embedding
Planar graphs
Approximation algorithms
Theory of computation~Design and analysis of algorithms
70:1-70:15
Track A: Algorithms, Complexity and Games
All the authors were partially supported by NSF grant CCF 1535972. Additional funding information appears below.
A full version of the paper is available at https://arxiv.org/abs/1904.11946.
We would like to thank Seffi Naor for helpful discussions on the problems considered in this paper.
Samuel
Haney
Samuel Haney
Duke University, Durham, NC, USA
Mehraneh
Liaee
Mehraneh Liaee
Northeastern University, Boston, MA, USA
Bruce M.
Maggs
Bruce M. Maggs
Duke University, Durham, NC, USA
Akamai Technologies, Cambridge, MA, USA
Debmalya
Panigrahi
Debmalya Panigrahi
Duke University, Durham, NC, USA
NSF grant CCF 1527084, an NSF CAREER Award CCF 1750140, and the Joint Indo-US Networked Center for Algorithms under Uncertainty.
Rajmohan
Rajaraman
Rajmohan Rajaraman
Northeastern University, Boston, MA, USA
Ravi
Sundaram
Ravi Sundaram
Northeastern University, Boston, MA, USA
NSF grant CNS 1718286.
10.4230/LIPIcs.ICALP.2019.70
Mark Anthony Armstrong. Basic Topology. Springer Science &Business Media, 2013.
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Samuel Haney, Mehraneh Liaee, Bruce M. Maggs, Debmalya Panigrahi, Rajmohan Rajaraman, and Ravi Sundaram
Creative Commons Attribution 3.0 Unported license
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On Adaptive Algorithms for Maximum Matching
In the fundamental Maximum Matching problem the task is to find a maximum cardinality set of pairwise disjoint edges in a given undirected graph. The fastest algorithm for this problem, due to Micali and Vazirani, runs in time O(sqrt{n}m) and stands unbeaten since 1980. It is complemented by faster, often linear-time, algorithms for various special graph classes. Moreover, there are fast parameterized algorithms, e.g., time O(km log n) relative to tree-width k, which outperform O(sqrt{n}m) when the parameter is sufficiently small.
We show that the Micali-Vazirani algorithm, and in fact any algorithm following the phase framework of Hopcroft and Karp, is adaptive to beneficial input structure. We exhibit several graph classes for which such algorithms run in linear time O(n+m). More strongly, we show that they run in time O(sqrt{k}m) for graphs that are k vertex deletions away from any of several such classes, without explicitly computing an optimal or approximate deletion set; before, most such bounds were at least Omega(km). Thus, any phase-based matching algorithm with linear-time phases obliviously interpolates between linear time for k=O(1) and the worst case of O(sqrt{n}m) when k=Theta(n). We complement our findings by proving that the phase framework by itself still allows Omega(sqrt{n}) phases, and hence time Omega(sqrt{n}m), even on paths, cographs, and bipartite chain graphs.
Matchings
Adaptive Analysis
Parameterized Complexity
Mathematics of computing~Matchings and factors
Theory of computation~Graph algorithms analysis
Theory of computation~Parameterized complexity and exact algorithms
71:1-71:16
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1904.11244.
Falko
Hegerfeld
Falko Hegerfeld
Humboldt-Universität zu Berlin, Germany
Stefan
Kratsch
Stefan Kratsch
Humboldt-Universität zu Berlin, Germany
10.4230/LIPIcs.ICALP.2019.71
Amir Abboud, Virginia Vassilevska Williams, and Joshua Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete Algorithms, pages 377-391. SIAM, 2016.
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http://dx.doi.org/10.1145/2746539.2746594
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Jérémy Barbay and Gonzalo Navarro. On compressing permutations and adaptive sorting. Theor. Comput. Sci., 513:109-123, 2013. URL: http://dx.doi.org/10.1016/j.tcs.2013.10.019.
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Jean-Luc Fouquet, Igor Parfenoff, and Henri Thuillier. An 𝒪(n) time algorithm for maximum matching in P₄-tidy graphs. Information Processing Letters, 62(6):281-287, 1997.
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Falko Hegerfeld and Stefan Kratsch
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits
There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets S_1,...,S_k subset [n] is balancing if for every subset X subset {1,2,...,n} of size n/2, there is an i in [k] so that |S_i cap X| = |S_i|/2. We extend and simplify the framework developed by Hegedűs for proving lower bounds on the size of balancing set families. We prove that if n=2p for a prime p, then k >= p. For arbitrary values of n, we show that k >= n/2 - o(n).
We then exploit the connection between balancing families and depth-2 threshold circuits. This connection helps resolve a question raised by Kulikov and Podolskii on the fan-in of depth-2 majority circuits computing the majority function on n bits. We show that any depth-2 threshold circuit that computes the majority on n bits has at least one gate with fan-in at least n/2 - o(n). We also prove a sharp lower bound on the fan-in of depth-2 threshold circuits computing a specific weighted threshold function.
Balancing sets
depth-2 threshold circuits
polynomials
majority
weighted thresholds
Mathematics of computing~Combinatorics
Theory of computation~Circuit complexity
72:1-72:14
Track A: Algorithms, Complexity and Games
This work was done while the authors were visiting the Simons Institute for the Theory of Computing.
A full version of the paper is available at https://eccc.weizmann.ac.il/report/2019/026/.
Pavel
Hrubeš
Pavel Hrubeš
Institute of Mathematics of ASCR, Prague
Supported by ERC grant FEALORA 339691 and the GACR grant 19-27871X.
Sivaramakrishnan
Natarajan Ramamoorthy
Sivaramakrishnan Natarajan Ramamoorthy
Paul G. Allen School of Computer Science & Engineering, University of Washington, USA
Supported by the National Science Foundation under agreement CCF- 1420268.
Anup
Rao
Anup Rao
Paul G. Allen School of Computer Science & Engineering, University of Washington, USA
Supported by the National Science Foundation under agreement CCF- 1420268.
Amir
Yehudayoff
Amir Yehudayoff
Department of Mathematics, Technion-IIT, Haifa, Israel
Partially supported by ISF grant 1162/15.
10.4230/LIPIcs.ICALP.2019.72
M. Ajtai, J. Komlós, and E. Szemerédi. Sorting in c logn parallel steps. Combinatorica, 3(1):1-19, March 1983. URL: http://dx.doi.org/10.1007/BF02579338.
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http://arxiv.org/abs/1708.02037
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http://drops.dagstuhl.de/opus/volltexte/2018/9663
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Gábor Hegedűs. Balancing sets of vectors. Studia Scientiarum Mathematicarum Hungarica, 47(3):333-349, 2009.
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http://arxiv.org/abs/1711.10176
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http://arxiv.org/abs/1802.09121
Pavel Hrubeš, Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao, and Amir Yehudayoff
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Scalable and Jointly Differentially Private Packing
We introduce an (epsilon, delta)-jointly differentially private algorithm for packing problems. Our algorithm not only achieves the optimal trade-off between the privacy parameter epsilon and the minimum supply requirement (up to logarithmic factors), but is also scalable in the sense that the running time is linear in the number of agents n. Previous algorithms either run in cubic time in n, or require a minimum supply per resource that is sqrt{n} times larger than the best possible.
Joint differential privacy
packing
scalable algorithms
Theory of computation~Design and analysis of algorithms
Theory of computation~Packing and covering problems
73:1-73:12
Track A: Algorithms, Complexity and Games
A full version of the paper is available at [Zhiyi Huang and Xue Zhu, 2019], https://arxiv.org/abs/1905.00767.
Zhiyi
Huang
Zhiyi Huang
The University of Hong Kong
This work is supported in by a RGC grant HKU17203717E.
Xue
Zhu
Xue Zhu
The University of Hong Kong
10.4230/LIPIcs.ICALP.2019.73
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Rachel Cummings, Katrina Ligett, Jaikumar Radhakrishnan, Aaron Roth, and Zhiwei Steven Wu. Coordination complexity: Small information coordinating large populations. In ITCS, pages 281-290. ACM, 2016.
Rachel Cummings, David M Pennock, and Jennifer Wortman Vaughan. The possibilities and limitations of private prediction markets. In EC, pages 143-160. ACM, 2016.
John C Duchi, Michael I Jordan, and Martin J Wainwright. Local privacy and statistical minimax rates. In FOCS, pages 429-438. IEEE, 2013.
Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In TCC, pages 265-284. Springer, 2006.
Cynthia Dwork, Moni Naor, Toniann Pitassi, and Guy N Rothblum. Differential privacy under continual observation. In STOC, pages 715-724. ACM, 2010.
Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. Foundations and Trendsregistered in Theoretical Computer Science, 9(3-4):211-407, 2014.
Arpita Ghosh, Katrina Ligett, Aaron Roth, and Grant Schoenebeck. Buying private data without verification. In EC, pages 931-948. ACM, 2014.
Justin Hsu, Zhiyi Huang, Aaron Roth, Tim Roughgarden, and Zhiwei Steven Wu. Private matchings and allocations. SIAM Journal on Computing, 45(6):1953-1984, 2016.
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Justin Hsu, Aaron Roth, Tim Roughgarden, and Jonathan Ullman. Privately solving linear programs. In ICALP, pages 612-624. Springer, 2014.
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http://arxiv.org/abs/1905.00767
Michael Kearns, Mallesh Pai, Aaron Roth, and Jonathan Ullman. Mechanism design in large games: incentives and privacy. In ITCS, pages 403-410. ACM, 2014.
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Shang-Hua Teng. Scalable algorithms for data and network analysis. Foundations and Trendsregistered in Theoretical Computer Science, 12(1-2):1-274, 2016.
Zhiyi Huang and Xue Zhu
Creative Commons Attribution 3.0 Unported license
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Local Search Breaks 1.75 for Graph Balancing
Graph Balancing is the problem of orienting the edges of a weighted multigraph so as to minimize the maximum weighted in-degree. Since the introduction of the problem the best algorithm known achieves an approximation ratio of 1.75 and it is based on rounding a linear program with this exact integrality gap. It is also known that there is no (1.5 - epsilon)-approximation algorithm, unless P=NP. Can we do better than 1.75?
We prove that a different LP formulation, the configuration LP, has a strictly smaller integrality gap. Graph Balancing was the last one in a group of related problems from literature, for which it was open whether the configuration LP is stronger than previous, simple LP relaxations. We base our proof on a local search approach that has been applied successfully to the more general Restricted Assignment problem, which in turn is a prominent special case of makespan minimization on unrelated machines. With a number of technical novelties we are able to obtain a bound of 1.749 for the case of Graph Balancing. It is not clear whether the local search algorithm we present terminates in polynomial time, which means that the bound is non-constructive. However, it is a strong evidence that a better approximation algorithm is possible using the configuration LP and it allows the optimum to be estimated within a factor better than 1.75.
