29th International Symposium on Algorithms and Computation (ISAAC 2018), ISAAC 2018, December 16-19, 2018, Jiaoxi, Yilan, Taiwan
ISAAC 2018
December 16-19, 2018
Jiaoxi, Yilan, Taiwan
International Symposium on Algorithms and Computation
ISAAC
https://dblp.org/db/conf/isaac
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Wen-Lian
Hsu
Wen-Lian Hsu
Der-Tsai
Lee
Der-Tsai Lee
Chung-Shou
Liao
Chung-Shou Liao
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
123
2018
978-3-95977-094-1
https://www.dagstuhl.de/dagpub/978-3-95977-094-1
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter
Table of Contents
Preface
Conference Organization
0:i-0:xviii
Front Matter
Wen-Lian
Hsu
Wen-Lian Hsu
Der-Tsai
Lee
Der-Tsai Lee
Chung-Shou
Liao
Chung-Shou Liao
10.4230/LIPIcs.ISAAC.2018.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Going Beyond Traditional Characterizations in the Age of Big Data and Network Sciences (Invited Talk)
What are efficient algorithms? What are network models? Big Data and Network Sciences have fundamentally challenged the traditional polynomial-time characterization of efficiency and the conventional graph-theoretical characterization of networks.
More than ever before, it is not just desirable, but essential, that efficient algorithms should be scalable. In other words, their complexity should be nearly linear or sub-linear with respect to the problem size. Thus, scalability, not just polynomial-time computability, should be elevated as the central complexity notion for characterizing efficient computation.
For a long time, graphs have been widely used for defining the structure of social and information networks. However, real-world network data and phenomena are much richer and more complex than what can be captured by nodes and edges. Network data are multifaceted, and thus network science requires a new theory, going beyond traditional graph theory, to capture the multifaceted data.
In this talk, I discuss some aspects of these challenges. Using basic tasks in network analysis, social influence modeling, and machine learning as examples, I highlight the role of scalable algorithms and axiomatization in shaping our understanding of "effective solution concepts" in data and network sciences, which need to be both mathematically meaningful and algorithmically efficient.
scalable algorithms
axiomatization
graph sparsification
local algorithms
advanced sampling
big data
network sciences
machine learning
social influence
beyond graph theory
Theory of computation~Design and analysis of algorithms
Mathematics of computing~Discrete mathematics
Mathematics of computing~Probability and statistics
Information systems~World Wide Web
Information systems~Data mining
1:1-1:1
Invited Talk
Shang-Hua
Teng
Shang-Hua Teng
University of Southern California, Los Angeles, USA
10.4230/LIPIcs.ISAAC.2018.1
Shang-Hua Teng
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Approximate Matchings in Massive Graphs via Local Structure (Invited Talk)
Finding a maximum matching is a fundamental algorithmic problem and is fairly well understood in traditional sequential computing models. Some modern applications require that we handle massive graphs and hence we need to consider algorithms in models that do not allow the entire input graph to be held in the memory of one computer, or models in which the graph is evolving over time.
We introduce a new concept called an "Edge Degree Constrained Subgraph (EDCS)", which is a subgraph that is guaranteed to contain a large matching, and which can be identified via local conditions. We then show how to use an EDCS to find 1.5-approximate matchings in several different models including Map Reduce, streaming and distributed computing. We can also use an EDCS to maintain a 1.5-optimal matching in a dynamic graph.
This work is joint with Sepehr Asadi, Aaron Bernstein, Mohammad Hossein Bateni and Vahab Marrokni.
matching
dynamic algorithms
parallel algorithms
approximation algorithms
Theory of computation~Parallel algorithms
Theory of computation~Online algorithms
2:1-2:1
Invited Talk
Clifford
Stein
Clifford Stein
Columbia University, New York City, USA
10.4230/LIPIcs.ISAAC.2018.2
Clifford Stein
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Exploiting Sparsity for Bipartite Hamiltonicity
We present a Monte Carlo algorithm that detects the presence of a Hamiltonian cycle in an n-vertex undirected bipartite graph of average degree delta >= 3 almost surely and with no false positives, in (2-2^{1-delta})^{n/2}poly(n) time using only polynomial space. With the exception of cubic graphs, this is faster than the best previously known algorithms. Our method is a combination of a variant of Björklund's 2^{n/2}poly(n) time Monte Carlo algorithm for Hamiltonicity detection in bipartite graphs, SICOMP 2014, and a simple fast solution listing algorithm for very sparse CNF-SAT formulas.
Hamiltonian cycle
bipartite graph
Mathematics of computing~Graph algorithms
3:1-3:11
Regular Paper
This work was supported in part by the Swedish Research Council grant VR-2016-03855, "Algebraic Graph Algorithms".
Andreas
Björklund
Andreas Björklund
Department of Computer Science, Lund University, Sweden
10.4230/LIPIcs.ISAAC.2018.3
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, Petteri Kaski, and Mikko Koivisto. The traveling salesman problem in bounded degree graphs. ACM Trans. Algorithms, 8(2):18:1-18:13, 2012. URL: http://dx.doi.org/10.1145/2151171.2151181.
http://dx.doi.org/10.1145/2151171.2151181
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
Marek Cygan, Stefan Kratsch, and Jesper Nederlof. Fast Hamiltonicity Checking Via Bases of Perfect Matchings. J. ACM, 65(3):12:1-12:46, 2018.
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 52nd Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 150-159. IEEE Computer Society, 2011.
Marek Cygan and Marcin Pilipczuk. Faster exponential-time algorithms in graphs of bounded average degree. Inf. Comput., 243:75-85, 2015.
Richard A. DeMillo and Richard J. Lipton. A Probabilistic Remark on Algebraic Program Testing. Inf. Process. Lett., 7(4):193-195, 1978.
David Eppstein. The Traveling Salesman Problem for Cubic Graphs. J. Graph Algorithms Appl., 11(1):61-81, 2007.
Fedor V. Fomin and Kjartan Høie. Pathwidth of cubic graphs and exact algorithms. Inf. Process. Lett., 97(5):191-196, 2006.
Kazuo Iwama and Takuya Nakashima. An Improved Exact Algorithm for Cubic Graph TSP. In COCOON, volume 4598 of Lecture Notes in Computer Science, pages 108-117. Springer, 2007.
Richard M. Karp. Reducibility Among Combinatorial Problems. In Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972.
Joachim Kneis, Daniel Mölle, Stefan Richter, and Peter Rossmanith. A Bound on the Pathwidth of Sparse Graphs with Applications to Exact Algorithms. SIAM J. Discrete Math., 23(1):407-427, 2009.
Marko Samer and Stefan Szeider. Algorithms for propositional model counting. J. Discrete Algorithms, 8(1):50-64, 2010.
Jacob T. Schwartz. Fast Probabilistic Algorithms for Verification of Polynomial Identities. J. ACM, 27(4):701-717, 1980.
Andrew G. Thomason. Hamiltonian Cycles and Uniquely Edge Colourable Graphs. In B. Bollobás, editor, Advances in Graph Theory, volume 3 of Annals of Discrete Mathematics, pages 259-268. Elsevier, 1978.
Craig A. Tovey. A simplified NP-complete satisfiability problem. Discrete Applied Mathematics, 8(1):85-89, 1984.
Leslie G. Valiant. Completeness for Parity Problems. In COCOON, volume 3595 of Lecture Notes in Computer Science, pages 1-8. Springer, 2005.
Andreas Björklund
Creative Commons Attribution 3.0 Unported license
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Opinion Forming in Erdös-Rényi Random Graph and Expanders
Assume for a graph G=(V,E) and an initial configuration, where each node is blue or red, in each discrete-time round all nodes simultaneously update their color to the most frequent color in their neighborhood and a node keeps its color in case of a tie. We study the behavior of this basic process, which is called majority model, on the Erdös-Rényi random graph G_{n,p} and regular expanders. First we consider the behavior of the majority model on G_{n,p} with an initial random configuration, where each node is blue independently with probability p_b and red otherwise. It is shown that in this setting the process goes through a phase transition at the connectivity threshold, namely (log n)/n. Furthermore, we say a graph G is lambda-expander if the second-largest absolute eigenvalue of its adjacency matrix is lambda. We prove that for a Delta-regular lambda-expander graph if lambda/Delta is sufficiently small, then the majority model by starting from (1/2-delta)n blue nodes (for an arbitrarily small constant delta>0) results in fully red configuration in sub-logarithmically many rounds. Roughly speaking, this means the majority model is an "efficient" and "fast" density classifier on regular expanders. As a by-product of our results, we show regular Ramanujan graphs are asymptotically optimally immune, that is for an n-node Delta-regular Ramanujan graph if the initial number of blue nodes is s <= beta n, the number of blue nodes in the next round is at most cs/Delta for some constants c,beta>0. This settles an open problem by Peleg [Peleg, 2014].
majority model
random graph
expander graphs
dynamic monopoly
bootstrap percolation
Theory of computation
4:1-4:13
Regular Paper
https://arxiv.org/abs/1805.12172
Ahad
N. Zehmakan
Ahad N. Zehmakan
ETH Zurich, Switzerland
10.4230/LIPIcs.ISAAC.2018.4
Noga Alon. Eigenvalues and expanders. Combinatorica, 6(2):83-96, 1986.
Paul Balister, Béla Bollobás, J Robert Johnson, and Mark Walters. Random majority percolation. Random Structures &Algorithms, 36(3):315-340, 2010.
József Balogh and Gábor Pete. Random disease on the square grid. Random Structures &Algorithms, 13(3-4):409-422, 1998.
József Balogh and Boris G Pittel. Bootstrap percolation on the random regular graph. Random Structures &Algorithms, 30(1-2):257-286, 2007.
Eli Berger. Dynamic monopolies of constant size. Journal of Combinatorial Theory, Series B, 83(2):191-200, 2001.
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Willliam Feller. An introduction to probability theory and its applications, volume 2. John Wiley &Sons, 2008.
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Nikolaos Fountoulakis, Anna Huber, and Konstantinos Panagiotou. Reliable broadcasting in random networks and the effect of density. In INFOCOM, 2010 Proceedings IEEE, pages 1-9. IEEE, 2010.
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Péter Gács, Georgy L Kurdyumov, and Leonid Anatolevich Levin. One-dimensional uniform arrays that wash out finite islands. Problemy Peredachi Informatsii, 14(3):92-96, 1978.
Bernd Gärtner and Ahad N Zehmakan. (Biased) Majority Rule Cellular Automata. arXiv preprint arXiv:1711.10920, 2017.
Bernd Gärtner and Ahad N Zehmakan. Color war: Cellular automata with majority-rule. In International Conference on Language and Automata Theory and Applications, pages 393-404. Springer, 2017.
Bernd Gärtner and Ahad N Zehmakan. Majority Model on Random Regular Graphs. In Latin American Symposium on Theoretical Informatics, pages 572-583. Springer, 2018.
Eric Goles and J Olivos. Comportement périodique des fonctions à seuil binaires et applications. Discrete Applied Mathematics, 3(2):93-105, 1981.
Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bulletin of the American Mathematical Society, 43(4):439-561, 2006.
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Clemens Jeger and Ahad N Zehmakan. Dynamic Monopolies in Reversible Bootstrap Percolation. arXiv preprint arXiv:1805.07392, 2018.
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Moshe Morgenstern. Existence and explicit constructions of q+ 1 regular Ramanujan graphs for every prime power q. Journal of Combinatorial Theory, Series B, 62(1):44-62, 1994.
Nabil H Mustafa and Aleksandar Pekec. Majority consensus and the local majority rule. In International Colloquium on Automata, Languages, and Programming, pages 530-542. Springer, 2001.
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David Peleg. Size bounds for dynamic monopolies. Discrete Applied Mathematics, 86(2-3):263-273, 1998.
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Svatopluk Poljak and Daniel Turzík. On pre-periods of discrete influence systems. Discrete Applied Mathematics, 13(1):33-39, 1986.
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Ahad N. Zehmakan
Creative Commons Attribution 3.0 Unported license
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Colouring (P_r+P_s)-Free Graphs
The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u) subseteq {1,...,k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. We prove that List 3-Colouring is polynomial-time solvable for (P_2+P_5)-free graphs and for (P_3+P_4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices. We also prove that 5-Colouring is NP-complete for (P_3+P_5)-free graphs.
vertex colouring
H-free graph
linear forest
Mathematics of computing~Graph theory
5:1-5:13
Regular Paper
T. Masařík, J. Novotná and V. Slívová were supported by the project GAUK 1277018 and the grant SVV–2017–260452. T. Klimošová was supported by the Center of Excellence – ITI, project P202/12/G061 of GA ČR, by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004), and by the project GAUK 1277018. T. Masařík was also partly supported by the Center of Excellence – ITI, project P202/12/G061 of GA ČR. V. Slívová was partly supported by the project 17-09142S of GA ČR and Charles University project PRIMUS/17/SCI/9. D. Paulusma was supported by the Leverhulme Trust (RPG-2016-258).
https://arxiv.org/abs/1804.11091v2
Tereza
Klimosová
Tereza Klimosová
Department of Applied Mathematics, Charles University, Prague, Czech Republic
Josef
Malík
Josef Malík
Czech Technical University in Prague, Czech Republic
Tomás
Masarík
Tomás Masarík
Department of Applied Mathematics, Charles University, Prague, Czech Republic
Jana
Novotná
Jana Novotná
Department of Applied Mathematics, Charles University, Prague, Czech Republic
Daniël
Paulusma
Daniël Paulusma
Department of Computer Science, Durham University, Durham, UK
Veronika
Slívová
Veronika Slívová
Computer Science Institute of Charles University, Prague, Czech Republic
10.4230/LIPIcs.ISAAC.2018.5
Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, and Daniël Paulusma. Independent feedback vertex set for P₅-free graphs. Algorithmica, to appear.
Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Oliver Schaudt, Maya Stein, and Mingxian Zhong. Three-coloring and list three-coloring of graphs without induced paths on seven vertices. Combinatorica, (in press).
Hajo Broersma, Fedor V. Fomin, Petr A. Golovach, and Daniël Paulusma. Three complexity results on coloring P_k-free graphs. European Journal of Combinatorics, 34(3):609-619, 2013.
Hajo Broersma, Petr A. Golovach, Daniël Paulusma, and Jian Song. Updating the complexity status of coloring graphs without a fixed induced linear forest. TCS, 414(1):9-19, 2012.
Maria Chudnovsky. Coloring graphs with forbidden induced subgraphs. Proc. ICM 2014, IV:291-302, 2014.
Maria Chudnovsky, Peter Maceli, Juraj Stacho, and Mingxian Zhong. 4-Coloring P₆-free graphs with no induced 5-cycles. Journal of Graph Theory, 84(3):262-285, 2017.
Maria Chudnovsky, Sophie Spirkl, and Mingxian Zhong. Four-coloring P₆-free graphs. I. Extending an excellent precoloring. CoRR, 1802.02282, 2018.
Maria Chudnovsky, Sophie Spirkl, and Mingxian Zhong. Four-coloring P₆-free graphs. II. Finding an excellent precoloring. CoRR, 1802.02283, 2018.
Maria Chudnovsky and Juraj Stacho. 3-colorable subclasses of P₈-free graphs. Man., 2017.
Jean-François Couturier, Petr A. Golovach, Dieter Kratsch, and Daniël Paulusma. List Coloring in the Absence of a Linear Forest. Algorithmica, 71(1):21-35, 2015.
Konrad K. Dabrowski and Daniël Paulusma. On colouring (2P₂,H)-free and (P5,H)-free graphs. Information Processing Letters, 131:26-32, 2018.
Keith Edwards. The complexity of colouring problems on dense graphs. TCS, 43:337-343, 1986.
Thomas Emden-Weinert, Stefan Hougardy, and Bernd Kreuter. Uniquely Colourable Graphs and the Hardness of Colouring Graphs of Large Girth. Combinatorics, Probability and Computing, 7(04):375-386, 1998.
Petr A. Golovach, Matthew Johnson, Daniël Paulusma, and Jian Song. A Survey on the Computational Complexity of Colouring Graphs with Forbidden Subgraphs. Journal of Graph Theory, 84(4):331-363, 2017.
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Carla Groenland, Karolina Okrasa, Pawel Rzążewski, Alex Scott, Paul Seymour, and Sophie Spirkl. H-colouring P_t-free graphs in subexponential time. CoRR, 1803.05396, 2018.
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Chính T. Hoàng, Marcin Kamiński, Vadim V. Lozin, Joe Sawada, and Xiao Shu. Deciding k-Colorability of P₅-Free Graphs in Polynomial Time. Algorithmica, 57(1):74-81, 2010.
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Tereza Klimošová, Josef Malík, Tomáš Masařík, Jana Novotná, Daniël Paulusma, and Veronika Slívová. Colouring (P_r+ P_s)-Free Graphs. CoRR, 2018. URL: http://arxiv.org/abs/1804.11091v2.
http://arxiv.org/abs/1804.11091v2
Daniel Král', Jan Kratochvíl, Zsolt Tuza, and Gerhard J. Woeginger. Complexity of Coloring Graphs without Forbidden Induced Subgraphs. Proc. WG 2001, LNCS, 2204:254-262, 2001.
Jan Kratochvíl, Zsolt Tuza, and Margit Voigt. New trends in the theory of graph colorings: choosability and list coloring. Proc. DIMATIA-DIMACS Conference, 49:183-197, 1999.
Van Bang Le, Bert Randerath, and Ingo Schiermeyer. On the complexity of 4-coloring graphs without long induced paths. TCS, 389(1-2):330-335, 2007.
Daniel Leven and Zvi Galil. NP completeness of finding the chromatic index of regular graphs. Journal of Algorithms, 4(1):35-44, 1983.
László Lovász. Coverings and coloring of hypergraphs. Congr. Numer., VIII:3-12, 1973.
Daniël Paulusma. Open problems on graph coloring for special graph classes. Proc. WG 2015, LNCS, 9224:16-30, 2015.
Bert Randerath and Ingo Schiermeyer. 3-Colorability in P for P_6-free graphs. Discrete Applied Mathematics, 136(2-3):299-313, 2004.
Bert Randerath and Ingo Schiermeyer. Vertex Colouring and Forbidden Subgraphs - A Survey. Graphs and Combinatorics, 20(1):1-40, 2004.
Bert Randerath, Ingo Schiermeyer, and Meike Tewes. Three-colourability and forbidden subgraphs. II: polynomial algorithms. Discrete Mathematics, 251(1-3):137-153, 2002.
Thomas J. Schaefer. The Complexity of Satisfiability Problems. STOC, pages 216-226, 1978.
Zsolt Tuza. Graph colorings with local constraints - a survey. Discussiones Mathematicae Graph Theory, 17(2):161-228, 1997.
Gerhard J. Woeginger and Jiří Sgall. The complexity of coloring graphs without long induced paths. Acta Cybernetica, 15(1):107-117, 2001.
Tereza Klimošová, Josef Malík, Tomáš Masařík, Jana Novotná, Daniël Paulusma, and Veronika Slívová
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Use of a Pruned Modular Decomposition for Maximum Matching Algorithms on Some Graph Classes
We address the following general question: given a graph class C on which we can solve Maximum Matching in (quasi) linear time, does the same hold true for the class of graphs that can be modularly decomposed into C? As a way to answer this question for distance-hereditary graphs and some other superclasses of cographs, we study the combined effect of modular decomposition with a pruning process over the quotient subgraphs. We remove sequentially from all such subgraphs their so-called one-vertex extensions (i.e., pendant, anti-pendant, twin, universal and isolated vertices). Doing so, we obtain a "pruned modular decomposition", that can be computed in quasi linear time. Our main result is that if all the pruned quotient subgraphs have bounded order then a maximum matching can be computed in linear time. The latter result strictly extends a recent framework in (Coudert et al., SODA'18). Our work is the first to explain why the existence of some nice ordering over the modules of a graph, instead of just over its vertices, can help to speed up the computation of maximum matchings on some graph classes.
maximum matching
FPT in P
modular decomposition
pruned graphs
one-vertex extensions
P_4-structure
Mathematics of computing~Graph theory
Theory of computation~Design and analysis of algorithms
6:1-6:13
Regular Paper
This work was supported by the Institutional research programme PN 1819 "Advanced IT resources to support digital transformation processes in the economy and society - RESINFO-TD" (2018), project PN 1819-01-01 "Modeling, simulation, optimization of complex systems and decision support in new areas of IT&C research", funded by the Ministry of Research and Innovation, Romania.
https://arxiv.org/abs/1804.09407
Guillaume
Ducoffe
Guillaume Ducoffe
ICI – National Institute for Research and Development in Informatics, Bucharest, Romania , The Research Institute of the University of Bucharest ICUB, Bucharest, Romania
Alexandru
Popa
Alexandru Popa
University of Bucharest, Bucharest, Romania , ICI – National Institute for Research and Development in Informatics, Bucharest, Romania
10.4230/LIPIcs.ISAAC.2018.6
H.-J. Bandelt and H. Mulder. Distance-hereditary graphs. J. of Combinatorial Theory, Series B, 41(2):182-208, 1986.
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Guillaume Ducoffe and Alexandru Popa
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Novel Algorithm for the All-Best-Swap-Edge Problem on Tree Spanners
Given a 2-edge connected, unweighted, and undirected graph G with n vertices and m edges, a sigma-tree spanner is a spanning tree T of G in which the ratio between the distance in T of any pair of vertices and the corresponding distance in G is upper bounded by sigma. The minimum value of sigma for which T is a sigma-tree spanner of G is also called the stretch factor of T. We address the fault-tolerant scenario in which each edge e of a given tree spanner may temporarily fail and has to be replaced by a best swap edge, i.e. an edge that reconnects T-e at a minimum stretch factor. More precisely, we design an O(n^2) time and space algorithm that computes a best swap edge of every tree edge. Previously, an O(n^2 log^4 n) time and O(n^2+m log^2n) space algorithm was known for edge-weighted graphs [Bilò et al., ISAAC 2017]. Even if our improvements on both the time and space complexities are of a polylogarithmic factor, we stress the fact that the design of a o(n^2) time and space algorithm would be considered a breakthrough.
Transient edge failure
best swap edges
tree spanner
Theory of computation~Graph algorithms analysis
7:1-7:12
Regular Paper
Davide
Bilò
Davide Bilò
Department of Humanities and Social Sciences, University of Sassari, Italy
https://orcid.org/0000-0003-3169-4300
Kleitos
Papadopoulos
Kleitos Papadopoulos
InSPIRE, Agamemnonos 20, Nicosia, 1041, Cyprus
https://orcid.org/0000-0002-7086-0335
10.4230/LIPIcs.ISAAC.2018.7
Michael A. Bender and Martin Farach-Colton. The Level Ancestor Problem simplified. Theor. Comput. Sci., 321(1):5-12, 2004. URL: http://dx.doi.org/10.1016/j.tcs.2003.05.002.
http://dx.doi.org/10.1016/j.tcs.2003.05.002
Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, and Guido Proietti. A Faster Computation of All the Best Swap Edges of a Tree Spanner. In Christian Scheideler, editor, Structural Information and Communication Complexity - 22nd International Colloquium, SIROCCO 2015, Montserrat, Spain, July 14-16, 2015, Post-Proceedings, volume 9439 of Lecture Notes in Computer Science, pages 239-253. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-25258-2_17.
http://dx.doi.org/10.1007/978-3-319-25258-2_17
Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, and Guido Proietti. An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner. In Yoshio Okamoto and Takeshi Tokuyama, editors, 28th International Symposium on Algorithms and Computation, ISAAC 2017, December 9-12, 2017, Phuket, Thailand, volume 92 of LIPIcs, pages 14:1-14:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ISAAC.2017.14.
http://dx.doi.org/10.4230/LIPIcs.ISAAC.2017.14
Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, and Guido Proietti. Effective Edge-Fault-Tolerant Single-Source Spanners via Best (or Good) Swap Edges. In Shantanu Das and Sébastien Tixeuil, editors, Structural Information and Communication Complexity - 24th International Colloquium, SIROCCO 2017, Porquerolles, France, June 19-22, 2017, Revised Selected Papers, volume 10641 of Lecture Notes in Computer Science, pages 303-317. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-72050-0_18.
http://dx.doi.org/10.1007/978-3-319-72050-0_18
Davide Bilò, Luciano Gualà, and Guido Proietti. Finding Best Swap Edges Minimizing the Routing Cost of a Spanning Tree. Algorithmica, 68(2):337-357, 2014. URL: http://dx.doi.org/10.1007/s00453-012-9674-y.
http://dx.doi.org/10.1007/s00453-012-9674-y
Davide Bilò, Luciano Gualà, and Guido Proietti. A Faster Computation of All the Best Swap Edges of a Shortest Paths Tree. Algorithmica, 73(3):547-570, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9912-6.
http://dx.doi.org/10.1007/s00453-014-9912-6
Shantanu Das, Beat Gfeller, and Peter Widmayer. Computing All Best Swaps for Minimum-Stretch Tree Spanners. J. Graph Algorithms Appl., 14(2):287-306, 2010. URL: http://jgaa.info/accepted/2010/DasGfellerWidmayer2010.14.2.pdf.
http://jgaa.info/accepted/2010/DasGfellerWidmayer2010.14.2.pdf
Paola Flocchini, Antonio Mesa Enriques, Linda Pagli, Giuseppe Prencipe, and Nicola Santoro. Efficient Protocols for Computing the Optimal Swap Edges of a Shortest Path Tree. In Exploring New Frontiers of Theoretical Informatics, IFIP 18th World Computer Congress, TC1 3rd International Conference on Theoretical Computer Science (TCS2004), 22-27 August 2004, Toulouse, France, pages 153-166, 2004. URL: http://dx.doi.org/10.1007/1-4020-8141-3_14.
http://dx.doi.org/10.1007/1-4020-8141-3_14
Paola Flocchini, Antonio Mesa Enriques, Linda Pagli, Giuseppe Prencipe, and Nicola Santoro. Point-of-Failure Shortest-Path Rerouting: Computing the Optimal Swap Edges Distributively. IEICE Transactions, 89-D(2):700-708, 2006. URL: http://dx.doi.org/10.1093/ietisy/e89-d.2.700.
http://dx.doi.org/10.1093/ietisy/e89-d.2.700
Paola Flocchini, Linda Pagli, Giuseppe Prencipe, Nicola Santoro, and Peter Widmayer. Computing all the best swap edges distributively. J. Parallel Distrib. Comput., 68(7):976-983, 2008. URL: http://dx.doi.org/10.1016/j.jpdc.2008.03.002.
http://dx.doi.org/10.1016/j.jpdc.2008.03.002
Dov Harel and Robert Endre Tarjan. Fast Algorithms for Finding Nearest Common Ancestors. SIAM J. Comput., 13(2):338-355, 1984. URL: http://dx.doi.org/10.1137/0213024.
http://dx.doi.org/10.1137/0213024
Giuseppe F. Italiano and Rajiv Ramaswami. Maintaining Spanning Trees of Small Diameter. Algorithmica, 22(3):275-304, 1998. URL: http://dx.doi.org/10.1007/PL00009225.
http://dx.doi.org/10.1007/PL00009225
Hiro Ito, Kazuo Iwama, Yasuo Okabe, and Takuya Yoshihiro. Polynomial-Time Computable Backup Tables for Shortest-Path Routing. In Proc. of the 10th Intl. Colloquium Structural Information and Communication Complexity, pages 163-177, 2003.
Enrico Nardelli, Guido Proietti, and Peter Widmayer. A faster computation of the most vital edge of a shortest path. Inf. Process. Lett., 79(2):81-85, 2001. URL: http://dx.doi.org/10.1016/S0020-0190(00)00175-7.
http://dx.doi.org/10.1016/S0020-0190(00)00175-7
Seth Pettie. Sensitivity Analysis of Minimum Spanning Trees in Sub-inverse-Ackermann Time. In Proc. of the 16th Intl. Symposium on Algorithms and Computation, pages 964-973, 2005. URL: http://dx.doi.org/10.1007/11602613_96.
http://dx.doi.org/10.1007/11602613_96
Guido Proietti. Dynamic Maintenance Versus Swapping: An Experimental Study on Shortest Paths Trees. In Proc. of the 4th Intl. Workshop on Algorithm Engineering, pages 207-217, 2000. URL: http://dx.doi.org/10.1007/3-540-44691-5_18.
http://dx.doi.org/10.1007/3-540-44691-5_18
Aleksej Di Salvo and Guido Proietti. Swapping a failing edge of a shortest paths tree by minimizing the average stretch factor. Theor. Comput. Sci., 383(1):23-33, 2007. URL: http://dx.doi.org/10.1016/j.tcs.2007.03.046.
http://dx.doi.org/10.1016/j.tcs.2007.03.046
Bang Ye Wu, Chih-Yuan Hsiao, and Kun-Mao Chao. The Swap Edges of a Multiple-Sources Routing Tree. Algorithmica, 50(3):299-311, 2008. URL: http://dx.doi.org/10.1007/s00453-007-9080-z.
http://dx.doi.org/10.1007/s00453-007-9080-z
Davide Bilò and Kleitos Papadopoulos
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Efficient Enumeration of Dominating Sets for Sparse Graphs
A dominating set D of a graph G is a set of vertices such that any vertex in G is in D or its neighbor is in D. Enumeration of minimal dominating sets in a graph is one of central problems in enumeration study since enumeration of minimal dominating sets corresponds to enumeration of minimal hypergraph transversal. However, enumeration of dominating sets including non-minimal ones has not been received much attention. In this paper, we address enumeration problems for dominating sets from sparse graphs which are degenerate graphs and graphs with large girth, and we propose two algorithms for solving the problems. The first algorithm enumerates all the dominating sets for a k-degenerate graph in O(k) time per solution using O(n + m) space, where n and m are respectively the number of vertices and edges in an input graph. That is, the algorithm is optimal for graphs with constant degeneracy such as trees, planar graphs, H-minor free graphs with some fixed H. The second algorithm enumerates all the dominating sets in constant time per solution for input graphs with girth at least nine.
Enumeration algorithm
polynomial amortized time
dominating set
girth
degeneracy
Mathematics of computing~Graph algorithms
8:1-8:13
Regular Paper
Kazuhiro
Kurita
Kazuhiro Kurita
IST, Hokkaido University, Sapporo, Japan
Kunihiro
Wasa
Kunihiro Wasa
National Institute of Informatics, Tokyo, Japan
https://orcid.org/0000-0001-9822-6283
Hiroki
Arimura
Hiroki Arimura
IST, Hokkaido University, Sapporo, Japan
Takeaki
Uno
Takeaki Uno
National Institute of Informatics, Tokyo, Japan
10.4230/LIPIcs.ISAAC.2018.8
David Avis and Komei Fukuda. Reverse search for enumeration. Discrete Appl. Math., 65(1):21-46, 1996.
Etienne Birmelé, Rui A. Ferreira, Roberto Grossi, Andrea Marino, Nadia Pisanti, Romeo Rizzi, and Gustavo Sacomoto. Optimal Listing of Cycles and st-Paths in Undirected Graphs. In Proc. SODA 2013 ACM, pages 1884-1896, 2013.
Endre Boros, Khaled Elbassioni, Vladimir Gurvich, and Leonid Khachiyan. Generating maximal independent sets for hypergraphs with bounded edge-intersections. In Proc. LATIN 2004, pages 488-498. Springer, 2004.
Boštjan Brešar, Michael A Henning, and Douglas F Rall. RAINBOW DOMINATION IN GRAPHS. Taiwanese J. Math., 12(1):213-225, 2008.
L Sunil Chandran and CR Subramanian. Girth and treewidth. J. Combin. Theory Ser. B, 93(1):23-32, 2005.
Alessio Conte, Roberto Grossi, Andrea Marino, and Luca Versari. Sublinear-Space Bounded-Delay Enumeration for Massive Network Analytics: Maximal Cliques. In Proc. ICALP 2016, volume 55 of LIPIcs, pages 148:1-148:15. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.148.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.148
Bruno Courcelle. Linear delay enumeration and monadic second-order logic. Discrete Appl. Math., 157(12):2675-2700, 2009.
Nadia Creignou, Markus Kröll, Reinhard Pichler, Sebastian Skritek, and Heribert Vollmer. On the Complexity of Hard Enumeration Problems. In Proc. LATA 2017, volume 10168 of LNCS, pages 183-195. Springer, 2017.
Jean-Alexandre Anglès d’Auriac, Csilia Bujtás, Hakim El Maftouhi, Marek Karpinski, Yannis Manoussakis, Leandro Montero, Narayanan Narayanan, Laurent Rosaz, Johan Thapper, and Zsolt Tuza. Tropical Dominating Sets in Vertex-Coloured Graphs. In Proc. WALCOM 2016, pages 17-27. Springer, 2016.
Thomas Eiter, Georg Gottlob, and Kazuhisa Makino. New Results on Monotone Dualization and Generating Hypergraph Transversals. SIAM J. Comput., 32(2):514-537, 2003. URL: http://dx.doi.org/10.1137/S009753970240639X.
http://dx.doi.org/10.1137/S009753970240639X
David Eppstein, Maarten Löffler, and Darren Strash. Listing All Maximal Cliques in Large Sparse Real-World Graphs. J. Exp. Algorithmics, 18:3.1:3.1-3.1:3.21, November 2013. URL: http://dx.doi.org/10.1145/2543629.
http://dx.doi.org/10.1145/2543629
Michael R. Garey and David S. Johnson. Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman &Co., New York, NY, USA, 1990.
Petr A Golovach, Pinar Heggernes, Mamadou M Kanté, Dieter Kratsch, and Yngve Villanger. Enumerating minimal dominating sets in chordal bipartite graphs. Discrete Appl. Math., 199(30):30-36, 2016.
Petr A Golovach, Pinar Heggernes, Mamadou Moustapha Kanté, Dieter Kratsch, Sigve H Sæther, and Yngve Villanger. Output-Polynomial Enumeration on Graphs of Bounded (Local) Linear MIM-Width. Algorithmica, 80(2):714-741, 2018.
Petr A. Golovach, Pinar Heggernes, Dieter Kratsch, and Yngve Villanger. An Incremental Polynomial Time Algorithm to Enumerate All Minimal Edge Dominating Sets. Algorithmica, 72(3):836-859, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9875-7.
http://dx.doi.org/10.1007/s00453-014-9875-7
Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, and Lhouari Nourine. Enumeration of Minimal Dominating Sets and Variants. In Proc. FCT 2011, pages 298-309. Springer, 2011.
Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, and Lhouari Nourine. On the Neighbourhood Helly of Some Graph Classes and Applications to the Enumeration of Minimal Dominating Sets. In Proc. ISAAC 2012, volume 7676, pages 289-298. Springer, 2012.
Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, and Lhouari Nourine. On the Enumeration of Minimal Dominating Sets and Related Notions. SIAM J. Discrete Math., 28(4):1916-1929, 2014.
Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, Lhouari Nourine, and Takeaki Uno. A polynomial delay algorithm for enumerating minimal dominating sets in chordal graphs. In Proc. WG 2015, pages 138-153. Springer, 2015.
Mamadou Moustapha Kanté, Vincent Limouzy, Arnaud Mary, Lhouari Nourine, and Takeaki Uno. Polynomial Delay Algorithm for Listing Minimal Edge Dominating Sets in Graphs. In Proc. WADS 2015, volume 9214 of LNCS, pages 446-457. Springer Berlin Heidelberg, 2015.
E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan. Generating All Maximal Independent Sets: NP-Hardness and Polynomial-Time Algorithms. SIAM J. Comput., 9(3):558-565, 1980. URL: http://dx.doi.org/10.1137/0209042.
http://dx.doi.org/10.1137/0209042
Don R Lick and Arthur T White. k-DEGENERATE GRAPHS. Canadian J. Math., 22:1082-1096, 1970.
Kazuhisa Makino and Takeaki Uno. New Algorithms for Enumerating All Maximal Cliques. In Proc. SWAT 2004, volume 3111 of LNCS, pages 260-272. Springer, 2004.
David W Matula and Leland L Beck. Smallest-last ordering and clustering and graph coloring algorithms. J. ACM, 30(3):417-427, 1983.