A particularly interesting aspect of our techniques is the way we handle small edges in the local search. We manage to exploit the configuration constraints enforced on small edges in the LP. This may be of interest to other problems such as Restricted Assignment as well.
graph
approximation algorithm
scheduling
integrality gap
local search
Theory of computation~Approximation algorithms analysis
74:1-74:14
Track A: Algorithms, Complexity and Games
Research was supported by German Research Foundation (DFG) project JA 612/15-2.
A full version can be found at https://arxiv.org/abs/1811.00955.
Klaus
Jansen
Klaus Jansen
Department of Computer Science, Christian-Albrechts-Universität, Kiel, Germany
Lars
Rohwedder
Lars Rohwedder
Department of Computer Science, Christian-Albrechts-Universität, Kiel, Germany
10.4230/LIPIcs.ICALP.2019.74
Chidambaram Annamalai. Lazy Local Search Meets Machine Scheduling. CoRR, abs/1611.07371, 2016. URL: http://arxiv.org/abs/1611.07371.
http://arxiv.org/abs/1611.07371
Chidambaram Annamalai, Christos Kalaitzis, and Ola Svensson. Combinatorial Algorithm for Restricted Max-Min Fair Allocation. ACM Transactions on Algorithms, 13(3):37:1-37:28, 2017. Previously appeared in SODA'15. URL: http://dx.doi.org/10.1145/3070694.
http://dx.doi.org/10.1145/3070694
Arash Asadpour, Uriel Feige, and Amin Saberi. Santa Claus meets hypergraph matchings. ACM Transactions on Algorithms, 8(3):24:1-24:9, 2012. See Asadpour’s homepage for the bound of 4 instead of 5 as in the paper; Previously appeared in APPROX'08. URL: http://dx.doi.org/10.1145/2229163.2229168.
http://dx.doi.org/10.1145/2229163.2229168
Yuichi Asahiro, Jesper Jansson, Eiji Miyano, Hirotaka Ono, and Kouhei Zenmyo. Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree. Journal of Combinatorial Optimization, 22(1):78-96, 2011. URL: http://dx.doi.org/10.1007/s10878-009-9276-z.
http://dx.doi.org/10.1007/s10878-009-9276-z
Nikhil Bansal and Maxim Sviridenko. The Santa Claus problem. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 31-40, 2006. Previously appeared in STOC'06. URL: http://dx.doi.org/10.1145/1132516.1132522.
http://dx.doi.org/10.1145/1132516.1132522
Deeparnab Chakrabarty, Julia Chuzhoy, and Sanjeev Khanna. On Allocating Goods to Maximize Fairness. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA, pages 107-116, 2009. URL: http://dx.doi.org/10.1109/FOCS.2009.51.
http://dx.doi.org/10.1109/FOCS.2009.51
Deeparnab Chakrabarty and Kirankumar Shiragur. Graph Balancing with Two Edge Types. CoRR, abs/1604.06918, 2016. URL: http://arxiv.org/abs/1604.06918.
http://arxiv.org/abs/1604.06918
Siu-Wing Cheng and Yuchen Mao. Integrality Gap of the Configuration LP for the Restricted Max-Min Fair Allocation. CoRR, abs/1807.04152, 2018. URL: http://arxiv.org/abs/1807.04152.
http://arxiv.org/abs/1807.04152
Tomáš Ebenlendr, Marek Krčál, and Jiří Sgall. Graph Balancing: A Special Case of Scheduling Unrelated Parallel Machines. Algorithmica, 68(1):62-80, 2014. Previously appeared in SODA'08. URL: http://dx.doi.org/10.1007/s00453-012-9668-9.
http://dx.doi.org/10.1007/s00453-012-9668-9
Chien-Chung Huang and Sebastian Ott. A Combinatorial Approximation Algorithm for Graph Balancing with Light Hyper Edges. In 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, pages 49:1-49:15, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2016.49.
http://dx.doi.org/10.4230/LIPIcs.ESA.2016.49
Klaus Jansen, Kati Land, and Marten Maack. Estimating The Makespan of The Two-Valued Restricted Assignment Problem. In 15th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2016, June 22-24, 2016, Reykjavik, Iceland, pages 24:1-24:13, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.SWAT.2016.24.
http://dx.doi.org/10.4230/LIPIcs.SWAT.2016.24
Klaus Jansen and Lars Rohwedder. A Quasi-Polynomial Approximation for the Restricted Assignment Problem. In Integer Programming and Combinatorial Optimization - 19th International Conference, IPCO 2017, Waterloo, ON, Canada, June 26-28, 2017, Proceedings, pages 305-316, 2017. URL: http://dx.doi.org/10.1007/978-3-319-59250-3_25.
http://dx.doi.org/10.1007/978-3-319-59250-3_25
Klaus Jansen and Lars Rohwedder. On the Configuration-LP of the Restricted Assignment Problem. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2670-2678, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.176.
http://dx.doi.org/10.1137/1.9781611974782.176
Klaus Jansen and Lars Rohwedder. A note on the integrality gap of the configuration LP for restricted Santa Claus. CoRR, abs/1807.03626, 2018. URL: http://arxiv.org/abs/1807.03626.
http://arxiv.org/abs/1807.03626
Klaus Jansen and Lars Rohwedder. Compact LP Relaxations for Allocation Problems. In 1st Symposium on Simplicity in Algorithms, SOSA 2018, January 7-10, 2018, New Orleans, LA, USA, pages 11:1-11:19, 2018. URL: http://dx.doi.org/10.4230/OASIcs.SOSA.2018.11.
http://dx.doi.org/10.4230/OASIcs.SOSA.2018.11
Jan Karel Lenstra, David B. Shmoys, and Éva Tardos. Approximation Algorithms for Scheduling Unrelated Parallel Machines. Math. Program., 46:259-271, 1990. URL: http://dx.doi.org/10.1007/BF01585745.
http://dx.doi.org/10.1007/BF01585745
Daniel R. Page and Roberto Solis-Oba. A 3/2-Approximation Algorithm for the Graph Balancing Problem with Two Weights. Algorithms, 9(2):38, 2016. URL: http://dx.doi.org/10.3390/a9020038.
http://dx.doi.org/10.3390/a9020038
Lukás Polácek and Ola Svensson. Quasi-Polynomial Local Search for Restricted Max-Min Fair Allocation. ACM Transactions on Algorithms, 12(2):13:1-13:13, 2016. Previously appeared in ICALP'12. URL: http://dx.doi.org/10.1145/2818695.
http://dx.doi.org/10.1145/2818695
Ola Svensson. Santa Claus schedules jobs on unrelated machines. SIAM Journal on Computing, 41(5):1318-1341, 2012. Previously appeared in STOC'11. URL: http://dx.doi.org/10.1137/110851201.
http://dx.doi.org/10.1137/110851201
José Verschae and Andreas Wiese. On the configuration-LP for scheduling on unrelated machines. Journal of Scheduling, 17(4):371-383, 2014. Previously appeared in ESA'11. URL: http://dx.doi.org/10.1007/s10951-013-0359-4.
http://dx.doi.org/10.1007/s10951-013-0359-4
Chao Wang and René Sitters. On some special cases of the restricted assignment problem. Information Processing Letters, 116(11):723-728, 2016. URL: http://dx.doi.org/10.1016/j.ipl.2016.06.007.
http://dx.doi.org/10.1016/j.ipl.2016.06.007
Klaus Jansen and Lars Rohwedder
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Near-Linear Time Algorithm for n-fold ILPs via Color Coding
We study an important case of ILPs max {c^Tx | Ax = b, l <= x <= u, x in Z^{n t}} with n * t variables and lower and upper bounds l, u in Z^{nt}. In n-fold ILPs non-zero entries only appear in the first r rows of the matrix A and in small blocks of size s x t along the diagonal underneath. Despite this restriction many optimization problems can be expressed in this form. It is known that n-fold ILPs can be solved in FPT time regarding the parameters s, r, and Delta, where Delta is the greatest absolute value of an entry in A. The state-of-the-art technique is a local search algorithm that subsequently moves in an improving direction. Both, the number of iterations and the search for such an improving direction take time Omega(n), leading to a quadratic running time in n. We introduce a technique based on Color Coding, which allows us to compute these improving directions in logarithmic time after a single initialization step. This leads to the first algorithm for n-fold ILPs with a running time that is near-linear in the number nt of variables, namely (rs Delta)^{O(r^2s + s^2)} L^2 * nt log^{O(1)}(nt), where L is the encoding length of the largest integer in the input. In contrast to the algorithms in recent literature, we do not need to solve the LP relaxation in order to handle unbounded variables. Instead, we give a structural lemma to introduce appropriate bounds. If, on the other hand, we are given such an LP solution, the running time can be decreased by a factor of L.
Near-Linear Time Algorithm
n-fold ILP
Color Coding
Mathematics of computing~Integer programming
75:1-75:13
Track A: Algorithms, Complexity and Games
This work was supported by DFG project JA 612/20-1.
A full version can be found at https://arxiv.org/abs/1811.00950.
Klaus
Jansen
Klaus Jansen
Department of Computer Science, Kiel University, Kiel, Germany
Alexandra
Lassota
Alexandra Lassota
Department of Computer Science, Kiel University, Kiel, Germany
Lars
Rohwedder
Lars Rohwedder
Department of Computer Science, Kiel University, Kiel, Germany
10.4230/LIPIcs.ICALP.2019.75
Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. Journal of the ACM, 42(4):844-856, 1995. Previously appeared in STOC 1994.
Katerina Altmanová, Dušan Knop, and Martin Koutecký. Evaluating and Tuning n-fold Integer Programming. In 17th International Symposium on Experimental Algorithms, volume 103 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1-10:14, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Alexandr Andoni and Krzysztof Onak. Approximating edit distance in near-linear time. SIAM Journal on Computing, 41(6):1635-1648, 2012. Previously appeared in STOC 2009.
William Cook, Jean Fonlupt, and Alexander Schrijver. An integer analogue of Carathéodory’s theorem. Journal of Combinatorial Theory, Series B, 40(1):63-70, 1986.
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Friedrich Eisenbrand and Robert Weismantel. Proximity results and faster algorithms for integer programming using the steinitz lemma. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 808-816. Society for Industrial and Applied Mathematics, 2018.
Michael L. Fredman, János Komlós, and Endre Szemerédi. Storing a Sparse Table with O(1) Worst Case Access Time. Journal of the ACM, 31(3):538-544, 1984. Previously appeared in FOCS 1982.
Raymond Hemmecke, Shmuel Onn, and Lyubov Romanchuk. N-fold integer programming in cubic time. Mathematical Programming, pages 1-17, 2013.