Akiyoshi Shioura, Akihisa Tamura, and Takeaki Uno. An Optimal Algorithm for Scanning All Spanning Trees of Undirected Graphs. SIAM J. Comput., 26(3):678-692, 1997.
Andrew Thomason. The Extremal Function for Complete Minors. Journal of Combinatorial Theory, Series B, 81(2):318-338, 2001.
Kunihiro Wasa, Hiroki Arimura, and Takeaki Uno. Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph. In Proc. ISAAC 2014, volume 8889 of LNCS, pages 94-102. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-13075-0_8.
http://dx.doi.org/10.1007/978-3-319-13075-0_8
Kazuhiro Kurita, Kunihiro Wasa, Hiroki Arimura, and Takeaki Uno
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Complexity of Unordered CNF Games
The classic TQBF problem is to determine who has a winning strategy in a game played on a given CNF formula, where the two players alternate turns picking truth values for the variables in a given order, and the winner is determined by whether the CNF gets satisfied. We study variants of this game in which the variables may be played in any order, and each turn consists of picking a remaining variable and a truth value for it.
- For the version where the set of variables is partitioned into two halves and each player may only pick variables from his/her half, we prove that the problem is PSPACE-complete for 5-CNFs and in P for 2-CNFs. Previously, it was known to be PSPACE-complete for unbounded-width CNFs (Schaefer, STOC 1976).
- For the general unordered version (where each variable can be picked by either player), we also prove that the problem is PSPACE-complete for 5-CNFs and in P for 2-CNFs. Previously, it was known to be PSPACE-complete for 6-CNFs (Ahlroth and Orponen, MFCS 2012) and PSPACE-complete for positive 11-CNFs (Schaefer, STOC 1976).
CNF
Games
PSPACE-complete
SAT
Linear Time
Theory of computation~Computational complexity and cryptography
Theory of computation~Problems, reductions and completeness
Theory of computation
9:1-9:12
Regular Paper
This work was supported by NSF grant CCF-1657377.
https://eccc.weizmann.ac.il/report/2018/039/
Md Lutfar
Rahman
Md Lutfar Rahman
The University of Memphis, Memphis, TN, USA
Thomas
Watson
Thomas Watson
The University of Memphis, Memphis, TN, USA
10.4230/LIPIcs.ISAAC.2018.9
Lauri Ahlroth and Pekka Orponen. Unordered Constraint Satisfaction Games. In Proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 64-75. Springer, 2012.
Bengt Aspvall, Michael Plass, and Robert Tarjan. A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas. Information Processing Letters, 8(3):121-123, 1979.
Kyle Burke, Erik Demaine, Harrison Gregg, Robert Hearn, Adam Hesterberg, Michael Hoffmann, Hiro Ito, Irina Kostitsyna, Jody Leonard, Maarten Löffler, Aaron Santiago, Christiane Schmidt, Ryuhei Uehara, Yushi Uno, and Aaron Williams. Single-Player and Two-Player Buttons & Scissors Games. In Proceedings of the 18th Japan Conference on Discrete and Computational Geometry and Graphs (JCDCGG), pages 60-72. Springer, 2015.
William Burley and Sandy Irani. On Algorithm Design for Metrical Task Systems. Algorithmica, 18(4):461-485, 1997.
Jesper Byskov. Maker-Maker and Maker-Breaker Games Are PSPACE-Complete. Technical Report RS-04-14, BRICS, Department of Computer Science, Aarhus University, 2004.
Chris Calabro. 2-TQBF Is in P, 2008. Unpublished. URL: https://cseweb.ucsd.edu/~ccalabro/essays/complexity_of_2tqbf.pdf.
https://cseweb.ucsd.edu/~ccalabro/essays/complexity_of_2tqbf.pdf
Aviezri Fraenkel and Elisheva Goldschmidt. PSPACE-hardness of some combinatorial games. Journal of Combinatorial Theory, Series A, 46(1):21-38, 1987.
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Md. Lutfar Rahman and Thomas Watson
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Half-Duplex Communication Complexity
Suppose Alice and Bob are communicating in order to compute some function f, but instead of a classical communication channel they have a pair of walkie-talkie devices. They can use some classical communication protocol for f where in each round one player sends a bit and the other one receives it. The question is whether talking via walkie-talkie gives them more power? Using walkie-talkies instead of a classical communication channel allows players two extra possibilities: to speak simultaneously (but in this case they do not hear each other) and to listen at the same time (but in this case they do not transfer any bits). The motivation for this kind of a communication model comes from the study of the KRW conjecture. We show that for some definitions this non-classical communication model is, in fact, more powerful than the classical one as it allows to compute some functions in a smaller number of rounds. We also prove lower bounds for these models using both combinatorial and information theoretic methods.
communication complexity
half-duplex channel
information theory
Theory of computation~Communication complexity
10:1-10:12
Regular Paper
https://eccc.weizmann.ac.il/report/2018/089
Kenneth
Hoover
Kenneth Hoover
University of California San Diego, USA
Russell
Impagliazzo
Russell Impagliazzo
University of California San Diego, USA
Ivan
Mihajlin
Ivan Mihajlin
University of California San Diego, USA
Alexander V.
Smal
Alexander V. Smal
St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Russia
10.4230/LIPIcs.ISAAC.2018.10
Thomas M. Cover and Joy A. Thomas. Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing). Wiley-Interscience, New York, USA, 2006.
Jeff Edmonds, Russell Impagliazzo, Steven Rudich, and Jirí Sgall. Communication complexity towards lower bounds on circuit depth. Computational Complexity, 10(3):210-246, 2001. URL: http://dx.doi.org/10.1007/s00037-001-8195-x.
http://dx.doi.org/10.1007/s00037-001-8195-x
Klim Efremenko, Gillat Kol, and Raghuvansh Saxena. Interactive Coding Over the Noisy Broadcast Channel. Electronic Colloquium on Computational Complexity (ECCC), 24:93, 2017. URL: https://eccc.weizmann.ac.il/report/2017/093.
https://eccc.weizmann.ac.il/report/2017/093
Dmitry Gavinsky, Or Meir, Omri Weinstein, and Avi Wigderson. Toward Better Formula Lower Bounds: The Composition of a Function and a Universal Relation. SIAM J. Comput., 46(1):114-131, 2017. URL: http://dx.doi.org/10.1137/15M1018319.
http://dx.doi.org/10.1137/15M1018319
Johan Håstad and Avi Wigderson. Composition of the Universal Relation. In Jin-Yi Cai, editor, Advances In Computational Complexity Theory, Proceedings of a DIMACS Workshop, New Jersey, USA, December 3-7, 1990, volume 13 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 119-134. DIMACS/AMS, 1990. URL: http://dimacs.rutgers.edu/Volumes/Vol13.html.
http://dimacs.rutgers.edu/Volumes/Vol13.html
Kenneth Hoover, Russell Impagliazzo, Ivan Mihajlin, and Alexander Smal. Half-duplex communication complexity. Electronic Colloquium on Computational Complexity (ECCC), 25:89, 2018. URL: https://eccc.weizmann.ac.il/report/2018/089.
https://eccc.weizmann.ac.il/report/2018/089
Mauricio Karchmer, Ran Raz, and Avi Wigderson. Super-Logarithmic Depth Lower Bounds Via the Direct Sum in Communication Complexity. Computational Complexity, 5(3/4):191-204, 1995. URL: http://dx.doi.org/10.1007/BF01206317.
http://dx.doi.org/10.1007/BF01206317
Mauricio Karchmer and Avi Wigderson. Monotone Circuits for Connectivity Require Super-logarithmic Depth. In Janos Simon, editor, Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2-4, 1988, Chicago, Illinois, USA, pages 539-550. ACM, 1988. URL: http://dx.doi.org/10.1145/62212.62265.
http://dx.doi.org/10.1145/62212.62265
V. M. Khrapchenko. A method of obtaining lower bounds for the complexity of π-schemes. Mathematical Notes Academy of Sciences USSR, 10:474-479, 1972.
Eyal Kushilevitz and Noam Nisan. Communication complexity. Cambridge University Press, 1997.
Andrew Chi-Chih Yao. Some Complexity Questions Related to Distributive Computing(Preliminary Report). In Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC '79, pages 209-213, New York, NY, USA, 1979. ACM. URL: http://dx.doi.org/10.1145/800135.804414.
http://dx.doi.org/10.1145/800135.804414
Kenneth Hoover, Russell Impagliazzo, Ivan Mihajlin, and Alexander V. Smal
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On the Complexity of Stable Fractional Hypergraph Matching
In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system. Aharoni and Fleiner proved that there exists a stable fractional matching in every hypergraphic preference system. Furthermore, Kintali, Poplawski, Rajaraman, Sundaram, and Teng proved that the problem of finding a stable fractional matching in a hypergraphic preference system is PPAD-complete. In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is bounded by some constant. The proof by Kintali, Poplawski, Rajaraman, Sundaram, and Teng implies the PPAD-completeness of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is 5. In this paper, we prove that (i) this problem is PPAD-complete even if the maximum degree is 3, and (ii) if the maximum degree is 2, then this problem can be solved in polynomial time. Furthermore, we prove that the problem of finding an approximate stable fractional matching in a hypergraphic preference system is PPAD-complete.
fractional hypergraph matching
stable matching
PPAD-completeness
Theory of computation~Design and analysis of algorithms
11:1-11:12
Regular Paper
Takashi
Ishizuka
Takashi Ishizuka
Graduate School of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
Naoyuki
Kamiyama
Naoyuki Kamiyama
Institute of Mathematics for Industry, Kyushu University, JST, PRESTO, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
This research was supported by JST PRESTO Grant Number JPMJPR1753, Japan.
10.4230/LIPIcs.ISAAC.2018.11
R. Aharoni and T. Fleiner. On a lemma of Scarf. Journal of Combinatorial Theory, Series B, 87(1):72-80, 2003.
P. Biró and T. Fleiner. Fractional solutions for capacitated NTU-games, with applications to stable matchings. Discrete Optimization, 22:241-254, 2016.
P. Biró, T. Fleiner, and R. W. Irving. Matching couples with Scarf’s algorithm. Annals of Mathematics and Artificial Intelligence, 77(3-4):303-316, 2016.
P. Biró and F. Klijn. MATCHING WITH COUPLES: A MULTIDISCIPLINARY SURVEY. International Game Theory Review, 15(02):1340008, 2013.
X. Chen, X. Deng, and S.-H. Teng. Settling the Complexity of Computing Two-player Nash Equilibria. Journal of the ACM, 56(3):14:1-14:57, 2009.
C. Daskalakis, P. W. Goldberg, and C. H. Papadimitriou. The Complexity of Computing a Nash Equilibrium. SIAM Journal on Computing, 39(1):195-259, 2009.
D. Gale and L. S. Shapley. College Admissions and the Stability of Marriage. The American Mathematical Monthly, 69(1):9-15, 1962.
D. Gusfield and R. W. Irving. The Stable Marriage Problem: Structure and Algorithm. MIT Press, 1989.
L. G. Khachiyan. Polynomial algorithms in linear programming. USSR Computational Mathematics and Mathematical Physics, 20(1):53-72, 1980.
S. Kintali, L. J. Poplawski, R. Rajaraman, R. Sundaram, and S.-H. Teng. Reducibility among Fractional Stability Problems. SIAM Journal on Computing, 42(6):2063-2113, 2013.
N. Megiddo and C. H. Papadimitriou. On total functions, existence theorems and computational complexity. Theoretical Computer Science, 81(2):317-324, 1991.
J. Nash. Equilibrium points in n-person games. Proceedings of the National Academy of Science, 36(1):48-49, 1950.
T. Nguyen and R. Vohra. Near Feasible Stable Matchings. In Proceedings of the 16th ACM Conference on Economics and Computation, pages 41-42, 2015.
C. H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Sciences, 48(3):498-532, 1994.
C. H. Papadimitriou. The Complexity of Finding Nash Equilibria. In N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani, editors, Algorithmic game theory, pages 29-52. Cambridge university press, 2007.
H. E. Scarf. The Core of an N Person Game. Econometrica, 35(1):50-69, 1967.
Takashi Ishizuka and Naoyuki Kamiyama
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Deciding the Closure of Inconsistent Rooted Triples Is NP-Complete
Interpreting three-leaf binary trees or rooted triples as constraints yields an entailment relation, whereby binary trees satisfying some rooted triples must also thus satisfy others, and thence a closure operator, which is known to be polynomial-time computable. This is extended to inconsistent triple sets by defining that a triple is entailed by such a set if it is entailed by any consistent subset of it.
Determining whether the closure of an inconsistent rooted triple set can be computed in polynomial time was posed as an open problem in the Isaac Newton Institute's "Phylogenetics" program in 2007. It appears (as NC4) in a collection of such open problems maintained by Mike Steel, and it is the last of that collection's five problems concerning computational complexity to have remained open. We resolve the complexity of computing this closure, proving that its decision version is NP-Complete.
In the process, we also prove that detecting the existence of any acyclic B-hyperpath (from specified source to destination) is NP-Complete, in a significantly narrower special case than the version whose minimization problem was recently proven NP-hard by Ritz et al. This implies it is NP-hard to approximate (our special case of) their minimization problem to within any factor.
phylogenetic trees
rooted triple entailment
NP-Completeness
directed hypergraphs
acyclic induced subgraphs
computational complexity
Mathematics of computing~Trees
Mathematics of computing~Hypergraphs
Theory of computation~Problems, reductions and completeness
Applied computing~Molecular evolution
12:1-12:13
Regular Paper
Matthew P.
Johnson
Matthew P. Johnson
Department of Computer Science, Lehman College, Ph.D. Program in Computer Science, The Graduate Center, City University of New York, USA
10.4230/LIPIcs.ISAAC.2018.12
Alfred V. Aho, Yehoshua Sagiv, Thomas G. Szymanski, and Jeffrey D. Ullman. Inferring a tree from lowest common ancestors with an application to the optimization of relational expressions. SIAM Journal on Computing, 10(3):405-421, 1981.
Giorgio Ausiello, Roberto Giaccio, Giuseppe F Italiano, and Umberto Nanni. Optimal traversal of directed hypergraphs. Technical Report TR-92-073, International Computer Science Institute, Berkeley, CA, September 1992.
Jørgen Bang-Jensen, Frédéric Havet, and Nicolas Trotignon. Finding an induced subdivision of a digraph. Theoretical Computer Science, 443:10-24, 2012.
Dan Bienstock. On the complexity of testing for odd holes and induced odd paths. Discrete Mathematics, 90(1):85-92, 1991.
David Bryant. Building Trees, Hunting for Trees, and Comparing Trees: Theory and Methods in Phylogenetic Analysis. PhD thesis, University of Canterbury, 1997.
David Bryant and Mike Steel. Extension operations on sets of leaf-labeled trees. Advances in Applied Mathematics, 16(4):425-453, 1995.
Giorgio Gallo, Giustino Longo, Stefano Pallottino, and Sang Nguyen. Directed hypergraphs and applications. Discrete Applied Mathematics, 42(2-3):177-201, 1993.
Oded Goldreich. On promise problems: A survey. In Theoretical Computer Science: Essays in Memory of Shimon Even, pages 254-290. Springer, 2006.
Daniel Huson, Vincent Moulton, and Mike Steel. Final Report for the `Phylogenetics' Programme. Technical report, Isaac Newton Institute for Mathematical Sciences, February 2008.
Lars Relund Nielsen, Daniele Pretolani, and K Andersen. A remark on the definition of a B-hyperpath. Technical report, Department of Operations Research, University of Aarhus, 2001.
Anna Ritz, Brendan Avent, and T Murali. Pathway analysis with signaling hypergraphs. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 14(5):1042-1055, September 2017.
Charles Semple and Mike Steel. Phylogenetics, volume 24 of Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, 2003.
Mike Steel. Phylogenetics: Challenges and Conjectures, updated in July 2018. URL: http://www.math.canterbury.ac.nz/~m.steel/Non_UC/files/research/conjectures_updated.pdf.
http://www.math.canterbury.ac.nz/~m.steel/Non_UC/files/research/conjectures_updated.pdf
Mayur Thakur and Rahul Tripathi. Linear connectivity problems in directed hypergraphs. Theoretical Computer Science, 410(27-29):2592-2618, 2009.
Matthew P. Johnson
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Computing Vertex-Disjoint Paths in Large Graphs Using MAOs
We consider the problem of computing k in N internally vertex-disjoint paths between special vertex pairs of simple connected graphs. For general vertex pairs, the best deterministic time bound is, since 42 years, O(min{k,sqrt{n}}m) for each pair by using traditional flow-based methods.
The restriction of our vertex pairs comes from the machinery of maximal adjacency orderings (MAOs). Henzinger showed for every MAO and every 1 <= k <= delta (where delta is the minimum degree of the graph) the existence of k internally vertex-disjoint paths between every pair of the last delta-k+2 vertices of this MAO. Later, Nagamochi generalized this result by using the machinery of mixed connectivity. Both results are however inherently non-constructive.
We present the first algorithm that computes these k internally vertex-disjoint paths in linear time O(m), which improves the previously best time O(min{k,sqrt{n}}m). Due to the linear running time, this algorithm is suitable for large graphs. The algorithm is simple, works directly on the MAO structure, and completes a long history of purely existential proofs with a constructive method. We extend our algorithm to compute several other path systems and discuss its impact for certifying algorithms.
Computing Disjoint Paths
Large Graphs
Vertex-Connectivity
Linear-Time
Maximal Adjacency Ordering
Maximum Cardinality Search
Big Data
Certifying Algorithm
Mathematics of computing~Graph theory
Mathematics of computing~Graph algorithms
13:1-13:12
Regular Paper
This research is supported by the grant SCHM 3186/1-1 (270450205) from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).
Johanna E.
Preißer
Johanna E. Preißer
Institute of Mathematics, TU Ilmenau, Germany
Jens M.
Schmidt
Jens M. Schmidt
Institute of Mathematics, TU Ilmenau, Germany
10.4230/LIPIcs.ISAAC.2018.13
S. R. Arikati and K. Mehlhorn. A correctness certificate for the Stoer-Wagner min-cut algorithm. Information Processing Letters, 70(5):251-254, 1999.
R. Diestel. Graph Theory. Springer, fourth edition, 2010.
S. Even and R. E. Tarjan. Network Flow and Testing Graph Connectivity. SIAM Journal on Computing, 4(4):507-518, 1975.
A. Frank. On the edge-connectivity algorithm of Nagamochi and Ibaraki. Laboratoire Artemis, IMAG, Université J. Fourier, Grenoble, March 1994.
Z. Galil. Finding the vertex connectivity of graphs. SIAM Journal on Computing, 9(1):197-199, 1980.
M. Rauch Henzinger. A Static 2-Approximation Algorithm for Vertex Connectivity and Incremental Approximation Algorithms for Edge and Vertex Connectivity. Journal of Algorithms, 24:194-220, 1997.
A. V. Karzanov. O nakhozhdenii maksimal'nogo potoka v setyakh spetsial'nogo vida i nekotorykh prilozheniyakh (in Russian; On finding a maximum flow in a network with special structure and some applications). Matematicheskie Voprosy Upravleniya Proizvodstvom, 5:81-94, 1973.
N. Linial, L. Lovász, and A. Wigderson. Rubber bands, convex embeddings and graph connectivity. Combinatorica, 8(1):91-102, 1988.
W. Mader. Existenz gewisser Konfigurationen in n-gesättigten Graphen und in Graphen genügend großer Kantendichte. Mathematische Annalen, 194:295-312, 1971.
W. Mader. Grad und lokaler Zusammenhang in endlichen Graphen. Mathematische Annalen, 205:9-11, 1973.
R. M. McConnell, K. Mehlhorn, S. Näher, and P. Schweitzer. Certifying algorithms. Computer Science Review, 5(2):119-161, 2011.
K. Menger. Zur allgemeinen Kurventheorie. Fundamenta Mathematicae, 10:96-115, 1927.
H. Nagamochi. Sparse connectivity certificates via MA orderings in graphs. Discrete Applied Mathematics, 154(16):2411-2417, 2006.
H. Nagamochi and T. Ibaraki. Computing Edge-Connectivity in Multigraphs and Capacitated Graphs. SIAM Journal on Discrete Mathematics, 5(1):54-66, 1992.
H. Nagamochi and T. Ibaraki. Algorithmic Aspects of Graph Connectivity. Cambridge University Press, 2008.
J. M. Schmidt. Contractions, Removals and Certifying 3-Connectivity in Linear Time. SIAM Journal on Computing, 42(2):494-535, 2013.
M. Stoer and F. Wagner. A simple min-cut algorithm. Journal of the ACM, 44(4):585-591, 1997.
R. E. Tarjan and M. Yannakakis. Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal on Computing, 13(3):566-579, 1984.
H. Whitney. Non-separable and planar graphs. Transactions of the American Mathematical Society, 34(1):339-362, 1932.
Johanna E. Preißer and Jens M. Schmidt
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
An O(n^2 log^2 n) Time Algorithm for Minmax Regret Minsum Sink on Path Networks
We model evacuation in emergency situations by dynamic flow in a network. We want to minimize the aggregate evacuation time to an evacuation center (called a sink) on a path network with uniform edge capacities. The evacuees are initially located at the vertices, but their precise numbers are unknown, and are given by upper and lower bounds. Under this assumption, we compute a sink location that minimizes the maximum "regret." We present the first sub-cubic time algorithm in n to solve this problem, where n is the number of vertices. Although we cast our problem as evacuation, our result is accurate if the "evacuees" are fluid-like continuous material, but is a good approximation for discrete evacuees.
Facility location
minsum sink
evacuation problem
minmax regret
dynamic flow path network
Networks~Network algorithms
Mathematics of computing~Graph algorithms
Applied computing~Transportation
14:1-14:13
Regular Paper
https://arxiv.org/abs/1806.00814
Binay
Bhattacharya
Binay Bhattacharya
School of Computing Science, Simon Fraser University, Burnaby, Canada
Yuya
Higashikawa
Yuya Higashikawa
School of Business Administration, University of Hyogo, Kobe, Japan
Tsunehiko
Kameda
Tsunehiko Kameda
School of Computing Science, Simon Fraser University, Burnaby, Canada
Naoki
Katoh
Naoki Katoh
School of Science and Technology, Kwansei Gakuin University, Sanda, Japan
10.4230/LIPIcs.ISAAC.2018.14
Guru Prakash Arumugam, John Augustine, Mordecai Golin, and Prashanth Srikanthan. A Polynomial Time Algorithm for Minimax-Regret Evacuation on a Dynamic Path. arXiv:1404,5448 v1 [cs.DS] 22 April, 2014.
R. Benkoczi, B. Bhattacharya, Y. Higashikawa, T. Kameda, and N. Katoh. Minsum k-sink on dynamic flow path network. In Combinatorial Algorithms (Iliopoulos, Costas, Leong, Hon Wai, Sung, Wing-Kin, Eds.), Springer-Verlag LNCS 110979, pages 78-89, 2018.
Binay Bhattacharya, Yuya Higashikawa, Tiko Kameda, and Naoki Katoh. Minmax regret 1-sink for aggregate evacuation time on path networks. arXiv:1806.00814 Oct 2018.
Binay Bhattacharya and Tsunehiko Kameda. Improved algorithms for computing minmax regret sinks on path and tree networks. Theoretical Computer Science, 607:411-425, November 2015.
Siu-Wing Cheng, Yuya Higashikawa, Naoki Katoh, Guanqun Ni, Bing Su, and Yinfeng Xu. Minimax regret 1-sink location problem in dynamic path networks. In Proc. Annual Conf. on Theory and Applications of Models of Computation (T-H.H. Chan, L.C. Lau, and L. Trevisan, Eds.), Springer-Verlag, LNCS 7876, pages 121-132, 2013.
L.R. Ford and D.R. A. Fulkerson. Constructing maximal dynamic flows from static flows. Operations Research, 6(3):419-433, 1958.
Mordecai Golin and Sai Sandeep. Minmax-regret k-sink location on a dynamic tree network with uniform capacities. arXiv:1806.03814v1 [cs.DS] 11 June, 2018.
H.W. Hamacher and S.A. Tjandra. Mathematical modelling of evacuation problems: a state of the art. in: Pedestrian and Evacuation Dynamics, Springer Verlag,, pages 227-266, 2002.
J. Hershberger. Finding the upper envelope of n line segments in O(nlog n) time. Information Processing Letters, 33(4):169-174, 1989.
Yuya Higashikawa, John Augustine, Siu-Wing Cheng, Mordecai J. Golin, Naoki Katoh, Guanqun Ni, Bing Su, and Yinfeng Xu. Minimax regret 1-sink location problem in dynamic path networks. Theoretical Computer Science, 588(11):24-36, 2015.
Yuya Higashikawa, Siu-Wing Cheng, Tsunehiko Kameda, Naoki Katoh, and Shun Saburi. Minimax regret 1-median problem in dynamic path networks. Theory of Computing Systems, 62(6):1392-1408, August 2018.
Yuya Higashikawa, Mordecai J. Golin, and Naoki Katoh. Minimax regret sink location problem in dynamic tree networks with uniform capacity. Journal of Graph Algorithms and Applications, 18(4):539-555, 2014.
Yuya Higashikawa, Mordecai J. Golin, and Naoki Katoh. Multiple sink location problems in dynamic path networks. Theoretical Computer Science, 607(1):2-15, 2015.
Oded Kariv and S.Louis Hakimi. An algorithmic approach to network location problems, Part II: The p-median. SIAM J. Appl. Math., 37:539-560, 1979.
P. Kouvelis and G. Yu. Robust Discrete Optimization and its Applications. Kluwer Academic Publishers, London, 1997.
Satoko Mamada, Takeaki Uno, Kazuhisa Makino, and Satoru Fujishige. An O(nlog² n) algorithm for a sink location problem in dynamic tree networks. Discrete Applied Mathematics, 154:2387-2401, 2006.
Haitao Wang. Minmax Regret 1-Facility Location on Uncertain Path Networks. European J. of Operational Research, 239(3):636-643, 2014.
Binay Bhattacharya, Yuya Higashikawa, Tsunehiko Kameda, and Naoki Katoh
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Computing Optimal Shortcuts for Networks
We study augmenting a plane Euclidean network with a segment, called shortcut, to minimize the largest distance between any two points along the edges of the resulting network. Questions of this type have received considerable attention recently, mostly for discrete variants of the problem. We study a fully continuous setting, where all points on the network and the inserted segment must be taken into account. We present the first results on the computation of optimal shortcuts for general networks in this model, together with several results for networks that are paths, restricted to two types of shortcuts: shortcuts with a fixed orientation and simple shortcuts.
graph augmentation
shortcut
diameter
geometric graph
Theory of computation~Computational geometry
15:1-15:12
Regular Paper
D. G. and R. S. were supported by project MTM2015-63791-R. R. S. was also supported by Gen. Cat. 2017SGR1640 and MINECO through the Ramón y Cajal program. A. M. was supported by project BFU2016-74975-P.
https://arxiv.org/abs/1807.10093
Delia
Garijo
Delia Garijo
Departamento de Matemática Aplicada I, Universidad de Sevilla, Spain
Alberto
Márquez
Alberto Márquez
Departamento de Matemática Aplicada I, Universidad de Sevilla, Spain
Natalia
Rodríguez
Natalia Rodríguez
Departamento de Computación, Universidad de Buenos Aires, Argentina
Rodrigo I.
Silveira
Rodrigo I. Silveira
Departament de Matemàtiques, Universitat Politècnica de Catalunya, Spain
10.4230/LIPIcs.ISAAC.2018.15
S. W. Bae, M. de Berg, O. Cheong, J. Gudmundsson, and C. Levcopoulos. Shortcuts for the Circle. In Proceedings ISAAC 2017, pages 9:1-9:13, 2017.
M. A. Bender, M. Farach-Colton, G. Pemmasani, S. Skiena, and P. Sumazin. Lowest common ancestors in trees and directed acyclic graphs. J. Algorithms, 57(2):75-94, 2005.
O. Berkman and U. Vishkin. Recursive Star-Tree Parallel Data Structure. SIAM J. Comput., 22(2):221-242, 1993.
G. S. Brodal and R. Jacob. Dynamic Planar Convex Hull. In Proceedings FOCS '02, pages 617-626, Washington, DC, USA, 2002. IEEE Computer Society.
J. Cáceres, D. Garijo, A. González, A. Márquez, M. L. Puertas, and P. Ribeiro. Shortcut sets for the locus of plane Euclidean networks. Appl. Math. Comp., 334:192-205, 2018.
J. L. De Carufel, C. Grimm, A. Maheshwari, and M. Smid. Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts. In Proceedings SWAT 2016, pages 27:1-27:14, 2016.
J. L. De Carufel, C. Grimm, S. Schirra, and M. Smid. Minimizing the Continuous Diameter when Augmenting a Tree with a Shortcut. In Algorithms and Data Structures. WADS 2017. LNCS 10389, pages 301-312, 2017.
C. E. Chen and R. S. Garfinkel. The generalized diameter of a graph. Networks, 12:335-340, 1982.
M. de Berg, O. Cheong, M. J. van Kreveld, and M. H. Overmars. Computational geometry: algorithms and applications, 3rd Edition. Springer, 2008.
Greg N. Frederickson. Fast Algorithms for Shortest Paths in Planar Graphs, with Applications. SIAM J. Comput., 16(6):1004-1022, December 1987.
K. Y. Fung, T. M. Nicholl, R. E. Tarjan, and C. J. Van Wyk. Simplified linear-time jordan sorting and polygon clipping. Information Processing Letters, 35(2):85-92, 1990.
D. Garijo, A. Márquez, N. Rodríguez, and R. I. Silveira. Computing optimal shortcuts for networks. Preprint, 2018. https://arxiv.org/abs/1807.10093.
P. Hansen, M. Labbé, and B. Nicolas. The continuous center set of a network. Discrete Applied Mathematics, 30:181-195, 1991.
F. Hurtado and C. D. Tóth. Plane Geometric Graph Augmentation: A Generic Perspective. In Pach J. (eds) Thirty Essays on Geometric Graph Theory, pages 327-354. Springer, 2013.
B. Yang. Euclidean Chains and Their Shortcuts. Theor. Comput. Sci., 497:55-67, July 2013.
H. Yang and M. G. H. Bell. Models and algorithms for road network design: a review and some new developments. Transport Reviews, 18(3):257-278, 1998.
Delia Garijo, Alberto Márquez, Natalia Rodríguez, and Rodrigo I. Silveira
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Algorithmic Channel Design
Payment networks, also known as channels, are a most promising solution to the throughput problem of cryptocurrencies. In this paper we study the design of capital-efficient payment networks, offline as well as online variants. We want to know how to compute an efficient payment network topology, how capital should be assigned to the individual edges, and how to decide which transactions to accept. Towards this end, we present a flurry of interesting results, basic but generally applicable insights on the one hand, and hardness results and approximation algorithms on the other hand.
blockchain
payment channels
layer 2 solution
network design
payment hubs
routing
Theory of computation~Graph algorithms analysis
16:1-16:12
Regular Paper
https://github.com/zetavar/Channel-Network-Design/blob/master/algorithmic-channel-design.pdf
Georgia
Avarikioti
Georgia Avarikioti
ETH Zurich, Switzerland
Yuyi
Wang
Yuyi Wang
ETH Zurich, Switzerland
Roger
Wattenhofer
Roger Wattenhofer
ETH Zurich, Switzerland
10.4230/LIPIcs.ISAAC.2018.16
Ethereum white paper. URL: https://github.com/ethereum/wiki/wiki/White-Paper.
https://github.com/ethereum/wiki/wiki/White-Paper
Raiden network, 2017. URL: http://raiden.network/.
http://raiden.network/
Georgia Avarikioti, Gerrit Janssen, Yuyi Wang, and Roger Wattenhofer. Payment Network Design with Fees, 2018. URL: https://github.com/zetavar/Payment-Network-Design-with-Fees/blob/master/Payment_Network_Design_with_Fees-Full_Version.pdf.
https://github.com/zetavar/Payment-Network-Design-with-Fees/blob/master/Payment_Network_Design_with_Fees-Full_Version.pdf
Adam Back, Matt Corallo, Luke Dashjr, Mark Friedenbach, Gregory Maxwell, Andrew Miller, Andrew Poelstra, Jorge Timón, and Pieter Wuille. Enabling Blockchain Innovations with Pegged Sidechains, 2014. URL: https://www.blockstream.com/sidechains.pdf.
https://www.blockstream.com/sidechains.pdf
Conrad Burchert, Christian Decker, and Roger Wattenhofer. Scalable Funding of Bitcoin Micropayment Channel Networks. In 19th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS), Boston, Massachusetts, USA, November 2017.
Kyle Croman, Christian Decker, Ittay Eyal, Adem Efe Gencer, Ari Juels, Ahmed Kosba, Andrew Miller, Prateek Saxena, Elaine Shi, Emin Gün Sirer, Dawn Song, and Roger Wattenhofer. On Scaling Decentralized Blockchains. In Financial Cryptography and Data Security, pages 106-125. Springer Berlin Heidelberg, 2016.
Christian Decker and Roger Wattenhofer. A Fast and Scalable Payment Network with Bitcoin Duplex Micropayment Channels. In Andrzej Pelc and Alexander A. Schwarzmann, editors, Stabilization, Safety, and Security of Distributed Systems, pages 3-18, Cham, 2015. Springer International Publishing.
Matthew Green and Ian Miers. Bolt: Anonymous Payment Channels for Decentralized Currencies. In Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, CCS '17, pages 473-489. ACM, 2017.
Ethan Heilman, Leen Alshenibr, Foteini Baldimtsi, Alessandra Scafuro, and Sharon Goldberg. TumbleBit: An Untrusted Bitcoin-Compatible Anonymous Payment Hub. In Network and Distributed Systems Security Symposium 2017 (NDSS), February 2017, 2017.
T. Hu. Optimum Communication Spanning Trees. SIAM Journal on Computing, 3(3):188-195, 1974. URL: http://dx.doi.org/10.1137/0203015.
http://dx.doi.org/10.1137/0203015
D. S. Johnson, J. K. Lenstra, and A. H. G. Kan Rinnooy. The complexity of the network design problem. Networks, 8(4):279-285, 1978. URL: http://dx.doi.org/10.1002/net.3230080402.
http://dx.doi.org/10.1002/net.3230080402
Eleftherios Kokoris-Kogias, Philipp Jovanovic, Linus Gasser, Nicolas Gailly, Ewa Syta, and Bryan Ford. OmniLedger: A Secure, Scale-Out, Decentralized Ledger via Sharding, 2017.
Loi Luu, Viswesh Narayanan, Chaodong Zheng, Kunal Baweja, Seth Gilbert, and Prateek Saxena. A secure sharding protocol for open blockchains. In Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pages 17-30. ACM, 2016.
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Satoshi Nakamoto. Bitcoin: A peer-to-peer electronic cash system, 2008.
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Georgia Avarikioti, Yuyi Wang, and Roger Wattenhofer
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Counting Connected Subgraphs with Maximum-Degree-Aware Sieving
We study the problem of counting the isomorphic occurrences of a k-vertex pattern graph P as a subgraph in an n-vertex host graph G. Our specific interest is on algorithms for subgraph counting that are sensitive to the maximum degree Delta of the host graph.
Assuming that the pattern graph P is connected and admits a vertex balancer of size b, we present an algorithm that counts the occurrences of P in G in O ((2 Delta-2)^{(k+b)/2} 2^{-b} n/(Delta) k^2 log n) time. We define a balancer as a vertex separator of P that can be represented as an intersection of two equal-size vertex subsets, the union of which is the vertex set of P, and both of which induce connected subgraphs of P.
A corollary of our main result is that we can count the number of k-vertex paths in an n-vertex graph in O((2 Delta-2)^{floor[k/2]} n k^2 log n) time, which for all moderately dense graphs with Delta <= n^{1/3} improves on the recent breakthrough work of Curticapean, Dell, and Marx [STOC 2017], who show how to count the isomorphic occurrences of a q-edge pattern graph as a subgraph in an n-vertex host graph in time O(q^q n^{0.17q}) for all large enough q. Another recent result of Brand, Dell, and Husfeldt [STOC 2018] shows that k-vertex paths in a bounded-degree graph can be approximately counted in O(4^kn) time. Our result shows that the exact count can be recovered at least as fast for Delta<10.