Raymond Hemmecke, Shmuel Onn, and Robert Weismantel. N-fold integer programming and nonlinear multi-transshipment. Optimization Letters, 5(1):13-25, 2011.
Klaus Jansen, Kim-Manuel Klein, Marten Maack, and Malin Rau. Empowering the Configuration-IP - New PTAS Results for Scheduling with Setups Times. In ITCS, volume 124 of LIPIcs, pages 44:1-44:19. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019.
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 Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, pages 217-226. SIAM, 2014.
Dušan Knop and Martin Kouteckỳ. Scheduling meets n-fold integer programming. Journal of Scheduling, pages 1-11, 2017.
Dušan Knop, Martin Koutecký, and Matthias Mnich. Combinatorial n-fold Integer Programming and Applications. In 25th Annual European Symposium on Algorithms, ESA 2017, September 4-6, 2017, Vienna, Austria, pages 54:1-54:14, 2017.
Martin Koutecký, Asaf Levin, and Shmuel Onn. A Parameterized Strongly Polynomial Algorithm for Block Structured Integer Programs. In 45th International Colloquium on Automata, Languages, and Programming, volume 107 of Leibniz International Proceedings in Informatics (LIPIcs), pages 85:1-85:14, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Moni Naor, Leonard J. Schulman, and Aravind Srinivasan. Splitters and Near-Optimal Derandomization. In 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, USA, 23-25 October 1995, pages 182-191, 1995.
Moni Naor, Leonard J. Schulman, and Aravind Srinivasan. Splitters and near-optimal derandomization. In Proceedings of IEEE 36th Annual Foundations of Computer Science, pages 182-191. IEEE, 1995.
Shmuel Onn and Pauline Sarrabezolles. Huge Unimodular n-Fold Programs. SIAM J. Discrete Math., 29(4):2277-2283, 2015.
Éva Tardos. A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs. Operations Research, 34(2):250-256, 1986. URL: http://dx.doi.org/10.1287/opre.34.2.250.
http://dx.doi.org/10.1287/opre.34.2.250
Klaus Jansen, Alexandra Lassota, and Lars Rohwedder
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
An Improved FPTAS for 0-1 Knapsack
The 0-1 knapsack problem is an important NP-hard problem that admits fully polynomial-time approximation schemes (FPTASs). Previously the fastest FPTAS by Chan (2018) with approximation factor 1+epsilon runs in O~(n + (1/epsilon)^{12/5}) time, where O~ hides polylogarithmic factors. In this paper we present an improved algorithm in O~(n+(1/epsilon)^{9/4}) time, with only a (1/epsilon)^{1/4} gap from the quadratic conditional lower bound based on (min,+)-convolution. Our improvement comes from a multi-level extension of Chan’s number-theoretic construction, and a greedy lemma that reduces unnecessary computation spent on cheap items.
approximation algorithms
knapsack
subset sum
Theory of computation~Algorithm design techniques
76:1-76:14
Track A: Algorithms, Complexity and Games
Part of this research was done while visiting Harvard University. I would like to thank Professor Jelani Nelson for introducing this problem to me, advising this project, and giving many helpful comments on my writeup.
Ce
Jin
Ce Jin
Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
10.4230/LIPIcs.ICALP.2019.76
Alok Aggarwal, Maria M. Klawe, Shlomo Moran, Peter Shor, and Robert Wilber. Geometric applications of a matrix-searching algorithm. Algorithmica, 2(1):195-208, November 1987. URL: http://dx.doi.org/10.1007/BF01840359.
http://dx.doi.org/10.1007/BF01840359
David Bremner, Timothy M. Chan, Erik D. Demaine, Jeff Erickson, Ferran Hurtado, John Iacono, Stefan Langerman, Mihai Pǎtraşcu, and Perouz Taslakian. Necklaces, Convolutions, and X+Y. Algorithmica, 69(2):294-314, June 2014. URL: http://dx.doi.org/10.1007/s00453-012-9734-3.
http://dx.doi.org/10.1007/s00453-012-9734-3
Timothy M. Chan. Approximation Schemes for 0-1 Knapsack. In Proceedings of the 1st Symposium on Simplicity in Algorithms (SOSA), pages 5:1-5:12, 2018. URL: http://dx.doi.org/10.4230/OASIcs.SOSA.2018.5.
http://dx.doi.org/10.4230/OASIcs.SOSA.2018.5
Timothy M. Chan and Ryan Williams. Deterministic APSP, Orthogonal Vectors, and More: Quickly Derandomizing Razborov-Smolensky. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1246-1255, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch87.
http://dx.doi.org/10.1137/1.9781611974331.ch87
Marek Cygan, Marcin Mucha, Karol Węgrzycki, and Michał Włodarczyk. On Problems Equivalent to (Min,+)-Convolution. ACM Trans. Algorithms, 15(1):14:1-14:25, January 2019. URL: http://dx.doi.org/10.1145/3293465.
http://dx.doi.org/10.1145/3293465
Oscar H. Ibarra and Chul E. Kim. Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems. Journal of the ACM (JACM), 22(4):463-468, October 1975. URL: http://dx.doi.org/10.1145/321906.321909.
http://dx.doi.org/10.1145/321906.321909
Klaus Jansen and Stefan E.J. Kraft. A faster FPTAS for the Unbounded Knapsack Problem. European Journal of Combinatorics, 68:148-174, 2018. URL: http://dx.doi.org/10.1016/j.ejc.2017.07.016.
http://dx.doi.org/10.1016/j.ejc.2017.07.016
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http://dx.doi.org/10.1007/978-1-4684-2001-2_9
Hans Kellerer, Renata Mansini, Ulrich Pferschy, and Maria Grazia Speranza. An efficient fully polynomial approximation scheme for the Subset-Sum Problem. Journal of Computer and System Sciences, 66(2):349-370, 2003. URL: http://dx.doi.org/10.1016/S0022-0000(03)00006-0.
http://dx.doi.org/10.1016/S0022-0000(03)00006-0
Hans Kellerer and Ulrich Pferschy. A New Fully Polynomial Time Approximation Scheme for the Knapsack Problem. Journal of Combinatorial Optimization, 3(1):59-71, July 1999. URL: http://dx.doi.org/10.1023/A:1009813105532.
http://dx.doi.org/10.1023/A:1009813105532
Hans Kellerer and Ulrich Pferschy. Improved Dynamic Programming in Connection with an FPTAS for the Knapsack Problem. Journal of Combinatorial Optimization, 8(1):5-11, March 2004. URL: http://dx.doi.org/10.1023/B:JOCO.0000021934.29833.6b.
http://dx.doi.org/10.1023/B:JOCO.0000021934.29833.6b
Marvin Künnemann, Ramamohan Paturi, and Stefan Schneider. On the Fine-Grained Complexity of One-Dimensional Dynamic Programming. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP), pages 21:1-21:15, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.21.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.21
Eugene L. Lawler. Fast Approximation Algorithms for Knapsack Problems. Mathematics of Operations Research, 4(4):339-356, 1979. URL: http://dx.doi.org/10.1287/moor.4.4.339.
http://dx.doi.org/10.1287/moor.4.4.339
Marcin Mucha, Karol Węgrzycki, and Michał Włodarczyk. Subquadratic Approximation Scheme for Partition. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 70-88, 2019. URL: http://dx.doi.org/10.1137/1.9781611975482.5.
http://dx.doi.org/10.1137/1.9781611975482.5
Donguk Rhee. Faster fully polynomial approximation schemes for knapsack problems. Master’s thesis, Massachusetts Institute of Technology, 2015. URL: http://hdl.handle.net/1721.1/98564.
http://hdl.handle.net/1721.1/98564
Ryan Williams. Faster All-pairs Shortest Paths via Circuit Complexity. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC), pages 664-673, 2014. URL: http://dx.doi.org/10.1145/2591796.2591811.
http://dx.doi.org/10.1145/2591796.2591811
Ce Jin
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Testing the Complexity of a Valued CSP Language
A Valued Constraint Satisfaction Problem (VCSP) provides a common framework that can express a wide range of discrete optimization problems. A VCSP instance is given by a finite set of variables, a finite domain of labels, and an objective function to be minimized. This function is represented as a sum of terms where each term depends on a subset of the variables. To obtain different classes of optimization problems, one can restrict all terms to come from a fixed set Gamma of cost functions, called a language.
Recent breakthrough results have established a complete complexity classification of such classes with respect to language Gamma: if all cost functions in Gamma satisfy a certain algebraic condition then all Gamma-instances can be solved in polynomial time, otherwise the problem is NP-hard. Unfortunately, testing this condition for a given language Gamma is known to be NP-hard. We thus study exponential algorithms for this meta-problem. We show that the tractability condition of a finite-valued language Gamma can be tested in O(sqrt[3]{3}^{|D|}* poly(size(Gamma))) time, where D is the domain of Gamma and poly(*) is some fixed polynomial. We also obtain a matching lower bound under the Strong Exponential Time Hypothesis (SETH). More precisely, we prove that for any constant delta<1 there is no O(sqrt[3]{3}^{delta|D|}) algorithm, assuming that SETH holds.
Valued Constraint Satisfaction Problems
Exponential time algorithms
Exponential Time Hypothesis
Theory of computation~Parameterized complexity and exact algorithms
77:1-77:12
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1803.02289.
Vladimir
Kolmogorov
Vladimir Kolmogorov
Institute of Science and Technology Austria, Klosterneuburg, Austria
supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 616160.
10.4230/LIPIcs.ICALP.2019.77
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Andrei Bulatov. A dichotomy theorem for constraint satisfaction problems on a 3-element set. Journal of the ACM, 53(1):66-120, 2006. URL: http://dx.doi.org/10.1145/1120582.1120584.
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Tomás Feder and Moshe Y. Vardi. The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory. SIAM Journal on Computing, 28(1):57-104, 1998. URL: http://dx.doi.org/10.1137/S0097539794266766.
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Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. Journal of Computer and System Sciences, 62(2):367-375, 2001.
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? Journal of Computer and System Sciences, 63(4):512-530, 2001.
P. Jonsson, V. Lagerkvist, G. Nordh, , and B. Zanuttini. Strong partial clones and the time complexity of SAT problems. Journal of Computer and System Sciences, 84:52-78, 2017.
Peter Jonsson, Victor Lagerkvist, and Biman Roy. Time Complexity of Constraint Satisfaction via Universal Algebra. In Proceedings of the 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS), 2017.
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Marcin Kozik and Joanna Ochremiak. Algebraic Properties of Valued Constraint Satisfaction Problem. arXiv, 2015. Extended abstract published in ICALP'15. URL: http://arxiv.org/abs/1403.0476.