Our algorithm is based on the principle of inclusion and exclusion, and can be viewed as a sparsity-sensitive version of the "counting in halves"-approach explored by Björklund, Husfeldt, Kaski, and Koivisto [ESA 2009].
graph embedding
k-path
subgraph counting
maximum degree
Mathematics of computing~Graph algorithms
17:1-17:12
Regular Paper
This work was supported in part by the Swedish Research Council grant VR-2016-03855, "Algebraic Graph Algorithms". BARC, Basic Algorithms Research Copenhagen, is funded by the Villum Foundation grant 16582. The research leading to these results has received funding from 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".
Andreas
Björklund
Andreas Björklund
Department of Computer Science, Lund University, Sweden
Thore
Husfeldt
Thore Husfeldt
BARC, IT University of Copenhagen, Denmark and Lund University, Sweden
Petteri
Kaski
Petteri Kaski
Department of Computer Science, Aalto University, Finland
Mikko
Koivisto
Mikko Koivisto
Department of Computer Science, University of Helsinki, Finland
10.4230/LIPIcs.ISAAC.2018.17
Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. If the Current Clique Algorithms are Optimal, So is Valiant’s Parser. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS, pages 98-117, 2015.
Noga Alon and Shai Gutner. Balanced families of perfect hash functions and their applications. ACM Trans. Algorithms, 6(3):54:1-54:12, 2010.
Noga Alon, Raphael Yuster, and Uri Zwick. Finding and Counting Given Length Cycles. Algorithmica, 17(3):209-223, 1997.
Omid Amini, Fedor V. Fomin, and Saket Saurabh. Counting Subgraphs via Homomorphisms. SIAM J. Discrete Math., 26(2):695-717, 2012.
Per Austrin, Petteri Kaski, and Kaie Kubjas. Tensor network complexity of multilinear maps. CoRR, abs/1712.09630, 2017.
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.
Andreas Björklund, Petteri Kaski, and Lukasz Kowalik. Counting Thin Subgraphs via Packings Faster Than Meet-in-the-Middle Time. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 594-603, 2014.
Cornelius Brand, Holger Dell, and Thore Husfeldt. Extensor-Coding. In STOC '18: Symposium on Theory of Computing, June 23-27, 2018, Los Angeles, CA, USA, page 14. ACM, New York, NY, USA, 2018.
Yijia Chen and Jörg Flum. On Parameterized Path and Chordless Path Problems. In 22nd Annual IEEE Conference on Computational Complexity (CCC), pages 250-263, 2007.
Yijia Chen, Marc Thurley, and Mark Weyer. Understanding the Complexity of Induced Subgraph Isomorphisms. In Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Track A, pages 587-596, 2008.
Radu Curticapean. Counting Matchings of Size k Is #W[1]-Hard. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 352-363, 2013.
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, pages 210-223, 2017.
Radu Curticapean and Dániel Marx. Complexity of Counting Subgraphs: Only the Boundedness of the Vertex-Cover Number Counts. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 130-139, 2014.
Friedrich Eisenbrand and Fabrizio Grandoni. On the complexity of fixed parameter clique and dominating set. Theoret. Comput. Sci., 326(1-3):57-67, 2004.
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Jörg Flum and Martin Grohe. The Parameterized Complexity of Counting Problems. In 43rd Symposium on Foundations of Computer Science (FOCS 2002), 16-19 November 2002, Vancouver, BC, Canada, Proceedings, page 538, 2002.
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Andreas Björklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto
Creative Commons Attribution 3.0 Unported license
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Target Set Selection in Dense Graph Classes
In this paper we study the Target Set Selection problem from a parameterized complexity perspective. Here for a given graph and a threshold for each vertex the task is to find a set of vertices (called a target set) to activate at the beginning which activates the whole graph during the following iterative process. A vertex outside the active set becomes active if the number of so far activated vertices in its neighborhood is at least its threshold.
We give two parameterized algorithms for a special case where each vertex has the threshold set to the half of its neighbors (the so called Majority Target Set Selection problem) for parameterizations by the neighborhood diversity and the twin cover number of the input graph.
We complement these results from the negative side. We give a hardness proof for the Majority Target Set Selection problem when parameterized by (a restriction of) the modular-width - a natural generalization of both previous structural parameters. We show that the Target Set Selection problem parameterized by the neighborhood diversity when there is no restriction on the thresholds is W[1]-hard.
parameterized complexity
target set selection
dense graphs
Theory of computation~Parameterized complexity and exact algorithms
18:1-18:13
Regular Paper
https://arxiv.org/abs/1610.07530
Pavel
Dvorák
Pavel Dvorák
Computer Science Institute, Charles University, Prague, Czech Republic
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 616787.
Dusan
Knop
Dusan Knop
Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Berlin, Germany and Faculty of Information Technology, Czech Technical University in Prague, Czech Republic
Part of this work has been done while affiliated with Department of Informatics, University of Bergen, Norway and supported by the project NFR MULTIVAL.
Tomás
Toufar
Tomás Toufar
Computer Science Institute, Charles University, Prague, Czech Republic
10.4230/LIPIcs.ISAAC.2018.18
Eyal Ackerman, Oren Ben-Zwi, and Guy Wolfovitz. Combinatorial model and bounds for target set selection. Theoretical Computer Science, 411(44):4017-4022, 2010. URL: http://dx.doi.org/10.1016/j.tcs.2010.08.021.
http://dx.doi.org/10.1016/j.tcs.2010.08.021
József Balogh, Béla Bollobás, and Robert Morris. Bootstrap Percolation in High Dimensions. Combinatorics, Probability & Computing, 19(5-6):643-692, 2010. URL: http://dx.doi.org/10.1017/S0963548310000271.
http://dx.doi.org/10.1017/S0963548310000271
Oren Ben-Zwi, Danny Hermelin, Daniel Lokshtanov, and Ilan Newman. Treewidth governs the complexity of target set selection. Discrete Optimization, 8(1):87-96, 2011. Parameterized Complexity of Discrete Optimization.
Ning Chen. On the approximability of influence in social networks. SIAM Journal on Discrete Mathematics, 23(3):1400-1415, 2009.
Chun-Ying Chiang, Liang-Hao Huang, Bo-Jr Li, Jiaojiao Wu, and Hong-Gwa Yeh. Some results on the target set selection problem. Journal of Combinatorial Optimization, 25(4):702-715, 2013. URL: http://dx.doi.org/10.1007/s10878-012-9518-3.
http://dx.doi.org/10.1007/s10878-012-9518-3
Morgan Chopin, André Nichterlein, Rolf Niedermeier, and Mathias Weller. Constant Thresholds Can Make Target Set Selection Tractable. Theory Comput. Syst., 55(1):61-83, 2014. URL: http://dx.doi.org/10.1007/s00224-013-9499-3.
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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
Pedro Domingos and Matt Richardson. Mining the network value of customers. In ACM SIGKDD, pages 57-66. ACM, 2001.
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http://dx.doi.org/10.1007/978-3-319-03898-8_15
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http://dx.doi.org/10.1007/978-3-642-28050-4_21
Tim A. Hartmann. Target Set Selection Parameterized by Clique-Width and Maximum Threshold. In SOFSEM 2018, pages 137-149. Springer International Publishing, 2018.
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http://dx.doi.org/10.1007/978-3-540-70575-8_52
Pavel Dvořák, Dušan Knop, and Tomáš Toufar
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm
Given an undirected graph and two disjoint vertex pairs s_1,t_1 and s_2,t_2, the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting s_1 with t_1, and s_2 with t_2, respectively.
We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs.
Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs with arbitrary placement of the terminals. In contrast, the randomized polynomial time algorithm by Björklund and Husfeldt, ICALP 2014, for general graphs is much slower, is serial in nature, and cannot count the solutions.
Our results are built on an approach by Hirai and Namba, Algorithmica 2017, for a generalisation of S2DP, and fast algorithms for counting perfect matchings in planar graphs.
Shortest disjoint paths
Cubic planar graph
Mathematics of computing~Paths and connectivity problems
Mathematics of computing~Combinatorial algorithms
19:1-19:13
Regular Paper
This work was supported by Swedish Research Council grant VR-2016-03855 "Algebraic Graph Algorithms" and Villum Foundation grant 16582.
Andreas
Björklund
Andreas Björklund
Department of Computer Science, Lund University, Sweden
Thore
Husfeldt
Thore Husfeldt
BARC, IT University of Copenhagen, Denmark and Lund University, Sweden
10.4230/LIPIcs.ISAAC.2018.19
Isolde Adler, Stavros G. Kolliopoulos, and Dimitrios M. Thilikos. Planar Disjoint-Paths Completion. In Parameterized and Exact Computation - 6th International Symposium, IPEC 2011, Saarbrücken, Germany, September 6-8, 2011., pages 80-93, 2011. URL: http://dx.doi.org/10.1007/978-3-642-28050-4_7.
http://dx.doi.org/10.1007/978-3-642-28050-4_7
Nima Anari and Vijay V. Vazirani. Planar Graph Perfect Matching is in NC. CoRR, abs/1709.07822, 2017. URL: http://arxiv.org/abs/1709.07822.
http://arxiv.org/abs/1709.07822
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Andreas Björklund and Thore Husfeldt. Shortest Two Disjoint Paths in Polynomial Time. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 211-222, 2014. URL: http://dx.doi.org/10.1007/978-3-662-43948-7_18.
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http://arxiv.org/abs/1802.01338
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Andreas Björklund and Thore Husfeldt
Creative Commons Attribution 3.0 Unported license
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Data-Compression for Parametrized Counting Problems on Sparse Graphs
We study the concept of compactor, which may be seen as a counting-analogue of kernelization in counting parameterized complexity. For a function F:Sigma^* -> N and a parameterization kappa: Sigma^* -> N, a compactor (P,M) consists of a polynomial-time computable function P, called condenser, and a computable function M, called extractor, such that F=M o P, and the condensing P(x) of x has length at most s(kappa(x)), for any input x in Sigma^*. If s is a polynomial function, then the compactor is said to be of polynomial-size. Although the study on counting-analogue of kernelization is not unprecedented, it has received little attention so far. We study a family of vertex-certified counting problems on graphs that are MSOL-expressible; that is, for an MSOL-formula phi with one free set variable to be interpreted as a vertex subset, we want to count all A subseteq V(G) where |A|=k and (G,A) models phi. In this paper, we prove that every vertex-certified counting problems on graphs that is MSOL-expressible and treewidth modulable, when parameterized by k, admits a polynomial-size compactor on H-topological-minor-free graphs with condensing time O(k^2n^2) and decoding time 2^{O(k)}. This implies the existence of an FPT-algorithm of running time O(n^2 k^2)+2^{O(k)}. All aforementioned complexities are under the Uniform Cost Measure (UCM) model where numbers can be stored in constant space and arithmetic operations can be done in constant time.
Parameterized counting
compactor
protrusion decomposition
Theory of computation~Graph algorithms analysis
20:1-20:13
Regular Paper
http://arxiv.org/abs/1809.08160
Eun Jung
Kim
Eun Jung Kim
Université Paris-Dauphine, PSL Research University, CNRS/LAMSADE, 75016, Paris, France
Supported by project ESIGMA (ANR-17-CE23-0010).
Maria
Serna
Maria Serna
Computer Science Department & BGSMath, Universitat Politècnica de Catalunya, Barcelona, Spain
Partially funded by MINECO and FEDER funds under grants TIN2017-86727-C2-1-R (GRAMM) and MDM-2014-044 (BGSMath), and by AGAUR grant 2017SGR-786 (ALBCOM).
Dimitrios M.
Thilikos
Dimitrios M. Thilikos
AlGCo project-team, LIRMM, Université de Montpellier, CNRS, Montpellier, France
and, Department of Mathematics, National and Kapodistrian University of Athens, Greece
Supported by projects DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010).
10.4230/LIPIcs.ISAAC.2018.20
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Eun Jung Kim, Maria Serna, and Dimitrios M. Thilikos. Data-compression for parametrized counting problems on sparse graphs. CoRR, abs/1809.08160, 2018. URL: http://arxiv.org/abs/1809.08160.
http://arxiv.org/abs/1809.08160
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Eun Jung Kim, Maria Serna, and Dimitrios M. Thilikos
Creative Commons Attribution 3.0 Unported license
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Planar Maximum Matching: Towards a Parallel Algorithm
Perfect matchings in planar graphs have been extensively studied and understood in the context of parallel complexity [P W Kastelyn, 1967; Vijay Vazirani, 1988; Meena Mahajan and Kasturi R. Varadarajan, 2000; Datta et al., 2010; Nima Anari and Vijay V. Vazirani, 2017]. However, corresponding results for maximum matchings have been elusive. We partly bridge this gap by proving:
1) An SPL upper bound for planar bipartite maximum matching search.
2) Planar maximum matching search reduces to planar maximum matching decision.
3) Planar maximum matching count reduces to planar bipartite maximum matching count and planar maximum matching decision.
The first bound improves on the known [Thanh Minh Hoang, 2010] bound of L^{C_=L} and is adaptable to any special bipartite graph class with non-zero circulation such as bounded genus graphs, K_{3,3}-free graphs and K_5-free graphs. Our bounds and reductions non-trivially combine techniques like the Gallai-Edmonds decomposition [L. Lovász and M.D. Plummer, 1986], deterministic isolation [Datta et al., 2010; Samir Datta et al., 2012; Rahul Arora et al., 2016], and the recent breakthroughs in the parallel search for planar perfect matchings [Nima Anari and Vijay V. Vazirani, 2017; Piotr Sankowski, 2018].
maximum matching
planar graphs
parallel complexity
reductions
Theory of computation~Parallel algorithms
21:1-21:13
Regular Paper
Samir
Datta
Samir Datta
Chennai Mathematical Institute & UMI ReLaX, Chennai, India
The author was partially funded by a grant from Infosys foundation and SERB grant MTR/2017/000480.
Raghav
Kulkarni
Raghav Kulkarni
Chennai Mathematical Institute, Chennai, India
Ashish
Kumar
Ashish Kumar
Chennai Mathematical Institute, Chennai, India
Anish
Mukherjee
Anish Mukherjee
Chennai Mathematical Institute, Chennai, India
The author was partially supported by a grant from Infosys foundation and TCS PhD fellowship.
10.4230/LIPIcs.ISAAC.2018.21
E. Allender, K. Reinhardt, and S. Zhou. Isolation, Matching, and Counting: Uniform and Nonuniform Upper Bounds. Journal of Computer and System Sciences, 59:164-181, 1999.
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Rahul Arora, Ashu Gupta, Rohit Gurjar, and Raghunath Tewari. Derandomizing Isolation Lemma for K3,3-free and K5-free Bipartite Graphs. In 33rd Symposium on Theoretical Aspects of Computer Science, STACS, pages 10:1-10:15, 2016.
Samir Datta, Raghav Kulkarni, Nutan Limaye, and Meena Mahajan. Planarity, Determinants, Permanents, and (Unique) Matchings. ACM Trans. Comput. Theory, 1(3):1-20, 2010.
Samir Datta, Raghav Kulkarni, and Anish Mukherjee. Space-Efficient Approximation Scheme for Maximum Matching in Sparse Graphs. In 41st International Symposium on Mathematical Foundations of Computer Science, MFCS, pages 28:1-28:12, 2016.
Samir Datta, Raghav Kulkarni, and Sambuddha Roy. Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs. Theory of Computing Systems, 47:737-757, 2010.
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Shafi Goldwasser, Ofer Grossman, and Dhiraj Holden. Pseudo-Deterministic Proofs. In 9th Innovations in Theoretical Computer Science Conference, ITCS, pages 17:1-17:18, 2018.
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Piotr Sankowski. NC algorithms for weighted planar perfect matching and related problems. In 45th International Colloquium on Automata, Languages, and Programming, ICALP, pages 97:1-97:16, 2018.
Simon Straub, Thomas Thierauf, and Fabian Wagner. Counting the Number of Perfect Matchings in K 5-Free Graphs. Theory Comput. Syst., 59(3):416-439, 2016.
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Samir Datta, Raghav Kulkarni, Ashish Kumar, and Anish Mukherjee
Creative Commons Attribution 3.0 Unported license
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Distributed Approximation Algorithms for the Minimum Dominating Set in K_h-Minor-Free Graphs
In this paper we will give two distributed approximation algorithms (in the Local model) for the minimum dominating set problem. First we will give a distributed algorithm which finds a dominating set D of size O(gamma(G)) in a graph G which has no topological copy of K_h. The algorithm runs L_h rounds where L_h is a constant which depends on h only. This procedure can be used to obtain a distributed algorithm which given epsilon>0 finds in a graph G with no K_h-minor a dominating set D of size at most (1+epsilon)gamma(G). The second algorithm runs in O(log^*{|V(G)|}) rounds.
Distributed algorithms
minor-closed family of graphs
MDS
Theory of computation~Distributed computing models
22:1-22:12
Regular Paper
Andrzej
Czygrinow
Andrzej Czygrinow
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, 85287-1804, USA
Research supported in part by Simons Foundation Grant # 521777.
Michal
Hanckowiak
Michal Hanckowiak
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland
Wojciech
Wawrzyniak
Wojciech Wawrzyniak
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland
Marcin
Witkowski
Marcin Witkowski
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland
10.4230/LIPIcs.ISAAC.2018.22
E. Szymańska A. Czygrinow, M. Hańćkowiak. Fast Distributed Algorithm for the Maximum Matching Problem in Bounded Arboricity Graphs. Proc. of 20th International Symposium, ISAAC, LNCS 5878, pages 668-678, 2009.
W. Wawrzyniak A. Czygrinow, M. Hańćkowiak. Fast distributed approximations in planar graphs. International Symposium on Distributed Computing, DISC, Arcachon, France, September 2008, LNCS 5218, pages 78-92, 2008.
R. Wattenhofer C. Lenzen. Minimum Dominating Set Approximation in Graphs of Bounded Arboricity. International Symposium on Distributed Computing, DISC 2010, pages 510-524, 2010.
R. Wattenhofer C. Lenzen, Y. A. Oswald. Distributed minimum dominating set approximations in restricted families of graphs. Distrib. Comput., 26 (2), pages 119-137, 2013.
S. Gutner N. Alon. Kernels for the Dominating Set Problem on Graphs with an Excluded Minor. Electronic Colloquium on Computational Complexity, Report No. 66, 2008.
S. Siebertz S. A. Amiri, S. Schmid. A Local Constant Factor MDS Approximation for Bounded Genus Graphs. PODC 2016, Proceedings of the ACM Symp. on Principles of Distributed Computing, 227-233, 2016.
Andrzej Czygrinow, Michał Hanćkowiak, Wojciech Wawrzyniak, and Marcin Witkowski
Creative Commons Attribution 3.0 Unported license
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Proving the Turing Universality of Oritatami Co-Transcriptional Folding
We study the oritatami model for molecular co-transcriptional folding. In oritatami systems, the transcript (the "molecule") folds as it is synthesized (transcribed), according to a local energy optimisation process, which is similar to how actual biomolecules such as RNA fold into complex shapes and functions as they are transcribed. We prove that there is an oritatami system embedding universal computation in the folding process itself.
Our result relies on the development of a generic toolbox, which is easily reusable for future work to design complex functions in oritatami systems. We develop "low-level" tools that allow to easily spread apart the encoding of different "functions" in the transcript, even if they are required to be applied at the same geometrical location in the folding. We build upon these low-level tools, a programming framework with increasing levels of abstraction, from encoding of instructions into the transcript to logical analysis. This framework is similar to the hardware-to-algorithm levels of abstractions in standard algorithm theory. These various levels of abstractions allow to separate the proof of correctness of the global behavior of our system, from the proof of correctness of its implementation. Thanks to this framework, we were able to computerise the proof of correctness of its implementation and produce certificates, in the form of a relatively small number of proof trees, compact and easily readable/checkable by human, while encapsulating huge case enumerations. We believe this particular type of certificates can be generalised to other discrete dynamical systems, where proofs involve large case enumerations as well.
Molecular computing
Turing universality
co-transcriptional folding
Theory of computation~Models of computation
Applied computing~Life and medical sciences
Hardware~Biology-related information processing
23:1-23:13
Regular Paper
https://arxiv.org/abs/1508.00510
Cody
Geary
Cody Geary
California Institute of Technology, Pasadena, CA, USA
Pierre-Étienne
Meunier
Pierre-Étienne Meunier
Maynooth University, Ireland
Nicolas
Schabanel
Nicolas Schabanel
CNRS, ÉNS de Lyon (LIP, UMR 5668), France and IXXI, U. Lyon, France, http://perso.ens-lyon.fr/nicolas.schabanel/
Supported by CNRS grants MoPrExProgMol and AMARP.
Shinnosuke
Seki
Shinnosuke Seki
Oritatami Lab, University of Electro-Communications, Tokyo, Japan, http://www.sseki.lab.uec.ac.jp/
Supported by JST Program No. 6F36, and JSPS Grants Nos. 16H05854, 18K19779, and YB29004.
10.4230/LIPIcs.ISAAC.2018.23
J. Boyle, G. Robillard, and S. Kim. Sequential folding of transfer RNA. A nuclear magnetic resonance study of successively longer tRNA fragments with a common 5' endA nuclear magnetic resonance study of successively longer tRNA fragments with a common 5' end. J. Mol. Biol., 139:601-625, 1980.
M. Cook. Universality in Elementary Cellular Automata. Complex Systems, 15:1-40, 2004.
E.D. Demaine, J. Hendricks, M. Olsen, M.J. Patitz, T. Rogers N. Schabanel, S. Seki, and H. Thomas. Know When to Fold 'Em: Self-Assembly of Shapes by Folding in Oritatami. In DNA, 2018. To be published.
K. L. Frieda and S. M. Block. Direct observation of cotranscriptional folding in an adenine riboswitch. Science, 338(6105):397-400, 2012.
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C. Geary, P.-É. Meunier, N. Schabanel, and S. Seki. Programming Biomolecules that Fold Greedily during Transcription. In MFCS, volume LIPIcs 58, pages 43:1-43:14, 2016.
C. Geary, P-É Meunier, N. Schabanel, and S. Seki. Proving the Turing Universality of Oritatami Co-Transcriptional Folding (Full Text). CoRR, 2018. URL: http://arxiv.org/abs/1508.00510.
http://arxiv.org/abs/1508.00510
C. Geary, P. W. K. Rothemund, and E. S. Andersen. A Single-Stranded Architecture for Cotranscriptional Folding of RNA Nanostructures. Science, 345:799-804, 2014.
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M. Ota and S. Seki. Ruleset Design Problems for Oritatami Systems. Theor. Comput. Sci., 671:26-35, 2017.
D. Woods and T. Neary. On The Time Complexity of 2-tag Systems and Small Universal Turing Machines. In FOCS, pages 439-448, 2006.
Cody Geary, Pierre-Étienne Meunier, Nicolas Schabanel, and Shinnosuke Seki
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Cluster Editing in Multi-Layer and Temporal Graphs
Motivated by the recent rapid growth of research for algorithms to cluster multi-layer and temporal graphs, we study extensions of the classical Cluster Editing problem. In Multi-Layer Cluster Editing we receive a set of graphs on the same vertex set, called layers and aim to transform all layers into cluster graphs (disjoint unions of cliques) that differ only slightly. More specifically, we want to mark at most d vertices and to transform each layer into a cluster graph using at most k edge additions or deletions per layer so that, if we remove the marked vertices, we obtain the same cluster graph in all layers. In Temporal Cluster Editing we receive a sequence of layers and we want to transform each layer into a cluster graph so that consecutive layers differ only slightly. That is, we want to transform each layer into a cluster graph with at most k edge additions or deletions and to mark a distinct set of d vertices in each layer so that each two consecutive layers are the same after removing the vertices marked in the first of the two layers. We study the combinatorial structure of the two problems via their parameterized complexity with respect to the parameters d and k, among others. Despite the similar definition, the two problems behave quite differently: In particular, Multi-Layer Cluster Editing is fixed-parameter tractable with running time k^{O(k + d)} s^{O(1)} for inputs of size s, whereas Temporal Cluster Editing is W[1]-hard with respect to k even if d = 3.
Cluster Editing
Temporal Graphs
Multi-Layer Graphs
Fixed-Parameter Algorithms
Polynomial Kernels
Parameterized Complexity
Theory of computation~Fixed parameter tractability
24:1-24:13
Regular Paper
JC and MS supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement number 631163.11, the Israel Science Foundation (grant no. 551145/14), and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement numbers 677651 (JC) and 714704 (MS). HM supported by the DFG, project MATE (NI 369/17). OS supported by grant 17-20065S of the Czech Science Foundation.
https://arxiv.org/abs/1709.09100
Jiehua
Chen
Jiehua Chen
Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Poland
Hendrik
Molter
Hendrik Molter
Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Berlin, Germany
Manuel
Sorge
Manuel Sorge
Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Poland
Ondrej
Suchý
Ondrej Suchý
Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic
10.4230/LIPIcs.ISAAC.2018.24
Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation Clustering. Machine Learning, 56:89-113, 2004.
Matteo Barigozzi, Giorgio Fagiolo, and Giuseppe Mangioni. Identifying the Community Structure of the International-Trade Multi-Network. Physica A, 390(11):2051-2066, 2011.
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Sebastian Böcker and Jan Baumbach. Cluster Editing. In Proc. of CiE '13, volume 7921 of LNCS, pages 33-44. Springer, 2013.
Leizhen Cai and Junjie Ye. Dual Connectedness of Edge-Bicolored Graphs and Beyond. In Proc. of MFCS '14, volume 8635 of LNCS, pages 141-152. Springer, 2014.
Yixin Cao and Jianer Chen. Cluster Editing: Kernelization Based on Edge Cuts. Algorithmica, 64(1):152-169, 2012.
Jiehua Chen, Hendrik Molter, Manuel Sorge, and Ondrej Suchý. Cluster Editing in Multi-Layer and Temporal Graphs. CoRR, abs/1709.09100, 2018.
M. Cygan, F.V. Fomin, Ł. Kowalik, D. Lokshtanov, D. Marx, Ma. Pilipczuk, Mi. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015.
Frank Dehne, Mike Fellows, Frances Rosamond, and Peter Shaw. Greedy Localization, Iterative Compression, and Modeled Crown Reductions: New FPT Techniques, an Improved Algorithm for Set Splitting, and a Novel 2k Kernelization for Vertex Cover. In Proc. of IWPEC '04, volume 3162 of LNCS, pages 271-280. Springer, 2004.
T.K. Dey, A. Rossi, and A. Sidiropoulos. Temporal Clustering. In Proc. of ESA '17, volume 87 of LIPIcs, pages 34:1-34:14. Schloss Dagstuhl, 2017.
Martin Dörnfelder, Jiong Guo, Christian Komusiewicz, and Mathias Weller. On the Parameterized Complexity of Consensus Clustering. Theor. Comput. Sci., 542:71-82, 2014.
Fedor V. Fomin, Stefan Kratsch, Marcin Pilipczuk, Michał Pilipczuk, and Yngve Villanger. Tight Bounds for Parameterized Complexity of Cluster Editing with a Small Number of Clusters. J. Comput. Syst. Sci., 80(7):1430-1447, 2014.
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Christian Komusiewicz and Johannes Uhlmann. Cluster Editing with Locally Bounded Modifications. Discrete Appl. Math., 160:2259-2270, 2012.
Matthieu Latapy, Tiphaine Viard, and Clémence Magnien. Stream graphs and link streams for the modeling of interactions over time. CoRR, abs/1710.04073, 2017. URL: http://arxiv.org/abs/1710.04073.
http://arxiv.org/abs/1710.04073
Othon Michail. An introduction to temporal graphs: An algorithmic perspective. Internet Math., 12(4):239-280, 2016.
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Andrea Tagarelli, Alessia Amelio, and Francesco Gullo. Ensemble-Based Community Detection in Multilayer Networks. Data Min. Knowl. Discov., 31(5):1506-1543, 2017.
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C. Tantipathananandh and T. Y. Berger-Wolf. Finding Communities in Dynamic Social Networks. In Proc. of ICDM '11, pages 1236-1241. IEEE Computer Society, 2011.
Jiehua Chen, Hendrik Molter, Manuel Sorge, and Ondřej Suchý
Creative Commons Attribution 3.0 Unported license
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Parameterized Query Complexity of Hitting Set Using Stability of Sunflowers
In this paper, we study the query complexity of parameterized decision and optimization versions of Hitting-Set. We also investigate the query complexity of Packing. In doing so, we use generalizations to hypergraphs of an earlier query model, known as BIS introduced by Beame et al. in ITCS'18. The query models considered are the GPIS and GPISE oracles. The GPIS and GPISE oracles are used for the decision and optimization versions of the problems, respectively. We use color coding and queries to the oracles to generate subsamples from the hypergraph, that retain some structural properties of the original hypergraph. We use the stability of the sunflowers in a non-trivial way to do so.
Query complexity
Hitting set
Parameterized complexity
Theory of computation~Fixed parameter tractability
Theory of computation~Streaming, sublinear and near linear time algorithms
25:1-25:12
Regular Paper
https://arxiv.org/abs/1807.06272
Arijit
Bishnu
Arijit Bishnu
Indian Statistical Institute, Kolkata, India
Arijit
Ghosh
Arijit Ghosh
The Institute of Mathematical Sciences, Chennai, India
Sudeshna
Kolay
Sudeshna Kolay
Eindhoven University of Technology, Eindhoven, Netherlands
Gopinath
Mishra
Gopinath Mishra
Indian Statistical Institute, Kolkata, India
Saket
Saurabh
Saket Saurabh
The Institute of Mathematical Sciences, Chennai, India
10.4230/LIPIcs.ISAAC.2018.25
Noga Alon, Raphael Yuster, and Uri Zwick. Color Coding. In Encyclopedia of Alg., pages 335-338. Springer, 2016.
Paul Beame, Sariel Har-Peled, Sivaramakrishnan Natarajan Ramamoorthy, Cyrus Rashtchian, and Makrand Sinha. Edge Estimation with Independent Set Oracles. In ITCS, pages 38:1-38:21, 2018.
Arijit Bishnu, Arijit Ghosh, Sudeshna Kolay, Gopinath Mishra, and Saket Saurabh. Parameterized Query Complexity of Hitting Set using Stability of Sunflowers. CoRR, abs/1807.06272, 2018. URL: http://arxiv.org/abs/1807.06272.
http://arxiv.org/abs/1807.06272
Rajesh Chitnis, Graham Cormode, Hossein Esfandiari, MohammadTaghi Hajiaghayi, Andrew McGregor, Morteza Monemizadeh, and Sofya Vorotnikova. Kernelization via Sampling with Applications to Finding Matchings and Related Problems in Dynamic Graph Streams. In SODA, pages 1326-1344, 2016.
M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015.
P. Erdős and R. Rado. Intersection Theorems for Systems of Sets. Journal of the London Mathematical Society, s1-35(1):85-90, 1960.
Uriel Feige. On Sums of Independent Random Variables with Unbounded Variance and Estimating the Average Degree in a Graph. SIAM J. Comput., 35(4):964-984, 2006.
Oded Goldreich. Introduction to Property Testing. Cambridge University Press, 2017.
Oded Goldreich and Dana Ron. Approximating average parameters of graphs. Random Struct. Algorithms, 32(4):473-493, 2008.
Piotr Indyk, Sepideh Mahabadi, Ronitt Rubinfeld, Ali Vakilian, and Anak Yodpinyanee. Set Cover in Sub-linear Time. In SODA, pages 2467-2486, 2018.
Kazuo Iwama and Yuichi Yoshida. Parameterized Testability. TOCT, 9(4):16:1-16:16, 2018.
Krzysztof Onak, Dana Ron, Michal Rosen, and Ronitt Rubinfeld. A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size. In SODA, pages 1123-1131, 2012.
Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg. Computing Exact Minimum Cuts Without Knowing the Graph. In ITCS, pages 39:1-39:16, 2018.
Arijit Bishnu, Arijit Ghosh, Sudeshna Kolay, Gopinath Mishra, and Saket Saurabh
Creative Commons Attribution 3.0 Unported license
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Approximate Minimum-Weight Matching with Outliers Under Translation
Our goal is to compare two planar point sets by finding subsets of a given size such that a minimum-weight matching between them has the smallest weight. This can be done by a translation of one set that minimizes the weight of the matching. We give efficient algorithms (a) for finding approximately optimal matchings, when the cost of a matching is the L_p-norm of the tuple of the Euclidean distances between the pairs of matched points, for any p in [1,infty], and (b) for constructing small-size approximate minimization (or matching) diagrams: partitions of the translation space into regions, together with an approximate optimal matching for each region.
Minimum-weight partial matching
Pattern matching
Approximation
Mathematics of computing~Combinatorics
26:1-26:13
Regular Paper
Work on this paper was supported by grant 2012/229 from the U.S. - Israel Binational Science Foundation and by grant 1367/2016 from the German-Israeli Science Foundation (GIF).
Pankaj K.
Agarwal
Pankaj K. Agarwal
Department of Computer Science, Duke University, Durham, NC 27708, USA
Partially supported by NSF grants CCF-15-13816, CCF-15-46392, IIS-14-08846 and ARO grant W911NF-15-1-0408.
Haim
Kaplan
Haim Kaplan
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel
Geva
Kipper
Geva Kipper
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel
Wolfgang
Mulzer
Wolfgang Mulzer
Institut für Informatik, Freie Universität Berlin, 14195 Berlin, Germany
https://orcid.org/0000-0002-1948-5840
Partially supported by DFG grant MU/3501/1 and ERC STG 757609.
Günter
Rote
Günter Rote
Institut für Informatik, Freie Universität Berlin, 14195 Berlin, Germany
https://orcid.org/0000-0002-0351-5945
Micha
Sharir
Micha Sharir
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel
Partially supported by ISF Grant 892/13, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), by the Blavatnik Research Fund in Computer Science at Tel Aviv University, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University.
Allen
Xiao
Allen Xiao
Department of Computer Science, Duke University, Durham, NC 27708, USA
Partially supported by NSF CCF-15-13816, CCF-15-46392, IIS-14-08846 and ARO grant W911NF-15-1-0408.
10.4230/LIPIcs.ISAAC.2018.26
Pankaj K. Agarwal, Alon Efrat, and Micha Sharir. Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput., 29(3):912-953, 1999.
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Rinat Ben-Avraham, Matthias Henze, Rafel Jaume, Balázs Keszegh, Orit E. Raz, Micha Sharir, and Igor Tubis. Partial-Matching RMS Distance Under Translation: Combinatorics and Algorithms. Algorithmica, 80(8):2400-2421, 2018.
Sergio Cabello, Panos Giannopoulos, Christian Knauer, and Günter Rote. Matching point sets with respect to the Earth Mover’s Distance. Comput. Geom., 39(2):118-133, 2008.
Paz Carmi, Shlomi Dolev, Sariel Har-Peled, Matthew J. Katz, and Michael Segal. Geographic Quorum System Approximations. Algorithmica, 41(4):233-244, 2005.
Harold N. Gabow and Robert Endre Tarjan. Faster Scaling Algorithms for Network Problems. SIAM J. Comput., 18(5):1013-1036, 1989.
Andrew V. Goldberg, Sagi Hed, Haim Kaplan, and Robert E. Tarjan. Minimum-Cost Flows in Unit-Capacity Networks. Theory Comput. Syst., 61(4):987-1010, 2017.
Matthias Henze, Rafel Jaume, and Balázs Keszegh. On the complexity of the partial least-squares matching Voronoi diagram. In Proc. 29th European Workshop Comput. Geom. (EWCG), pages 193-196, 2013.
Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir. Dynamic Planar Voronoi Diagrams for General Distance Functions and their Algorithmic Applications. In Proc. 28th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 2495-2504, 2017.
Jeff M. Phillips and Pankaj K. Agarwal. On Bipartite Matching under the RMS Distance. In Proc. 18th Canad. Conf. Comput. Geom. (CCCG), pages 143-146, 2006.