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Vladimir Kolmogorov
Creative Commons Attribution 3.0 Unported license
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Towards Optimal Depth Reductions for Syntactically Multilinear Circuits
We show that any n-variate polynomial computable by a syntactically multilinear circuit of size poly(n) can be computed by a depth-4 syntactically multilinear (Sigma Pi Sigma Pi) circuit of size at most exp ({O (sqrt{n log n})}). For degree d = omega(n/log n), this improves upon the upper bound of exp ({O(sqrt{d}log n)}) obtained by Tavenas [Sébastien Tavenas, 2015] for general circuits, and is known to be asymptotically optimal in the exponent when d < n^{epsilon} for a small enough constant epsilon. Our upper bound matches the lower bound of exp ({Omega (sqrt{n log n})}) proved by Raz and Yehudayoff [Ran Raz and Amir Yehudayoff, 2009], and thus cannot be improved further in the exponent. Our results hold over all fields and also generalize to circuits of small individual degree.
More generally, we show that an n-variate polynomial computable by a syntactically multilinear circuit of size poly(n) can be computed by a syntactically multilinear circuit of product-depth Delta of size at most exp inparen{O inparen{Delta * (n/log n)^{1/Delta} * log n}}. It follows from the lower bounds of Raz and Yehudayoff [Ran Raz and Amir Yehudayoff, 2009] that in general, for constant Delta, the exponent in this upper bound is tight and cannot be improved to o inparen{inparen{n/log n}^{1/Delta}* log n}.
arithmetic circuits
multilinear circuits
depth reduction
lower bounds
Theory of computation~Circuit complexity
78:1-78:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1902.07063.
We are extremely thankful to Ben Rossman, who pointed us towards this question, and for many stimulating discussions at various stages of this work. We also thank Shubhangi Saraf, Amir Shpilka and Ben Lee Volk for many helpful conversations. Mrinal also thanks Prahladh Harsha for accommodating him in his apartment for a part of the visit to TIFR, where a part of this paper was written.
Mrinal
Kumar
Mrinal Kumar
University of Toronto, Canada
https://mrinalkr.bitbucket.io/
A part of this work was done during the postdoctoral stay at Harvard, during the lower bounds semester at Simons Institute for the Theory of Computing, Berkeley and while visiting TIFR, Mumbai.
Rafael
Oliveira
Rafael Oliveira
University of Toronto, Canada
http://www.cs.utoronto.ca/~rafael/
Part of this work was done while visiting the Simons Institute for the Theory of Computing.
Ramprasad
Saptharishi
Ramprasad Saptharishi
Tata Institute of Fundamental Research
https://www.tcs.tifr.res.in/~ramprasad/
Research supported by Ramanujan Fellowship of DST.
10.4230/LIPIcs.ICALP.2019.78
Manindra Agrawal and V. Vinay. Arithmetic Circuits: A Chasm at Depth Four. In Proceedings of the \ifnumcomp2008<1966\nth\intcalcSub20081959 Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 2008)\ifnumcomp2008<1975\nth\intcalcSub20081959 Annual Symposium on Switching and Automata Theory (SWAT 2008)\nth\intcalcSub20081959 Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008), pages 67-75, 2008. URL: http://dx.doi.org/10.1109/FOCS.2008.32.
http://dx.doi.org/10.1109/FOCS.2008.32
Eric Allender, Jia Jiao, Meena Mahajan, and V. Vinay. Non-Commutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds. Theoretical Computer Science, 209(1-2):47-86, 1998. URL: http://dx.doi.org/10.1016/S0304-3975(97)00227-2.
http://dx.doi.org/10.1016/S0304-3975(97)00227-2
Noga Alon, Mrinal Kumar, and Ben Lee Volk. Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits. In CCC 2018, pages 1-16, 2018. http://arxiv.org/abs/1708.02037. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2018.11.
http://dx.doi.org/10.4230/LIPIcs.CCC.2018.11
Suryajith Chillara, Mrinal Kumar, Ramprasad Saptharishi, and V. Vinay. The Chasm at Depth Four, and Tensor Rank : Old results, new insights. CoRR, 2016. URL: http://arxiv.org/abs/1606.04200.
http://arxiv.org/abs/1606.04200
Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Approaching the Chasm at Depth Four. Journal of the ACM, 61(6):33:1-33:16, 2014. Preliminary version in the \ifnumcomp2013<1996\nth\intcalcSub20131985 Annual Structure in Complexity Theory Conference (Structures 2013)\ifnumcomp2013<2015\nth\intcalcSub20131985 Annual IEEE Conference on Computational Complexity (CCC 2013)\nth\intcalcSub20131985 Annual Computational Complexity Conference (CCC 2013). URL: http://dx.doi.org/10.1145/2629541.
http://dx.doi.org/10.1145/2629541
Sumant Hegde and Chandan Saha. Improved Lower Bound for Multi-r-ic Depth Four Circuits as a Function of the Number of Input Variables. Proceedings of Indian National Science Academy, 83(4):907-922, 2017. URL: http://dx.doi.org/10.16943/ptinsa/2017/49224.
http://dx.doi.org/10.16943/ptinsa/2017/49224
Neeraj Kayal. An exponential lower bound for the sum of powers of bounded degree polynomials. In Electronic Colloquium on Computational Complexity (ECCC) TR12-081, 2012. URL: https://eccc.weizmann.ac.il/report/2012/081/.
https://eccc.weizmann.ac.il/report/2012/081/
Pascal Koiran. Arithmetic Circuits: The Chasm at Depth Four Gets Wider. Theoretical Computer Science, 448:56-65, 2012. URL: http://dx.doi.org/10.1016/j.tcs.2012.03.041.
http://dx.doi.org/10.1016/j.tcs.2012.03.041
Mrinal Kumar and Shubhangi Saraf. On the power of homogeneous depth 4 arithmetic circuits. In Proceedings of the \ifnumcomp2014<1966\nth\intcalcSub20141959 Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 2014)\ifnumcomp2014<1975\nth\intcalcSub20141959 Annual Symposium on Switching and Automata Theory (SWAT 2014)\nth\intcalcSub20141959 Annual IEEE Symposium on Foundations of Computer Science (FOCS 2014), pages 364-373, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.46.
http://dx.doi.org/10.1109/FOCS.2014.46
Ran Raz. Multi-Linear Formulas for Permanent and Determinant are of Super-Polynomial Size. J. ACM, 56(2):8:1-8:17, 2009. Preliminary version in the \nth\intcalcSub20041968 Annual ACM Symposium on Theory of Computing (STOC 2004). URL: http://dx.doi.org/10.1145/1502793.1502797.
http://dx.doi.org/10.1145/1502793.1502797
Ran Raz, Amir Shpilka, and Amir Yehudayoff. A lower bound for the size of syntactically multilinear arithmetic circuits. SIAM J. Comput., 38(4):1624-1647, 2008. Preliminary version in the \ifnumcomp2007<1966\nth\intcalcSub20071959 Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 2007)\ifnumcomp2007<1975\nth\intcalcSub20071959 Annual Symposium on Switching and Automata Theory (SWAT 2007)\nth\intcalcSub20071959 Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007). URL: http://dx.doi.org/10.1137/070707932.
http://dx.doi.org/10.1137/070707932
Ran Raz and Amir Yehudayoff. Lower Bounds and Separations for Constant Depth Multilinear Circuits. Computational Complexity, 18(2):171-207, 2009. Preliminary version in the \ifnumcomp2008<1996\nth\intcalcSub20081985 Annual Structure in Complexity Theory Conference (Structures 2008)\ifnumcomp2008<2015\nth\intcalcSub20081985 Annual IEEE Conference on Computational Complexity (CCC 2008)\nth\intcalcSub20081985 Annual Computational Complexity Conference (CCC 2008). URL: http://dx.doi.org/10.1007/s00037-009-0270-8.
http://dx.doi.org/10.1007/s00037-009-0270-8
Amir Shpilka and Amir Yehudayoff. Arithmetic Circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5:207-388, March 2010. URL: http://dx.doi.org/10.1561/0400000039.
http://dx.doi.org/10.1561/0400000039
Sébastien Tavenas. Improved bounds for reduction to depth 4 and depth 3. Inf. Comput., 240:2-11, 2015. Preliminary version in the \nth\intcalcSub20131975 Internationl Symposium on the Mathematical Foundations of Computer Science (MFCS 2013). URL: http://dx.doi.org/10.1016/j.ic.2014.09.004.
http://dx.doi.org/10.1016/j.ic.2014.09.004
Leslie G. Valiant, Sven Skyum, S. Berkowitz, and Charles Rackoff. Fast Parallel Computation of Polynomials Using Few Processors. SIAM J. Comput., 12(4):641-644, 1983. Preliminary version in the \nth\intcalcSub19811975 Internationl Symposium on the Mathematical Foundations of Computer Science (MFCS 1981). URL: http://dx.doi.org/10.1137/0212043.
http://dx.doi.org/10.1137/0212043
Mrinal Kumar, Rafael Mendes de Oliveira, and Ramprasad Saptharishi
Creative Commons Attribution 3.0 Unported license
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Sum-Of-Squares Bounds via Boolean Function Analysis
We introduce a method for proving bounds on the SoS rank based on Boolean Function Analysis and Approximation Theory. We apply our technique to improve upon existing results, thus making progress towards answering several open questions.
We consider two questions by Laurent. First, finding what is the SoS rank of the linear representation of the set with no integral points. We prove that the SoS rank is between ceil[n/2] and ceil[~ n/2 +sqrt{n log{2n}} ~]. Second, proving the bounds on the SoS rank for the instance of the Min Knapsack problem. We show that the SoS rank is at least Omega(sqrt{n}) and at most ceil[{n+ 4 ceil[sqrt{n} ~]}/2]. Finally, we consider the question by Bienstock regarding the instance of the Set Cover problem. For this problem we prove the SoS rank lower bound of Omega(sqrt{n}).
SoS certificate
SoS rank
hypercube optimization
semidefinite programming
Theory of computation~Semidefinite programming
Theory of computation~Convex optimization
79:1-79:15
Track A: Algorithms, Complexity and Games
Supported by SNSF project PZ00P2_174117.
I would like to express my gratitude to Markus Schweighofer for fruitful discussions.
Adam
Kurpisz
Adam Kurpisz
ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland
10.4230/LIPIcs.ICALP.2019.79
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Adam Kurpisz
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Dynamic Time Warping in Strongly Subquadratic Time: Algorithms for the Low-Distance Regime and Approximate Evaluation
Dynamic time warping distance (DTW) is a widely used distance measure between time series, with applications in areas such as speech recognition and bioinformatics. The best known algorithms for computing DTW run in near quadratic time, and conditional lower bounds prohibit the existence of significantly faster algorithms.
The lower bounds do not prevent a faster algorithm for the important special case in which the DTW is small, however. For an arbitrary metric space Sigma with distances normalized so that the smallest non-zero distance is one, we present an algorithm which computes dtw(x, y) for two strings x and y over Sigma in time O(n * dtw(x, y)). When dtw(x, y) is small, this represents a significant speedup over the standard quadratic-time algorithm.
Using our low-distance regime algorithm as a building block, we also present an approximation algorithm which computes dtw(x, y) within a factor of O(n^epsilon) in time O~(n^{2 - epsilon}) for 0 < epsilon < 1. The algorithm allows for the strings x and y to be taken over an arbitrary well-separated tree metric with logarithmic depth and at most exponential aspect ratio. Notably, any polynomial-size metric space can be efficiently embedded into such a tree metric with logarithmic expected distortion. Extending our techniques further, we also obtain the first approximation algorithm for edit distance to work with characters taken from an arbitrary metric space, providing an n^epsilon-approximation in time O~(n^{2 - epsilon}), with high probability.
Finally, we turn our attention to the relationship between edit distance and dynamic time warping distance. We prove a reduction from computing edit distance over an arbitrary metric space to computing DTW over the same metric space, except with an added null character (whose distance to a letter l is defined to be the edit-distance insertion cost of l). Applying our reduction to a conditional lower bound of Bringmann and Künnemann pertaining to edit distance over {0, 1}, we obtain a conditional lower bound for computing DTW over a three letter alphabet (with distances of zero and one). This improves on a previous result of Abboud, Backurs, and Williams, who gave a conditional lower bound for DTW over an alphabet of size five.
With a similar approach, we also prove a reduction from computing edit distance (over generalized Hamming Space) to computing longest-common-subsequence length (LCS) over an alphabet with an added null character. Surprisingly, this means that one can recover conditional lower bounds for LCS directly from those for edit distance, which was not previously thought to be the case.
dynamic time warping
edit distance
approximation algorithm
tree metrics
Theory of computation
Theory of computation~Design and analysis of algorithms
80:1-80:15
Track A: Algorithms, Complexity and Games
https://arxiv.org/abs/1904.09690
The author would like to thank Moses Charikar for his mentoring and advice throughout the project, Ofir Geri for his support and for many useful conversations, and Virginia Williams for suggesting the problem of reducing between edit distance and LCS.
William
Kuszmaul
William Kuszmaul
Massachusetts Institute of Technology, Cambridge, USA
Supported by an MIT Akamai Fellowship and a Fannie & John Hertz Foundation Fellowship. Also supported by NSF Grants 1314547 and 1533644. Parts of this research were performed during the Stanford CURIS research program.
10.4230/LIPIcs.ICALP.2019.80
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Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Quadratic-time hardness of LCS and other sequence similarity measures. arXiv preprint, 2015. URL: http://arxiv.org/abs/1501.07053.
http://arxiv.org/abs/1501.07053
Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight hardness results for LCS and other sequence similarity measures. In 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 59-78, 2015.
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Vladimir Braverman, Moses Charikar, William Kuszmaul, David Woodruff, and Lin Yang. The One-Way Communication Complexity of Dynamic Time Warping Distance. Manuscript submitted for publication, 2018.
Karl Bringmann and Marvin Künnemann. Quadratic conditional lower bounds for string problems and dynamic time warping. In 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 79-97, 2015.
Karl Bringmann and Wolfgang Mulzer. Approximability of the discrete Fréchet distance. Journal of Computational Geometry, 7(2):46-76, 2015.
EG Caiani, A Porta, G Baselli, M Turiel, S Muzzupappa, F Pieruzzi, C Crema, A Malliani, and S Cerutti. Warped-average template technique to track on a cycle-by-cycle basis the cardiac filling phases on left ventricular volume. In Computers in Cardiology 1998, pages 73-76, 1998.
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Moses Charikar, Ofir Geri, Michael P. Kim, and William Kuszmaul. On Estimating Edit Distance: Alignment, Dimension Reduction, and Embeddings. In 45th International Colloquium on Automata, Languages, and Programming (ICALP), pages 34:1-34:14, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.34.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.34
Alexander De Luca, Alina Hang, Frederik Brudy, Christian Lindner, and Heinrich Hussmann. Touch me once and i know it’s you!: implicit authentication based on touch screen patterns. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, pages 987-996, 2012.
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http://arxiv.org/abs/1904.09690
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William Kuszmaul
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Simple Gap-Producing Reduction for the Parameterized Set Cover Problem
Given an n-vertex bipartite graph I=(S,U,E), the goal of set cover problem is to find a minimum sized subset of S such that every vertex in U is adjacent to some vertex of this subset. It is NP-hard to approximate set cover to within a (1-o(1))ln n factor [I. Dinur and D. Steurer, 2014]. If we use the size of the optimum solution k as the parameter, then it can be solved in n^{k+o(1)} time [Eisenbrand and Grandoni, 2004]. A natural question is: can we approximate set cover to within an o(ln n) factor in n^{k-epsilon} time?
In a recent breakthrough result[Karthik et al., 2018], Karthik, Laekhanukit and Manurangsi showed that assuming the Strong Exponential Time Hypothesis (SETH), for any computable function f, no f(k)* n^{k-epsilon}-time algorithm can approximate set cover to a factor below (log n)^{1/poly(k,e(epsilon))} for some function e.
This paper presents a simple gap-producing reduction which, given a set cover instance I=(S,U,E) and two integers k<h <=(1-o(1))sqrt[k]{log |S|/log log |S|}, outputs a new set cover instance I'=(S,U',E') with |U'|=|U|^{h^k}|S|^{O(1)} in |U|^{h^k}* |S|^{O(1)} time such that
- if I has a k-sized solution, then so does I';
- if I has no k-sized solution, then every solution of I' must contain at least h vertices.
Setting h=(1-o(1))sqrt[k]{log |S|/log log |S|}, we show that assuming SETH, for any computable function f, no f(k)* n^{k-epsilon}-time algorithm can distinguish between a set cover instance with k-sized solution and one whose minimum solution size is at least (1-o(1))* sqrt[k]((log n)/(log log n)). This improves the result in [Karthik et al., 2018].
set cover
FPT inapproximability
gap-producing reduction
(n
k)-universal set
Theory of computation~Problems, reductions and completeness
81:1-81:15
Track A: Algorithms, Complexity and Games
This work was supported in part by the National Key R&D Program of China 2018YFB1003202 and JSPS KAKENHI Grant Number JP18H05291.
https://arxiv.org/abs/1902.03702
The author wishes to thank the anonymous referees for their detailed comments.
Bingkai
Lin
Bingkai Lin
National Institute of Informatics, Tokyo, Japan
Nanjing University, Nanjing, China
https://orcid.org/0000-0002-3444-6380
10.4230/LIPIcs.ICALP.2019.81
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Bingkai Lin. The Parameterized Complexity of the k-Biclique Problem. J. ACM, 65(5):34:1-34:23, 2018.
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Benjamin Rossman. On the constant-depth complexity of k-clique. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 721-730. ACM, 2008.
P. Slavík. A Tight Analysis of the Greedy Algorithm for Set Cover. Journal of Algorithms, 25(2):237-254, 1997.
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Bingkai Lin
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Maintaining Perfect Matchings at Low Cost
The min-cost matching problem suffers from being very sensitive to small changes of the input. Even in a simple setting, e.g., when the costs come from the metric on the line, adding two nodes to the input might change the optimal solution completely. On the other hand, one expects that small changes in the input should incur only small changes on the constructed solutions, measured as the number of modified edges. We introduce a two-stage model where we study the trade-off between quality and robustness of solutions. In the first stage we are given a set of nodes in a metric space and we must compute a perfect matching. In the second stage 2k new nodes appear and we must adapt the solution to a perfect matching for the new instance.
We say that an algorithm is (alpha,beta)-robust if the solutions constructed in both stages are alpha-approximate with respect to min-cost perfect matchings, and if the number of edges deleted from the first stage matching is at most beta k. Hence, alpha measures the quality of the algorithm and beta its robustness. In this setting we aim to balance both measures by deriving algorithms for constant alpha and beta. We show that there exists an algorithm that is (3,1)-robust for any metric if one knows the number 2k of arriving nodes in advance. For the case that k is unknown the situation is significantly more involved. We study this setting under the metric on the line and devise a (10,2)-robust algorithm that constructs a solution with a recursive structure that carefully balances cost and redundancy.
matchings
robust optimization
approximation algorithms
Mathematics of computing~Combinatorial algorithms
82:1-82:14
Track A: Algorithms, Complexity and Games
The authors were supported by Project Fondecyt Nr. 1181527, the Alexander von Humboldt Foundation with funds of the German Federal Ministry of Education and Research (BMBF), BAYLAT, and the Deutsche Forschungsgemeinschaft under grant BR 4744/2-1.
A full version of the paper is available at https://arxiv.org/abs/1811.10580.
We thank anonymous reviewers for their helpful feedback.
Jannik
Matuschke
Jannik Matuschke
Research Center for Operations Management, KU Leuven, Leuven, Belgium
Ulrike
Schmidt-Kraepelin
Ulrike Schmidt-Kraepelin
Institute of Software Engineering and Theoretical Computer Science, TU Berlin, Berlin, Germany
José
Verschae
José Verschae
Institute of Engineering Sciences, Universidad de O'Higgins, Rancagua, Chile
10.4230/LIPIcs.ICALP.2019.82
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Anupam Gupta and Kevin Lewi. The Online Metric Matching Problem for Doubling Metrics. In Proceedings of the Thirty-ninth International Colloquium on Automata, Languages and Programming (ICALP '12), pages 424-435, 2012.
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Samir Khuller, Stephen G. Mitchell, and Vijay V. Vazirani. On-line Algorithms for Weighted Bipartite Matching and Stable Marriages. Theoretical Computer Science, 127:255-267, 1994.
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K. Nayyar and S. Raghvendra. An Input Sensitive Online Algorithm for the Metric Bipartite Matching Problem. In Proceedings of the Fifty-eight Annual IEEE Symposium on Foundations of Computer Science (FOCS '17), pages 505-515, 2017.
Snjólfur Ólafsson. Weighted Matching in Chess Tournaments. The Journal of the Operational Research Society, 41:17-24, 1990.
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Sharath Raghvendra. Optimal Analysis of an Online Algorithm for the Bipartite Matching Problem on a Line, 2018. URL: http://arxiv.org/abs/1803.07206.
http://arxiv.org/abs/1803.07206
E. Reingold and R. Tarjan. On a Greedy Heuristic for Complete Matching. SIAM Journal on Computing, 10:676-681, 1981.
Jannik Matuschke, Ulrike Schmidt-Kraepelin, and José Verschae
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Minimum Cost Query Problem on Matroids with Uncertainty Areas
We study the minimum weight basis problem on matroid when elements' weights are uncertain. For each element we only know a set of possible values (an uncertainty area) that contains its real weight. In some cases there exist bases that are uniformly optimal, that is, they are minimum weight bases for every possible weight function obeying the uncertainty areas. In other cases, computing such a basis is not possible unless we perform some queries for the exact value of some elements.