Günter Rote. Partial least-squares point matching under translations. In Proc. 26th European Workshop Comput. Geom. (EWCG), pages 249-251, 2010.
R. Sharathkumar and Pankaj K. Agarwal. A near-linear time ε-approximation algorithm for geometric bipartite matching. In Proc. 44th Annu. ACM Sympos. Theory Comput. (STOC), pages 385-394, 2012.
R. Sharathkumar and Pankaj K. Agarwal. Algorithms for the transportation problem in geometric settings. In Proc. 23rd Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 306-317, 2012.
Kasturi R. Varadarajan and Pankaj K. Agarwal. Approximation Algorithms for Bipartite and Non-Bipartite Matching in the Plane. In Proc. 10th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 805-814, 1999.
Remco C. Veltkamp. Shape matching: Similarity measures and algorithms. In Proc. Intl. Conf. Shape Modeling and Applications, pages 188-197, 2001.
Pankaj K. Agarwal, Haim Kaplan, Geva Kipper, Wolfgang Mulzer, Günter Rote, Micha Sharir, and Allen Xiao
Creative Commons Attribution 3.0 Unported license
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New and Improved Algorithms for Unordered Tree Inclusion
The tree inclusion problem is, given two node-labeled trees P and T (the "pattern tree" and the "text tree"), to locate every minimal subtree in T (if any) that can be obtained by applying a sequence of node insertion operations to P. Although the ordered tree inclusion problem is solvable in polynomial time, the unordered tree inclusion problem is NP-hard. The currently fastest algorithm for the latter is from 1995 and runs in O(poly(m,n) * 2^{2d}) = O^*(2^{2d}) time, where m and n are the sizes of the pattern and text trees, respectively, and d is the maximum outdegree of the pattern tree. Here, we develop a new algorithm that improves the exponent 2d to d by considering a particular type of ancestor-descendant relationships and applying dynamic programming, thus reducing the time complexity to O^*(2^d). We then study restricted variants of the unordered tree inclusion problem where the number of occurrences of different node labels and/or the input trees' heights are bounded. We show that although the problem remains NP-hard in many such cases, it can be solved in polynomial time for c = 2 and in O^*(1.8^d) time for c = 3 if the leaves of P are distinctly labeled and each label occurs at most c times in T. We also present a randomized O^*(1.883^d)-time algorithm for the case that the heights of P and T are one and two, respectively.
parameterized algorithms
tree inclusion
unordered trees
dynamic programming
Theory of computation~Graph algorithms analysis
27:1-27:12
Regular Paper
Tatsuya
Akutsu
Tatsuya Akutsu
Bioinformatics Center, Institute for Chemical Research, Kyoto University, Kyoto 611-0011, Japan
JSPS KAKENHI #18H04113
Jesper
Jansson
Jesper Jansson
Department of Computing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
Ruiming
Li
Ruiming Li
Bioinformatics Center, Institute for Chemical Research, Kyoto University, Kyoto 611-0011, Japan
Atsuhiro
Takasu
Atsuhiro Takasu
National Institute of Informatics, Chiyoda-ku, Tokyo, 101-8430, Japan
Takeyuki
Tamura
Takeyuki Tamura
Bioinformatics Center, Institute for Chemical Research, Kyoto University, Kyoto 611-0011, Japan
JSPS KAKENHI #25730005
10.4230/LIPIcs.ISAAC.2018.27
Amir Abboud, Arturs Backurs, Thomas Dueholm Hansen, Virginia Vassilevska Williams, and Or Zamir. Subtree isomorphism revisited. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1256-1271. SIAM, 2018.
Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In Proceedings of the 41st International Colloquium on Automata, Languages, and Programming - Part 1, pages 39-51. Springer, 2014.
Tatsuya Akutsu, Daiji Fukagawa, Magnús M. Halldórsson, Atsuhiro Takasu, and Keisuke Tanaka. Approximation and parameterized algorithms for common subtrees and edit distance between unordered trees. Theoretical Computer Science, 470:10-22, 2013.
Tatsuya Akutsu, Daiji Fukagawa, Atsuhiro Takasu, and Takeyuki Tamura. Exact algorithms for computing the tree edit distance between unordered trees. Theoretical Computer Science, 412(4-5):352-364, 2011.
Tatsuya Akutsu, Takeyuki Tamura, Daiji Fukagawa, and Atsuhiro Takasu. Efficient exponential-time algorithms for edit distance between unordered trees. Journal of Discrete Algorithms, 25:79-93, 2014.
Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. Journal of the ACM, 42(4):844-856, 1995.
Philip Bille. A survey on tree edit distance and related problems. Theoretical Computer Science, 337(1):217-239, 2005.
Philip Bille and Inge Li Gørtz. The tree inclusion problem: In linear space and faster. ACM Transactions on Algorithms (TALG), 7(3):38, 2011.
Karl Bringmann, Pawel Gawrychowski, Shay Mozes, and Oren Weimann. Tree edit distance cannot be computed in strongly subcubic time (unless APSP can). In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1190-1206. SIAM, 2018.
Lijun Chang, Xuemin Lin, Wenjie Zhang, Jeffrey Xu Yu, Ying Zhang, and Lu Qin. Optimal enumeration: Efficient top-k tree matching. Proceedings of the VLDB Endowment, 8(5):533-544, 2015.
Sara Cohen and Nerya Or. A general algorithm for subtree similarity-search. In Data Engineering (ICDE), 2014 IEEE 30th International Conference on, pages 928-939. IEEE, 2014.
Erik D. Demaine, Shay Mozes, Benjamin Rossman, and Oren Weimann. An optimal decomposition algorithm for tree edit distance. ACM Transactions on Algorithms (TALG), 6(1):2, 2009.
Minoru Kanehisa, Susumu Goto, Yoko Sato, Masayuki Kawashima, Miho Furumichi, and Mao Tanabe. Data, information, knowledge and principle: back to metabolism in KEGG. Nucleic Acids Research, 42(D1):D199-D205, 2013.
Pekka Kilpeläinen and Heikki Mannila. Ordered and unordered tree inclusion. SIAM Journal on Computing, 24(2):340-356, 1995.
Hanna Köpcke, Andreas Thor, and Erhard Rahm. Evaluation of entity resolution approaches on real-world match problems. Proceedings of the VLDB Endowment, 3(1-2):484-493, 2010.
Jiwei Li, Thang Luong, Dan Jurafsky, and Eduard H. Hovy. When Are Tree Structures Necessary for Deep Learning of Representations? In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, EMNLP 2015, Lisbon, Portugal, September 17-21, 2015, pages 2304-2314, 2015. URL: http://aclweb.org/anthology/D/D15/D15-1278.pdf.
http://aclweb.org/anthology/D/D15/D15-1278.pdf
Jiří Matoušek and Robin Thomas. On the complexity of finding iso-and other morphisms for partial k-trees. Discrete Mathematics, 108(1-3):343-364, 1992.
Tomoya Mori, Atsuhiro Takasu, Jesper Jansson, Jaewook Hwang, Takeyuki Tamura, and Tatsuya Akutsu. Similar subtree search using extended tree inclusion. IEEE Transactions on Knowledge and Data Engineering, 27(12):3360-3373, 2015.
Dennis Shasha, Jason T. L. Wang, Kaizhong Zhang, and Frank Y. Shih. Exact and approximate algorithms for unordered tree matching. IEEE Transactions on Systems, Man, and Cybernetics, 24(4):668-678, 1994.
Kuo-Chung Tai. The tree-to-tree correction problem. Journal of the ACM (JACM), 26(3):422-433, 1979.
Gabriel Valiente. Constrained tree inclusion. Journal of Discrete Algorithms, 3(2):431-447, 2005.
Masaki Yamamoto. An improved O^∗(1.234^m)-time deterministic algorithm for SAT. In Proceedings of the 16th International Symposium on Algorithms and Computation, pages 644-653. Springer, 2005.
Mohammed Javeed Zaki. Efficiently mining frequent trees in a forest: Algorithms and applications. IEEE Transactions on Knowledge and Data Engineering, 17(8):1021-1035, 2005.
Kaizhong Zhang and Tao Jiang. Some MAX SNP-hard results concerning unordered labeled trees. Information Processing Letters, 49(5):249-254, 1994.
Kaizhong Zhang, Rick Statman, and Dennis Shasha. On the editing distance between unordered labeled trees. Information Processing Letters, 42(3):133-139, 1992.
Tatsuya Akutsu, Jesper Jansson, Ruiming Li, Atsuhiro Takasu, and Takeyuki Tamura
Creative Commons Attribution 3.0 Unported license
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Beyond-Planarity: Turán-Type Results for Non-Planar Bipartite Graphs
Beyond-planarity focuses on the study of geometric and topological graphs that are in some sense nearly planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations. Early research dates back to the 1960s (e.g., Avital and Hanani 1966) for extremal problems on geometric graphs, but is also related to graph drawing problems where visual clutter due to edge crossings should be minimized (e.g., Huang et al. 2018).
Most of the literature focuses on Turán-type problems, which ask for the maximum number of edges a beyond-planar graph can have. Here, we study this problem for bipartite topological graphs, considering several types of beyond-planar graphs, i.e. 1-planar, 2-planar, fan-planar, and RAC graphs. We prove bounds on the number of edges that are tight up to additive constants; some of them are surprising and not along the lines of the known results for non-bipartite graphs. Our findings lead to an improvement of the leading constant of the well-known Crossing Lemma for bipartite graphs, as well as to a number of interesting questions on topological graphs.
Bipartite topological graphs
beyond planarity
density
Crossing Lemma
Theory of computation~Computational geometry
Mathematics of computing~Graph theory
28:1-28:13
Regular Paper
This project was supported by DFG grant KA812/18-1.
https://arxiv.org/abs/1712.09855
Patrizio
Angelini
Patrizio Angelini
Wilhelm-Schickhard-Institut für Informatik, Universität Tübingen, Germany
Michael A.
Bekos
Michael A. Bekos
Wilhelm-Schickhard-Institut für Informatik, Universität Tübingen, Germany
Michael
Kaufmann
Michael Kaufmann
Wilhelm-Schickhard-Institut für Informatik, Universität Tübingen, Germany
Maximilian
Pfister
Maximilian Pfister
Wilhelm-Schickhard-Institut für Informatik, Universität Tübingen, Germany
Torsten
Ueckerdt
Torsten Ueckerdt
Fakultät für Informatik, KIT, Karlsruhe, Germany
10.4230/LIPIcs.ISAAC.2018.28
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Patrizio Angelini, Michael A. Bekos, Michael Kaufmann, Maximilian Pfister, and Torsten Ueckerdt
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Dichotomy Result for Cyclic-Order Traversing Games
Traversing game is a two-person game played on a connected undirected simple graph with a source node and a destination node. A pebble is placed on the source node initially and then moves autonomously according to some rules. Alice is the player who wants to set up rules for each node to determine where to forward the pebble while the pebble reaches the node, so that the pebble can reach the destination node. Bob is the second player who tries to deter Alice's effort by removing edges. Given access to Alice's rules, Bob can remove as many edges as he likes, while retaining the source and destination nodes connected. Under the guide of Alice's rules, if the pebble arrives at the destination node, then we say Alice wins the traversing game; otherwise the pebble enters an endless loop without passing through the destination node, then Bob wins. We assume that Alice and Bob both play optimally.
We study the problem: When will Alice have a winning strategy? This actually models a routing recovery problem in Software Defined Networking in which some links may be broken. In this paper, we prove a dichotomy result for certain traversing games, called cyclic-order traversing games. We also give a linear-time algorithm to find the corresponding winning strategy, if one exists.
st-planar graphs
biconnectivity
fault-tolerant routing algorithms
software defined network
Mathematics of computing~Graph theory
Theory of computation~Design and analysis of algorithms
Networks~Network reliability
29:1-29:13
Regular Paper
Yen-Ting
Chen
Yen-Ting Chen
Department of Computer Science, National Chiao Tung University, Hsinchu City, Taiwan
Meng-Tsung
Tsai
Meng-Tsung Tsai
Department of Computer Science, National Chiao Tung University, Hsinchu City, Taiwan
This research was supported in part by the Ministry of Science and Technology of Taiwan under contract MOST grant 107-2218-E-009- 026-MY3, and the Higher Education Sprout Project of National Chiao Tung University and Ministry of Education (MOE), Taiwan.
Shi-Chun
Tsai
Shi-Chun Tsai
Department of Computer Science, National Chiao Tung University, Hsinchu City, Taiwan
This research was supported in part by the Ministry of Science and Technology of Taiwan under contracts MOST 105-2622-8-009-008 and 105-2221-E-009-103-MY3.
10.4230/LIPIcs.ISAAC.2018.29
Fred S. Annexstein, Kenneth A. Berman, Tsan-Sheng Hsu, and Ram Swaminathan. A multi-tree routing scheme using acyclic orientations. Theoretical Computer Science, 240(2):487-494, 2000.
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Yen-Ting Chen, Meng-Tsung Tsai, and Shi-Chun Tsai
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The b-Matching Problem in Distance-Hereditary Graphs and Beyond
We make progress on the fine-grained complexity of Maximum-Cardinality Matching on graphs of bounded clique-width. Quasi linear-time algorithms for this problem have been recently proposed for the important subclasses of bounded-treewidth graphs (Fomin et al., SODA'17) and graphs of bounded modular-width (Coudert et al., SODA'18). We present such algorithm for bounded split-width graphs - a broad generalization of graphs of bounded modular-width, of which an interesting subclass are the distance-hereditary graphs. Specifically, we solve Maximum-Cardinality Matching in O((k log^2{k})*(m+n) * log{n})-time on graphs with split-width at most k. We stress that the existence of such algorithm was not even known for distance-hereditary graphs until our work. Doing so, we improve the state of the art (Dragan, WG'97) and we answer an open question of (Coudert et al., SODA'18). Our work brings more insights on the relationships between matchings and splits, a.k.a., join operations between two vertex-subsets in different connected components. Furthermore, our analysis can be extended to the more general (unit cost) b-Matching problem. On the way, we introduce new tools for b-Matching and dynamic programming over split decompositions, that can be of independent interest.
maximum-cardinality matching
b-matching
FPT in P
split decomposition
distance-hereditary graphs
Mathematics of computing~Graph theory
Theory of computation~Design and analysis of algorithms
30:1-30:13
Regular Paper
This work was supported by the Institutional research programme PN 1819 "Advanced IT resources to support digital transformation processes in the economy and society - RESINFO-TD" (2018), project PN 1819-01-01 "Modeling, simulation, optimization of complex systems and decision support in new areas of IT&C research", funded by the Ministry of Research and Innovation, Romania.
https://arxiv.org/abs/1804.09393
Guillaume
Ducoffe
Guillaume Ducoffe
ICI – National Institute for Research and Development in Informatics, Bucharest, Romania , The Research Institute of the University of Bucharest ICUB, Bucharest, Romania
Alexandru
Popa
Alexandru Popa
University of Bucharest, Bucharest, Romania , ICI – National Institute for Research and Development in Informatics, Bucharest, Romania
10.4230/LIPIcs.ISAAC.2018.30
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Guillaume Ducoffe and Alexandru Popa
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
New Algorithms for Edge Induced König-Egerváry Subgraph Based on Gallai-Edmonds Decomposition
König-Egerváry graphs form an important graph class which has been studied extensively in graph theory. Much attention has also been paid on König-Egerváry subgraphs and König-Egerváry graph modification problems. In this paper, we focus on one König-Egerváry subgraph problem, called the Maximum Edge Induced König Subgraph problem. By exploiting the classical Gallai-Edmonds decomposition, we establish connections between minimum vertex cover, Gallai-Edmonds decomposition structure, maximum matching, maximum bisection, and König-Egerváry subgraph structure. We obtain a new structural property of König-Egerváry subgraph: every graph G=(V, E) has an edge induced König-Egerváry subgraph with at least 2|E|/3 edges. Based on the new structural property proposed, an approximation algorithm with ratio 10/7 for the Maximum Edge Induced König Subgraph problem is presented, improving the current best ratio of 5/3. To the best of our knowledge, this paper is the first one establishing the connection between Gallai-Edmonds decomposition and König-Egerváry graphs. Using 2|E|/3 as a lower bound, we define the Edge Induced König Subgraph above lower bound problem, and give a kernel of at most 30k edges for the problem.
König-Egerváry graph
Gallai-Edmonds decomposition
Mathematics of computing~Graph algorithms
Mathematics of computing~Approximation algorithms
31:1-31:12
Regular Paper
This work is supported by the National Natural Science Foundation of China under Grants (61420106009, 61872450, 61828205, 61672536).
Qilong
Feng
Qilong Feng
School of Information Science and Engineering, Central South University, Changsha, P.R. China
Guanlan
Tan
Guanlan Tan
School of Information Science and Engineering, Central South University, Changsha, P.R. China
Senmin
Zhu
Senmin Zhu
School of Information Science and Engineering, Central South University, Changsha, P.R. China
Bin
Fu
Bin Fu
Department of Computer Science, University of Texas-Rio Grande Valley, USA
Jianxin
Wang
Jianxin Wang
School of Information Science and Engineering, Central South University, Changsha, P.R. China
10.4230/LIPIcs.ISAAC.2018.31
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http://arxiv.org/abs/1611.06795
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László Lovász. Ear-decompositions of matching-covered graphs. Combinatorica, 3(1):105-117, 1983.
László Lovász and Michael D. Plummer. Matching Theory. North-Holland, Amsterdam, 1986.
Eric Mcdermid. A 3/2-approximation algorithm for general stable marriage. In Proceedings of the 36th International Colloquium on Automata, Languages, and Programming(ICALP), pages 689-700, 2009.
Sounaka Mishra, Shijin Rajakrishnan, and Saket Saurabh. On approximability of optimization problems related to Red/Blue-split graphs. Theoretical Computer Science, 690:104-113, 2017.
Sounaka Mishra, Venkatesh Raman, Saket Saurabh, and Somnath Sikdar. König deletion sets and vertex covers above the matching size. In Proceedings of the 19th International Symposium on Algorithms and Computation(ISAAC), pages 836-847, 2008.