Our main result is a polynomial time algorithm for the following problem. Given a matroid with uncertainty areas and a query cost function on its elements, find the set of elements of minimum total cost that we need to simultaneously query such that, no matter their revelation, the resulting instance admits a uniformly optimal base. We also provide combinatorial characterizations of all uniformly optimal bases, when one exists; and of all sets of queries that can be performed so that after revealing the corresponding weights the resulting instance admits a uniformly optimal base.
Minimum spanning tree
matroids
uncertainty
queries
Mathematics of computing~Matroids and greedoids
83:1-83:14
Track A: Algorithms, Complexity and Games
Work supported by Conicyt via Fondecyt Grant 1181180 and PIA AFB-170001.
A full version of the paper is available at https://arxiv.org/abs/1904.11668.
Arturo I.
Merino
Arturo I. Merino
Dept. of Mathematical Engineering and CMM, Universidad de Chile & UMI-CNRS 2807, Santiago, Chile
https://orcid.org/0000-0002-1728-6936
José A.
Soto
José A. Soto
Dept. of Mathematical Engineering and CMM, Universidad de Chile & UMI-CNRS 2807, Santiago, Chile
www.dim.uchile.cl/~jsoto
https://orcid.org/0000-0003-2219-8401
10.4230/LIPIcs.ICALP.2019.83
Richard Bruce, Michael Hoffmann, Danny Krizanc, and Rajeev Raman. Efficient Update Strategies for Geometric Computing with Uncertainty. Theory Comput. Syst., 38(4):411-423, 2005. URL: http://dx.doi.org/10.1007/s00224-004-1180-4.
http://dx.doi.org/10.1007/s00224-004-1180-4
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http://dx.doi.org/10.1016/j.tcs.2015.11.025
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http://dx.doi.org/10.4230/LIPIcs.STACS.2008.1358
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http://dx.doi.org/10.1137/S0097539701395668
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http://dx.doi.org/10.1007/s00224-015-9664-y
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http://dx.doi.org/10.1016/j.ipl.2005.11.001
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http://dx.doi.org/10.1016/j.ejor.2005.12.033
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http://dx.doi.org/10.1016/0012-365X(77)90118-2
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http://dx.doi.org/10.1137/16M1088375
James G. Oxley. Matroid theory. Oxford University Press, 1992.
Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science &Business Media, 2003.
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http://dx.doi.org/10.1016/S0167-6377(01)00078-5
Arturo I. Merino and José A. Soto
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Short Proofs Are Hard to Find
We obtain a streamlined proof of an important result by Alekhnovich and Razborov [Michael Alekhnovich and Alexander A. Razborov, 2008], showing that it is hard to automatize both tree-like and general Resolution. Under a different assumption than [Michael Alekhnovich and Alexander A. Razborov, 2008], our simplified proof gives improved bounds: we show under ETH that these proof systems are not automatizable in time n^f(n), whenever f(n) = o(log^{1/7 - epsilon} log n) for any epsilon > 0. Previously non-automatizability was only known for f(n) = O(1). Our proof also extends fairly straightforwardly to prove similar hardness results for PCR and Res(r).
automatizability
Resolution
SAT solvers
proof complexity
Theory of computation~Proof complexity
Hardware~Theorem proving and SAT solving
84:1-84:16
Track A: Algorithms, Complexity and Games
A full version of the paper is available at http://www.cs.toronto.edu/~mertz/papers/mpw19.short_proofs_are_hard_to_find.pdf.
The authors thank Noah Fleming, Pravesh Kothari, Denis Pankratov, Robert Robere, Mika Göös, and Avi Wigderson for helpful conversations. We also thank Yijia Chen for providing more details on the construction in [Yijia Chen and Bingkai Lin, 2016], and Pasin Manurangsi for giving feedback on the parameters in [Parinya Chalermsook et al., 2017].
Ian
Mertz
Ian Mertz
University of Toronto, Canada
Research supported by NSERC.
Toniann
Pitassi
Toniann Pitassi
University of Toronto, Canada
Institute for Advanced Study, Princeton, NJ, USA
Research supported by NSERC.
Yuanhao
Wei
Yuanhao Wei
Carnegie Mellon University, Pittsburgh, PA, USA
Work done while a student at University of Toronto. Research supported by NSERC.
10.4230/LIPIcs.ICALP.2019.84
Michael Alekhnovich, Samuel R. Buss, Shlomo Moran, and Toniann Pitassi. Minimum Propositional Proof Length Is NP-Hard to Linearly Approximate. J. Symb. Log., 66(1):171-191, 2001.
Michael Alekhnovich and Alexander A. Razborov. Resolution Is Not Automatizable Unless W[P] Is Tractable. SIAM J. Comput., 38(4):1347-1363, 2008.
Noga Alon, Oded Goldreich, Johan Håstad, and René Peralta. Simple Construction of Almost k-wise Independent Random Variables. Random Struct. Algorithms, 3(3):289-304, 1992.
Albart Atserias and Moritz Müller. Automating Resolution is NP-Hard. CoRR, abs/1904.02991, 2019. URL: http://arxiv.org/abs/1904.02991.
http://arxiv.org/abs/1904.02991
Albert Atserias and Víctor Dalmau. A combinatorial characterization of resolution width. J. Comput. Syst. Sci., 74(3):323-334, 2008.
Albert Atserias, Massimo Lauria, and Jakob Nordström. Narrow Proofs May Be Maximally Long. ACM Trans. Comput. Log., 17(3):19:1-19:30, 2016.
Paul Beame, Russell Impagliazzo, Jan Krajíček, Toniann Pitassi, and Pavel Pudlák. Lower Bound on Hilbert’s Nullstellensatz and propositional proofs. In 35th Annual Symposium on Foundations of Computer Science, Santa Fe, New Mexico, USA, 20-22 November 1994, pages 794-806, 1994.
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Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow - resolution made simple. J. ACM, 48(2):149-169, 2001.
Maria Luisa Bonet, Carlos Domingo, Ricard Gavaldà, Alexis Maciel, and Toniann Pitassi. Non-Automatizability of Bounded-Depth Frege Proofs. Computational Complexity, 13(1-2):47-68, 2004.
Maria Luisa Bonet, Toniann Pitassi, and Ran Raz. On Interpolation and Automatization for Frege Systems. SIAM J. Comput., 29(6):1939-1967, 2000.
Michael Brickenstein and Alexander Dreyer. PolyBoRi: A Framework for GröBner-basis Computations with Boolean Polynomials. J. Symb. Comput., 44(9):1326-1345, 2009.
Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More. CoRR, abs/1708.04218, 2017. URL: http://arxiv.org/abs/1708.04218.
http://arxiv.org/abs/1708.04218
Yijia Chen and Bingkai Lin. The Constant Inapproximability of the Parameterized Dominating Set Problem. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 505-514, 2016.
Matthew Clegg, Jeff Edmonds, and Russell Impagliazzo. Using the Groebner Basis Algorithm to Find Proofs of Unsatisfiability. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 174-183, 1996.
Martin Davis, George Logemann, and Donald Loveland. A Machine Program for Theorem-Proving. J. ACM, 5(7):394-397, 1961.
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Irit Dinur. Mildly exponential reduction from gap 3SAT to polynomial-gap label-cover. Electronic Colloquium on Computational Complexity (ECCC), 23:128, 2016.
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Nicola Galesi and Massimo Lauria. On the Automatizability of Polynomial Calculus. Theory Comput. Syst., 47(2):491-506, 2010.
Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone Circuit Lower Bounds from Resolution. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 902-911, 2018.
Mika Göös, Toniann Pitassi, and Thomas Watson. Query-to-Communication Lifting for BPP. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, pages 132-143. IEEE Computer Society, 2017.
Russell Impagliazzo and Ramamohan Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001.
Stasys Jukna. Extremal Combinatorics: With Applications in Computer Science. Springer Publishing Company, Incorporated, 1st edition, 2010.
Jan Krajíček and Pavel Pudlák. Some Consequences of Cryptographical Conjectures for S^1_2 and EF. Inf. Comput., 140(1):82-94, 1998.
J. Lasserre. Global Optimization With Polynomials and the Problem of Moments. SIAM J. Optimization, 11(3):796-817, 2001.
Pasin Manurangsi and Prasad Raghavendra. A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 78:1-78:15, 2017.
Hugues Marchand, Alexander Martin, Robert Weismantel, and Laurence Wolsey. Cutting planes in integer and mixed integer programming. Discrete Applied Mathematics, 123(1):397-446, 2002.
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Jakob Nordström. On the Interplay Between Proof Complexity and SAT Solving. ACM SIGLOG News, 2(3):19-44, 2015.
Ryan O'Donnell. SOS is not obviously automatizable, even approximately. Electronic Colloquium on Computational Complexity (ECCC), 23:141, 2016.
Pablo Parrilo. Structured Semidefinite Programs and Semialgebraic Geometry. Methods in Robustness and Optimization, 2000.
K. Pipatsrisawat and A. Darwiche. On the power of clause-learning SAT solvers as resolution engines. Artificial Intelligence, 175(2):512-525, 2011.
Pavel Pudlák. Proofs as Games. The American Mathematical Monthly, 107(6):541-550, 2000.
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Prasad Raghavendra, Tselil Schramm, and David Steurer. High-dimensional estimation via sum-of-squares proofs. CoRR, 2018. URL: http://arxiv.org/abs/1807.11419.
http://arxiv.org/abs/1807.11419
Nathan Segerlind, Samuel R. Buss, and Russell Impagliazzo. A Switching Lemma for Small Restrictions and Lower Bounds for k-DNF Resolution. SIAM J. Comput., 33(5):1171-1200, 2004.
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http://arxiv.org/abs/1606.02577
M. Vinyals, J. Elffers, J. Giráldez-Cru, S. Gocht, and J Nordström. On the power of clause-learning SAT solvers as resolution engines. Artificial Intelligence, 175(2):512-525, 2011.
Ian Mertz, Toniann Pitassi, and Yuanhao Wei
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints
Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function subject to mixed packing and covering constraints. We present a tight approximation algorithm that for any constant epsilon >0 achieves a guarantee of 1-(1/e)-epsilon while violating only the covering constraints by a multiplicative factor of 1-epsilon. Our algorithm is based on a novel enumeration method, which unlike previously known enumeration techniques, can handle both packing and covering constraints. We extend the above main result by additionally handling a matroid independence constraint as well as finding (approximate) pareto set optimal solutions when multiple submodular objectives are present. Finally, we propose a novel and purely combinatorial dynamic programming approach. While this approach does not give tight bounds it yields deterministic and in some special cases also considerably faster algorithms. For example, for the well-studied special case of only packing constraints (Kulik et al. [Math. Oper. Res. `13] and Chekuri et al. [FOCS `10]), we are able to present the first deterministic non-trivial approximation algorithm. We believe our new combinatorial approach might be of independent interest.
submodular function
approximation algorithm
covering
packing
Theory of computation~Submodular optimization and polymatroids
Theory of computation~Approximation algorithms analysis
85:1-85:15
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1804.10947.
Joachim Spoerhase and Sumedha Uniyal thank an anonymous reviewer for pointing them to the fact that Theorem 6 also applies to polytopes that are not down-closed, which makes it possible to apply a randomized rounding approach.
Eyal
Mizrachi
Eyal Mizrachi
Computer Science Department, Technion, Haifa 32000, Israel
Roy
Schwartz
Roy Schwartz
Computer Science Department, Technion, Haifa 32000, Israel
Supported by ISF grant 1336/16 and NSF-BSF grant number 2016742.
Joachim
Spoerhase
Joachim Spoerhase
Department of Computer Science, Aalto University, Espoo, Finland
https://orcid.org/0000-0002-2601-6452
Supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant number 759557) and by Academy of Finland (grant number 310415).
Sumedha
Uniyal
Sumedha Uniyal
Department of Computer Science, Aalto University, Espoo, Finland
Partially supported by Academy of Finland under the grant agreement number 314284.
10.4230/LIPIcs.ICALP.2019.85
A. A. Ageev and M. I. Sviridenko. An 0.828 Approximation Algorithm for the Uncapacitated Facility Location Problem. Discrete Appl. Math., 93(2-3):149-156, July 1999.
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Per Austrin, Siavosh Benabbas, and Konstantinos Georgiou. Better Balance by Being Biased: A 0.8776-Approximation for Max Bisection. ACM Trans. Algorithms, 13(1):2:1-2:27, 2016.
Francis Bach. Learning with Submodular Functions: A Convex Optimization Perspective. Now Publishers Inc., Hanover, MA, USA, 2013.
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Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. Submodular Maximization with Cardinality Constraints. In Proc. 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'14), pages 1433-1452, 2014.
Gruia Calinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a Monotone Submodular Function Subject to a Matroid Constraint. SIAM J. Comput., 40(6):1740-1766, December 2011.
Chandra Chekuri and Sanjeev Khanna. A Polynomial Time Approximation Scheme for the Multiple Knapsack Problem. SIAM Journal on Computing, 35(3):713-728, 2005.
Chandra Chekuri, Jan Vondrák, and Rico Zenklusen. Dependent Randomized Rounding via Exchange Properties of Combinatorial Structures. In Proc. 51th Annual IEEE Symposium on Foundations of Computer Science (FOCS'10), pages 575-584, 2010.
Reuven Cohen, Liran Katzir, and Danny Raz. An Efficient Approximation for the Generalized Assignment Problem. Inf. Process. Lett., 100(4):162-166, November 2006.
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Uriel Feige. A Threshold of Ln N for Approximating Set Cover. J. ACM, 45(4):634-652, July 1998.
Lisa Fleischer, Michel X Goemans, Vahab S Mirrokni, and Maxim Sviridenko. Tight approximation algorithms for maximum general assignment problems. In Proc. 17th annual ACM-SIAM Symposium on Discrete Algorithm, (SODA'06), pages 611-620. SIAM, 2006.
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David Kempe, Jon Kleinberg, and Éva Tardos. Maximizing the Spread of Influence through a Social Network. Theory of Computing, 11(4):105-147, 2015.
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Ariel Kulik, Hadas Shachnai, and Tami Tamir. Approximations for Monotone and Nonmonotone Submodular Maximization with Knapsack Constraints. Mathematics of Operations Research, 38(4):729-739, 2013. preliminary version appeared in SODA'09.
Jon Lee, Vahab S Mirrokni, Viswanath Nagarajan, and Maxim Sviridenko. Non-monotone submodular maximization under matroid and knapsack constraints. In Proc. 41st Annual ACM Symposium on Theory Of Computing, (STOC'09), pages 323-332. ACM, 2009.
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Eyal Mizrachi, Roy Schwartz, Joachim Spoerhase, and Sumedha Uniyal. A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints. CoRR, abs/1804.10947, 2018. URL: http://arxiv.org/abs/1804.10947.
http://arxiv.org/abs/1804.10947
George L. Nemhauser, Laurence A. Wolsey, and Marshall L. Fisher. An analysis of approximations for maximizing submodular set functions - I. Math. Program., 14(1):265-294, 1978.
George L Nemhauser and Leonard A Wolsey. Best algorithms for approximating the maximum of a submodular set function. Mathematics of operations research, 3(3):177-188, 1978.
Andreas S. Schulz and Nelson A. Uhan. Approximating the least core value and least core of cooperative games with supermodular costs. Discrete Optimization, 10(2):163-180, 2013.
Maxim Sviridenko. A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett., 32(1):41-43, 2004.
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Eyal Mizrachi, Roy Schwartz, Joachim Spoerhase, and Sumedha Uniyal
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Scheduling to Approximate Minimization Objectives on Identical Machines
This paper considers scheduling on identical machines. The scheduling objective considered in this paper generalizes most scheduling minimization problems. In the problem, there are n jobs and each job j is associated with a monotonically increasing function g_j. The goal is to design a schedule that minimizes sum_{j in [n]} g_{j}(C_j) where C_j is the completion time of job j in the schedule. An O(1)-approximation is known for the single machine case. On multiple machines, this paper shows that if the scheduler is required to be either non-migratory or non-preemptive then any algorithm has an unbounded approximation ratio. Using preemption and migration, this paper gives a O(log log nP)-approximation on multiple machines, the first result on multiple machines. These results imply the first non-trivial positive results for several special cases of the problem considered, such as throughput minimization and tardiness.
Natural linear programs known for the problem have a poor integrality gap. The results are obtained by strengthening a natural linear program for the problem with a set of covering inequalities we call job cover inequalities. This linear program is rounded to an integral solution by building on quasi-uniform sampling and rounding techniques.
Scheduling
LP rounding
Approximation Algorithms
Theory of computation
Theory of computation~Approximation algorithms analysis
86:1-86:14
Track A: Algorithms, Complexity and Games
Benjamin
Moseley
Benjamin Moseley
Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA
Relational AI, Berkeley, CA, USA
Supported in part by a Google Research Award, a Infor Award and NSF Grants CCF-1824303, CCF-1733873 and CCF-1845146.
10.4230/LIPIcs.ICALP.2019.86
Antonios Antoniadis, Ruben Hoeksma, Julie Meißner, José Verschae, and Andreas Wiese. A QPTAS for the General Scheduling Problem with Identical Release Dates. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 31:1-31:14, 2017.
Nikhil Bansal and Kirk Pruhs. The Geometry of Scheduling. SIAM J. Comput., 43(5):1684-1698, 2014.
Robert D. Carr, Lisa 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, January 9-11, 2000, San Francisco, CA, USA., pages 106-115, 2000.
Timothy M. Chan, Elyot Grant, Jochen Könemann, and Malcolm Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1576-1585, 2012.
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Wiebke Höhn and Tobias Jacobs. On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost. ACM Trans. Algorithms, 11(4):25, 2015.
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Sungjin Im and Benjamin Moseley. Fair Scheduling via Iterative Quasi-Uniform Sampling. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2601-2615, 2017.
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http://dx.doi.org/10.1016/0167-6377(82)90022-0
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http://dx.doi.org/10.1145/2155620.2155650
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http://dx.doi.org/10.1109/CCGrid.2011.56
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Benjamin Moseley
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Computing Optimal Epsilon-Nets Is as Easy as Finding an Unhit Set
Given a set system (X, R) with VC-dimension d, the celebrated result of Haussler and Welzl (1987) showed that there exists an epsilon-net for (X, R) of size O(d/epsilon log 1/epsilon). Furthermore, the algorithm is simple: just take a uniform random sample from X! However, for many geometric set systems this bound is sub-optimal and since then, there has been much work presenting improved bounds and algorithms tailored to specific geometric set systems.
In this paper, we consider the following natural algorithm to compute an epsilon-net: start with an initial random sample N. Iteratively, as long as N is not an epsilon-net for R, pick any unhit set S in R (say, given by an Oracle), and add O(1) randomly chosen points from S to N.
We prove that the above algorithm computes, in expectation, epsilon-nets of asymptotically optimal size for all known cases of geometric set systems. Furthermore, it makes O(1/epsilon) calls to the Oracle. In particular, this implies that computing optimal-sized epsilon-nets are as easy as computing an unhit set in the given set system.
epsilon-nets
Geometric Set Systems
Theory of computation~Sketching and sampling
87:1-87:12
Track A: Algorithms, Complexity and Games
This research was supported by the grant ANR SAGA (JCJC-14-CE25-0016-01).
We thank the reviewers for insightful feedback that helped the content and presentation of this work.
Nabil H.
Mustafa
Nabil H. Mustafa
Université Paris-Est, Laboratoire d'Informatique Gaspard-Monge, ESIEE Paris, France
10.4230/LIPIcs.ICALP.2019.87
P. K. Agarwal. Range Searching. In J. E. Goodman, J. O'Rourke, and C. D. Tóth, editors, Handbook of Discrete and Computational Geometry. CRC Press LLC, 2017.
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B. Aronov, E. Ezra, and M. Sharir. Small-size ε-nets for axis-parallel rectangles and boxes. SIAM Journal on Computing, 39(7):3248-3282, 2010.
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T. M. Chan, E. Grant, J. Könemann, and M. Sharpe. Weighted Capacitated, Priority, and Geometric Set Cover via Improved Quasi-uniform Sampling. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1576-1585, 2012.
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N. H. Mustafa and K. Varadarajan. Epsilon-approximations and Epsilon-nets. In J. E. Goodman, J. O'Rourke, and C. D. Tóth, editors, Handbook of Discrete and Computational Geometry. CRC Press LLC, 2017.
J. Pach and P. K. Agarwal. Combinatorial geometry. John Wiley &Sons, 1995.
J. Phillips. Coresets and Sketches. In J. E. Goodman, J. O'Rourke, and C. D. Tóth, editors, Handbook of Discrete and Computational Geometry. CRC Press LLC, 2017.
E. Pyrga and S. Ray. New existence proofs for ε-nets. In Proceedings of the 24th Annual ACM Symposium on Computational Geometry (SoCG), pages 199-207, 2008.
K. R. Varadarajan. Weighted Geometric Set Cover via Quasi-uniform Sampling. In Proceedings of the 42nd Annual ACM Symposium on Theory of Computing (STOC), pages 641-648, 2010.
H. Yu, P. K. Agarwal, R. Poreddy, and K. R. Varadarajan. Practical Methods for Shape Fitting and Kinetic Data Structures using Coresets. Algorithmica, 52(3):378-402, 2008.
Nabil H. Mustafa
Creative Commons Attribution 3.0 Unported license
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Tight Bounds for Online Weighted Tree Augmentation
The Weighted Tree Augmentation problem (WTAP) is a fundamental problem in network design. In this paper, we consider this problem in the online setting. We are given an n-vertex spanning tree T and an additional set L of edges (called links) with costs. Then, terminal pairs arrive one-by-one and our task is to maintain a low-cost subset of links F such that every terminal pair that has arrived so far is 2-edge-connected in T cup F. This online problem was first studied by Gupta, Krishnaswamy and Ravi (SICOMP 2012) who used it as a subroutine for the online survivable network design problem. They gave a deterministic O(log^2 n)-competitive algorithm and showed an Omega(log n) lower bound on the competitive ratio of randomized algorithms. The case when T is a path is also interesting: it is exactly the online interval set cover problem, which also captures as a special case the parking permit problem studied by Meyerson (FOCS 2005). The contribution of this paper is to give tight results for online weighted tree and path augmentation problems. The main result of this work is a deterministic O(log n)-competitive algorithm for online WTAP, which is tight up to constant factors.
Online algorithms
competitive analysis
tree augmentation
network design
Theory of computation~Online algorithms
88:1-88:14
Track A: Algorithms, Complexity and Games
A full version of this paper is available at http://arxiv.org/abs/1904.11777.
This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing.
Joseph (Seffi)
Naor
Joseph (Seffi) Naor
Technion, Haifa, Israel
Supported in part by ISF grant 1585/15 and BSF grant 2014414.
Seeun William
Umboh
Seeun William Umboh
The University of Sydney, Australia
https://orcid.org/0000-0001-6984-4007
Supported in part by NWO grant 639.022.211 and ISF grant 1817/17. Part of this work was done while a postdoc at Eindhoven University of Technology, and while visiting the Hebrew University of Jerusalem and the Technion.
David P.
Williamson
David P. Williamson
Cornell University, Ithaca, NY, USA
http://www.davidpwilliamson.net/work
https://orcid.org/0000-0002-2884-0058
10.4230/LIPIcs.ICALP.2019.88
Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Naor. The Online Set Cover Problem. SIAM J. Comput., 39(2):361-370, 2009. URL: http://dx.doi.org/10.1137/060661946.
http://dx.doi.org/10.1137/060661946
Baruch Awerbuch, Yossi Azar, and Yair Bartal. On-line generalized Steiner problem. Theoretical Computer Science, 324:313-324, 2004.
Piotr Berman and Chris Coulston. On-Line Algorithms for Steiner Tree Problems. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages 344-353, 1997.
Sina Dehghani, Soheil Ehsani, MohammadTaghi Hajiaghayi, Vahid Liaghat, and Saeed Seddighin. Greedy Algorithms for Online Survivable Network Design. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 152:1-152:14, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.152.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.152
Guy Even and Shakhar Smorodinsky. Hitting sets online and unique-max coloring. Discrete Applied Mathematics, 178:71-82, 2014. URL: http://dx.doi.org/10.1016/j.dam.2014.06.019.
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http://dx.doi.org/10.1145/3188745.3188898
Anupam Gupta, Ravishankar Krishnaswamy, and R. Ravi. Online and Stochastic Survivable Network Design. SIAM J. Comput., 41(6):1649-1672, 2012. URL: http://dx.doi.org/10.1137/09076725X.
http://dx.doi.org/10.1137/09076725X
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Makoto Imase and Bernard M. Waxman. Dynamic Steiner Tree Problem. SIAM Journal on Discrete Mathematics, 4:369-384, 1991.
Adam Meyerson. The Parking Permit Problem. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pages 274-282, 2005.
Joseph Naor, Debmalya Panigrahi, and Mohit Singh. Online Node-Weighted Steiner Tree and Related Problems. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 210-219, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.65.
http://dx.doi.org/10.1109/FOCS.2011.65
Jiawei Qian, Seeun William Umboh, and David P. Williamson. Online Constrained Forest and Prize-Collecting Network Design. Algorithmica, 80(11):3335-3364, 2018.
Daniel Dominic Sleator and Robert Endre Tarjan. A Data Structure for Dynamic Trees. J. Comput. Syst. Sci., 26(3):362-391, 1983. URL: http://dx.doi.org/10.1016/0022-0000(83)90006-5.
http://dx.doi.org/10.1016/0022-0000(83)90006-5
Seeun Umboh. Online Network Design Algorithms via Hierarchical Decompositions. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1373-1387, 2015.
Joseph (Seffi) Naor, Seeun William Umboh, and David P. Williamson
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Optimal Short Cycle Decomposition in Almost Linear Time
Short cycle decomposition is an edge partitioning of an unweighted graph into edge-disjoint short cycles, plus a small number of extra edges not in any cycle. This notion was introduced by Chu et al. [FOCS'18] as a fundamental tool for graph sparsification and sketching. Clearly, it is most desirable to have a fast algorithm for partitioning the edges into as short as possible cycles, while omitting few edges.
The most naïve procedure for such decomposition runs in time O(m * n) and partitions the edges into O(log n)-length edge-disjoint cycles plus at most 2n edges. Chu et al. improved the running time considerably to m^{1+o(1)}, while increasing both the length of the cycles and the number of omitted edges by a factor of n^{o(1)}. Even more recently, Liu-Sachdeva-Yu [SODA'19] showed that for every constant delta in (0,1] there is an O(m * n^{delta})-time algorithm that provides, w.h.p., cycles of length O(log n)^{1/delta} and O(n) extra edges.
In this paper, we significantly improve upon these bounds. We first show an m^{1+o(1)}-time deterministic algorithm for computing nearly optimal cycle decomposition, i.e., with cycle length O(log^2 n) and an extra subset of O(n log n) edges not in any cycle. This algorithm is based on a reduction to low-congestion cycle covers, introduced by the authors in [SODA'19].
We also provide a simple deterministic algorithm that computes edge-disjoint cycles of length 2^{1/epsilon} with n^{1+epsilon}* 2^{1/epsilon} extra edges, for every epsilon in (0,1]. Combining this with Liu-Sachdeva-Yu [SODA'19] gives a linear time randomized algorithm for computing cycles of length poly(log n) and O(n) extra edges, for every n-vertex graphs with n^{1+1/delta} edges for some constant delta.
These decomposition algorithms lead to improvements in all the algorithmic applications of Chu et al. as well as to new distributed constructions.
Cycle decomposition
low-congestion cycle cover
graph sparsification
Theory of computation~Design and analysis of algorithms
89:1-89:14
Track A: Algorithms, Complexity and Games
Merav
Parter
Merav Parter
Weizmann IS, Rehovot, Israel
http://www.weizmann.ac.il/math/parter/home
The Israel Science Foundation grant no. 2084/18
Eylon
Yogev
Eylon Yogev
Technion, Haifa, Israel
https://www.eylonyogev.com/about
The European Union's Horizon 2020 research and innovation program under grant agreement No. 742754.
10.4230/LIPIcs.ICALP.2019.89
Alexandr Andoni, Jiecao Chen, Robert Krauthgamer, Bo Qin, David P Woodruff, and Qin Zhang. On sketching quadratic forms. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, pages 311-319. ACM, 2016.
Joshua Batson, Daniel A Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM Journal on Computing, 41(6):1704-1721, 2012.
Timothy Chu, Yu Gao, Richard Peng, Sushant Sachdeva, Saurabh Sawlani, and Junxing Wang. Graph Sparsification, Spectral Sketches, and Faster Resistance Computation, via Short Cycle Decompositions. In 59th Annual Symposium on Foundations of Computer Science, FOCS. IEEE Computer Society, 2018.
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Yang P. Liu, Sushant Sachdeva, and Zejun Yu. Short Cycles via Low-Diameter Decompositions. SODA, 2019.
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Merav Parter and Eylon Yogev. Distributed Computing Made Secure: A Graph Theoreric Approach. SODA, 2019.
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Merav Parter and Eylon Yogev
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Satisfiability Thresholds for Regular Occupation Problems
In the last two decades the study of random instances of constraint satisfaction problems (CSPs) has flourished across several disciplines, including computer science, mathematics and physics. The diversity of the developed methods, on the rigorous and non-rigorous side, has led to major advances regarding both the theoretical as well as the applied viewpoints. The two most popular types of such CSPs are the Erdős-Rényi and the random regular CSPs.
Based on a ceteris paribus approach in terms of the density evolution equations known from statistical physics, we focus on a specific prominent class of problems of the latter type, the so-called occupation problems. The regular r-in-k occupation problems resemble a basis of this class. By now, out of these CSPs only the satisfiability threshold - the largest degree for which the problem admits asymptotically a solution - for the 1-in-k occupation problem has been rigorously established. In the present work we take a general approach towards a systematic analysis of occupation problems. In particular, we discover a surprising and explicit connection between the 2-in-k occupation problem satisfiability threshold and the determination of contraction coefficients, an important quantity in information theory measuring the loss of information that occurs when communicating through a noisy channel. We present methods to facilitate the computation of these coefficients and use them to establish explicitly the threshold for the 2-in-k occupation problem for k=4. Based on this result, for general k >= 5 we formulate a conjecture that pins down the exact value of the corresponding coefficient, which, if true, is shown to determine the threshold in all these cases.
Constraint satisfaction problem
replica symmetric
contraction coefficient
first moment
second moment
small subgraph conditioning
Theory of computation~Randomness, geometry and discrete structures
Mathematics of computing~Discrete mathematics
Mathematics of computing~Graph theory
Mathematics of computing~Probability and statistics
Mathematics of computing~Probabilistic inference problems
Mathematics of computing~Probabilistic reasoning algorithms
Mathematics of computing~Information theory
90:1-90:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1811.00991.
Konstantinos
Panagiotou
Konstantinos Panagiotou
LMU München, Germany
The research leading to these results has received funding from the European Research Council, ERC Grant Agreement 772606–PTRCSP.
Matija
Pasch
Matija Pasch
LMU München, Germany
10.4230/LIPIcs.ICALP.2019.90
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A. Coja-Oghla, T. Kapetanopoulos, and N. Müller. The replica symmetric phase of random constraint satisfaction problems. arXiv e-prints, page arXiv:1802.09311, February 2018. URL: http://arxiv.org/abs/1802.09311.
http://arxiv.org/abs/1802.09311
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