Sounaka Mishra, Venkatesh Raman, Saket Saurabh, Somnath Sikdar, and C. R. Subramanian. The complexity of finding subgraphs whose matching number equals the vertex cover number. In Proceedings of the 18th International Symposium on Algorithms and Computation(ISAAC), pages 268-279, 2007.
Sounaka Mishra, Venkatesh Raman, Saket Saurabh, Somnath Sikdar, and C. R. Subramanian. The complexity of König subgraph problems and above-guarantee vertex cover. Algorithmica, 61(4):857-881, 2011.
Matthias Mnich and Rico Zenklusen. Bisections above tight lower bounds. In Proceedings of the 38th International Workshop on Graph-Theoretic Concepts in Computer Science(WG), pages 184-193, 2012.
N. S. Narayanaswamy, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. LP can be a cure for parameterized problems. In Proceedings of the 29th Internatinal Symposium on Theoretical Aspects of Computer Science(STACS)), pages 338-349, 2012.
Ojas Parekh. Edge dominating and hypomatchable sets. In Proceedings of the 13th annual ACM-SIAM Symposium on Discrete Algorithms(SODA), pages 287-291, 2002.
Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Paths, flowers and vertex cover. In Proceedings of the 19th Annual European Symposium on Algorithms(ESA), pages 382-393, 2011.
Igor Razgon and Barry O'Sullivan. Almost 2-SAT is fixed-parameter tractable. Journal of Computer and System Sciences, 75(8):435-450, 2009.
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C. R. Subramanian. Vertex covers: parameterizing above the requirement. IMSc Technical Report, 2001.
Qilong Feng, Guanlan Tan, Senmin Zhu, Bin Fu, and Jianxin Wang
Creative Commons Attribution 3.0 Unported license
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Computing Approximate Statistical Discrepancy
Consider a geometric range space (X,A) where X is comprised of the union of a red set R and blue set B. Let Phi(A) define the absolute difference between the fraction of red and fraction of blue points which fall in the range A. The maximum discrepancy range A^* = arg max_{A in (X,A)} Phi(A). Our goal is to find some A^ in (X,A) such that Phi(A^*) - Phi(A^) <= epsilon. We develop general algorithms for this approximation problem for range spaces with bounded VC-dimension, as well as significant improvements for specific geometric range spaces defined by balls, halfspaces, and axis-aligned rectangles. This problem has direct applications in discrepancy evaluation and classification, and we also show an improved reduction to a class of problems in spatial scan statistics.
Scan Statistics
Discrepancy
Rectangles
Theory of computation~Computational geometry
32:1-32:13
Regular Paper
https://arxiv.org/abs/1804.11287
Michael
Matheny
Michael Matheny
University of Utah, Salt Lake City, USA
Jeff M.
Phillips
Jeff M. Phillips
University of Utah, Salt Lake City, USA
Thanks to supported by NSF CCF-1350888, IIS-1251019, ACI-1443046, CNS-1514520, and CNS-1564287.
10.4230/LIPIcs.ISAAC.2018.32
Deepak Agarwal, Andrew McGregor, Jeff M. Phillips, Suresh Venkatasubramanian, and Zhengyuan Zhu. Spatial Scan Statistics: Approximations and Performance Study. In KDD, 2006.
Deepak Agarwal, Jeff M. Phillips, and Suresh Venkatasubramanian. The Hunting of the Bump: On Maximizing Statistical Discrepancy. SODA, 2006.
Arturs Backurs, Nishanth Dikkala, and Christos Tzamos. Tight Hardness Results for Maximum Weight Rectangles. In ICALP, 2016. URL: http://arxiv.org/abs/1602.05837.
http://arxiv.org/abs/1602.05837
Jérémy Barbay, Timothy M. Chan, Gonzalo Navarro, and Pablo Pérez-Lantero. Maximum-weight planar boxes in time (and better). Information Processing Letters, 114(8):437-445, 2014.
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Michael Matheny and Jeff M. Phillips. Computing Approximate Statistical Discrepancy. Technical report, arXiv, 2018. URL: http://arxiv.org/abs/1804.11287.
http://arxiv.org/abs/1804.11287
Michael Matheny and Jeff M. Phillips. Practical Low-Dimensional Halfspace Range Space Sampling. In European Symposium on Algorithms, 2018. URL: http://arxiv.org/abs/1804.11307.
http://arxiv.org/abs/1804.11307
Michael Matheny, Raghvendra Singh, Liang Zhang, Kaiqiang Wang, and Jeff M. Phillips. Scalable Spatial Scan Statistics Through Sampling. In SIGSPATIAL, 2016.
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Michael Matheny and Jeff Phillips
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Diversity Maximization in Doubling Metrics
Diversity maximization is an important geometric optimization problem with many applications in recommender systems, machine learning or search engines among others. A typical diversification problem is as follows: Given a finite metric space (X,d) and a parameter k in N, find a subset of k elements of X that has maximum diversity. There are many functions that measure diversity. One of the most popular measures, called remote-clique, is the sum of the pairwise distances of the chosen elements. In this paper, we present novel results on three widely used diversity measures: Remote-clique, remote-star and remote-bipartition.
Our main result are polynomial time approximation schemes for these three diversification problems under the assumption that the metric space is doubling. This setting has been discussed in the recent literature. The existence of such a PTAS however was left open.
Our results also hold in the setting where the distances are raised to a fixed power q >= 1, giving rise to more variants of diversity functions, similar in spirit to the variations of clustering problems depending on the power applied to the pairwise distances. Finally, we provide a proof of NP-hardness for remote-clique with squared distances in doubling metric spaces.
Remote-clique
remote-star
remote-bipartition
doubling dimension
grid rounding
epsilon-nets
polynomial time approximation scheme
facility location
information retrieval
Theory of computation~Facility location and clustering
33:1-33:12
Regular Paper
https://arxiv.org/abs/1809.09521
Alfonso
Cevallos
Alfonso Cevallos
Swiss Federal Institute of Technology (ETH), Switzerland
https://orcid.org/0000-0001-8622-5830
Friedrich
Eisenbrand
Friedrich Eisenbrand
École Polytechnique Fédérale de Lausanne (EPFL), Switzerland
https://orcid.org/0000-0001-7928-1076
The second author acknowledges support from the Swiss National Science Foundation grant 163071, "Convexity, geometry of numbers, and the complexity of integer programming".
Sarah
Morell
Sarah Morell
Technische Universität Berlin (TU Berlin), Germany
Work conducted while the third author was affiliated to EPFL, Switzerland.
10.4230/LIPIcs.ISAAC.2018.33
Z. Abbassi, V. S. Mirrokni, and M. Thakur. Diversity maximization under matroid constraints. In 19th Conference on Knowledge Discovery and Data Mining (SIGKDD), pages 32-40. ACM, 2013.
S. Aghamolaei, M. Farhadi, and H. Zarrabi-Zadeh. Diversity Maximization via Composable Coresets. In 27th Canadian Conference on Computational Geometry (CCCG), page 43, 2015.
N. Alon, S. Arora, R. Manokaran, D. Moshkovitz, and O. Weinstein. Inapproximability of densest κ-subgraph from average case hardness. Unpublished manuscript, 2011.
A. Bhaskara, M. Ghadiri, V. Mirrokni, and O. Svensson. Linear relaxations for finding diverse elements in metric spaces. In Advances in Neural Information Processing Systems, pages 4098-4106, 2016.
B. Birnbaum and K. J. Goldman. An improved analysis for a greedy remote-clique algorithm using factor-revealing LPs. Algorithmica, 55(1):42-59, 2009.
A. Borodin, H. C. Lee, and Y. Ye. Max-sum diversification, monotone submodular functions and dynamic updates. In Proceedings of the 31st Symposium on Principles of Database Systems, pages 155-166, 2012.
M. Ceccarello, A. Pietracaprina, G. Pucci, and E. Upfal. MapReduce and streaming algorithms for diversity maximization in metric spaces of bounded doubling dimension. Proceedings of the VLDB Endowment, 10(5):469-480, 2017.
A. Cevallos, F. Eisenbrand, and S. Morell. Diversity maximization in doubling metrics. arXiv preprint, 2018. URL: http://arxiv.org/abs/1809.09521.
http://arxiv.org/abs/1809.09521
A. Cevallos, F. Eisenbrand, and R. Zenklusen. Max-Sum Diversity via Convex Programming. In 32nd Annual Symposium on Computational Geometry (SoCG), pages 26:1-26:14, 2016.
A. Cevallos, F. Eisenbrand, and R. Zenklusen. Local Search for Max-Sum Diversification. In 28th Symposium on Discrete Algorithms (SODA), pages 130-142. SIAM, 2017.
B. Chandra and M. M. Halldórsson. Approximation algorithms for dispersion problems. Journal of algorithms, 38(2):438-465, 2001.
V. Cohen-Addad, P. N. Klein, and C. Mathieu. Local search yields approximation schemes for k-means and k-median in Euclidean and minor-free metrics. In 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 353-364. IEEE, 2016.
J. P. Cunningham and Z. Ghahramani. Linear dimensionality reduction: survey, insights, and generalizations. Journal of Machine Learning Research, 16(1):2859-2900, 2015.
S. Dasgupta and Y. Freund. Random projection trees and low dimensional manifolds. In Proceedings of the 40th Symposium on Theory of Computing, pages 537-546. ACM, 2008.
S. P. Fekete and H. Meijer. Maximum dispersion and geometric maximum weight cliques. Algorithmica, 38(3):501-511, 2004.
W. Fernandez de la Vega, M. Karpinski, and C. Kenyon. A Polynomial Time Approximation Scheme for Metric MIN-BISECTION. Electronic Colloquium on Computational Complexity (ECCC), pages 1-12, 2002.
S. Gollapudi and A. Sharma. An axiomatic approach for result diversification. In 18th International Conference on World Wide Web (WWW), pages 381-390. ACM, 2009.
L. A. Gottlieb and R. Krauthgamer. A nonlinear approach to dimension reduction. Discrete &Computational Geometry, 54(2):291-315, 2015.
S. Har-Peled. Geometric approximation algorithms, volume 173. American mathematical society Boston, 2011.
R. Hassin, S. Rubinstein, and A. Tamir. Approximation algorithms for maximum dispersion. Operations Research Letters, 21(3):133-137, 1997.
P. Indyk, S. Mahabadi, M. Mahdian, and V. S. Mirrokni. Composable core-sets for diversity and coverage maximization. In 33rd ACM Symposium on Principles of Database Systems, pages 100-108, 2014.
P. Indyk and A. Naor. Nearest-neighbor-preserving embeddings. ACM Transactions on Algorithms (TALG), 3(3):31, 2007.
L. Qin, J. X. Yu, and L. Chang. Diversifying top-k results. Proceedings of the VLDB Endowment, 5(11):1124-1135, 2012.
F. Radlinski and S. Dumais. Improving personalized web search using result diversification. In 29th SIGIR Conference on Research and Development in Information Retrieval, pages 691-692. ACM, 2006.
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J. B. Tenenbaum, V. de Silva, and J. C Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290(5500):2319-2323, 2000.
N. Vasconcelos. Feature selection by maximum marginal diversity. In Advances in Neural Information Processing Systems, pages 1375-1382, 2003.
M. R. Vieira, H. L. Razente, M. C. N. Barioni, M. Hadjieleftheriou, D. Srivastava, C. Traina, and V. J. Tsotras. On query result diversification. In 27th International Conference on Data Engineering (ICDE), pages 1163-1174. IEEE, 2011.
D.W. Wang and Y.S. Kuo. A study on two geometric location problems. Information processing letters, 28(6):281-286, 1988.
Alfonso Cevallos, Friedrich Eisenbrand, and Sarah Morell
Creative Commons Attribution 3.0 Unported license
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On Polynomial Time Constructions of Minimum Height Decision Tree
A decision tree T in B_m:={0,1}^m is a binary tree where each of its internal nodes is labeled with an integer in [m]={1,2,...,m}, each leaf is labeled with an assignment a in B_m and each internal node has two outgoing edges that are labeled with 0 and 1, respectively. Let A subset {0,1}^m. We say that T is a decision tree for A if (1) For every a in A there is one leaf of T that is labeled with a. (2) For every path from the root to a leaf with internal nodes labeled with i_1,i_2,...,i_k in[m], a leaf labeled with a in A and edges labeled with xi_{i_1},...,xi_{i_k}in {0,1}, a is the only element in A that satisfies a_{i_j}=xi_{i_j} for all j=1,...,k.
Our goal is to write a polynomial time (in n:=|A| and m) algorithm that for an input A subseteq B_m outputs a decision tree for A of minimum depth. This problem has many applications that include, to name a few, computer vision, group testing, exact learning from membership queries and game theory.
Arkin et al. and Moshkov [Esther M. Arkin et al., 1998; Mikhail Ju. Moshkov, 2004] gave a polynomial time (ln |A|)- approximation algorithm (for the depth). The result of Dinur and Steurer [Irit Dinur and David Steurer, 2014] for set cover implies that this problem cannot be approximated with ratio (1-o(1))* ln |A|, unless P=NP. Moshkov studied in [Mikhail Ju. Moshkov, 2004; Mikhail Ju. Moshkov, 1982; Mikhail Ju. Moshkov, 1982] the combinatorial measure of extended teaching dimension of A, ETD(A). He showed that ETD(A) is a lower bound for the depth of the decision tree for A and then gave an exponential time ETD(A)/log(ETD(A))-approximation algorithm and a polynomial time 2(ln 2)ETD(A)-approximation algorithm.
In this paper we further study the ETD(A) measure and a new combinatorial measure, DEN(A), that we call the density of the set A. We show that DEN(A) <=ETD(A)+1. We then give two results. The first result is that the lower bound ETD(A) of Moshkov for the depth of the decision tree for A is greater than the bounds that are obtained by the classical technique used in the literature. The second result is a polynomial time (ln 2)DEN(A)-approximation (and therefore (ln 2)ETD(A)-approximation) algorithm for the depth of the decision tree of A.
We then apply the above results to learning the class of disjunctions of predicates from membership queries [Nader H. Bshouty et al., 2017]. We show that the ETD of this class is bounded from above by the degree d of its Hasse diagram. We then show that Moshkov algorithm can be run in polynomial time and is (d/log d)-approximation algorithm. This gives optimal algorithms when the degree is constant. For example, learning axis parallel rays over constant dimension space.
Decision Tree
Minimal Depth
Approximation algorithms
Mathematics of computing~Combinatorial optimization
34:1-34:12
Regular Paper
https://arxiv.org/abs/1802.00233
Nader
H. Bshouty
Nader H. Bshouty
Department of Computer Science, Technion, Haifa, Israel
Waseem
Makhoul
Waseem Makhoul
Department of Computer Science, Technion, Haifa, Israel
10.4230/LIPIcs.ISAAC.2018.34
Dana Angluin. Queries and Concept Learning. Machine Learning, 2(4):319-342, 1988. URL: http://dx.doi.org/10.1023/A:1022821128753.
http://dx.doi.org/10.1023/A:1022821128753
Martin Anthony, Graham R. Brightwell, David A. Cohen, and John Shawe-Taylor. On Exact Specification by Examples. In Proceedings of the Fifth Annual ACM Conference on Computational Learning Theory, COLT 1992, Pittsburgh, PA, USA, July 27-29, 1992., pages 311-318, 1992. URL: http://dx.doi.org/10.1145/130385.130420.
http://dx.doi.org/10.1145/130385.130420
Esther M. Arkin, Michael T. Goodrich, Joseph S. B. Mitchell, David M. Mount, Christine D. Piatko, and Steven Skiena. Point Probe Decision Trees for Geometric Concept Classes. In Algorithms and Data Structures, Third Workshop, WADS '93, Montréal, Canada, August 11-13, 1993, Proceedings, pages 95-106, 1993. URL: http://dx.doi.org/10.1007/3-540-57155-8_239.
http://dx.doi.org/10.1007/3-540-57155-8_239
Esther M. Arkin, Henk Meijer, Joseph S. B. Mitchell, David Rappaport, and Steven Skiena. Decision trees for geometric models. Int. J. Comput. Geometry Appl., 8(3):343-364, 1998. URL: http://dx.doi.org/10.1142/S0218195998000175.
http://dx.doi.org/10.1142/S0218195998000175
Nader H. Bshouty, Dana Drachsler-Cohen, Martin T. Vechev, and Eran Yahav. Learning Disjunctions of Predicates. In Proceedings of the 30th Conference on Learning Theory, COLT 2017, Amsterdam, The Netherlands, 7-10 July 2017, pages 346-369, 2017. URL: http://proceedings.mlr.press/v65/bshouty17a.html.
http://proceedings.mlr.press/v65/bshouty17a.html
Nader H. Bshouty and Waseem Makhoul. On Polynomial time Constructions of Minimum Height Decision Tree. CoRR, abs/1802.00233, 2018. URL: http://arxiv.org/abs/1802.00233.
http://arxiv.org/abs/1802.00233
Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 624-633, 2014. URL: http://dx.doi.org/10.1145/2591796.2591884.
http://dx.doi.org/10.1145/2591796.2591884
M. R. Garey. Optimal Binary Identification Procedures. SIAM Journal on Applied Mathematics, 23(2):173-186, 1971. URL: http://epubs.siam.org/doi/abs/10.1137/0123019.
http://epubs.siam.org/doi/abs/10.1137/0123019
Sally A. Goldman and Michael J. Kearns. On the Complexity of Teaching. J. Comput. Syst. Sci., 50(1):20-31, 1995. URL: http://dx.doi.org/10.1006/jcss.1995.1003.
http://dx.doi.org/10.1006/jcss.1995.1003
Tibor Hegedüs. Generalized Teaching Dimensions and the Query Complexity of Learning. In Proceedings of the Eigth Annual Conference on Computational Learning Theory, COLT 1995, Santa Cruz, California, USA, July 5-8, 1995, pages 108-117, 1995. URL: http://dx.doi.org/10.1145/225298.225311.
http://dx.doi.org/10.1145/225298.225311
Laurent Hyafil and Ronald L. Rivest. Constructing Optimal Binary Decision Trees is NP-Complete. Inf. Process. Lett., 5(1):15-17, 1976. URL: http://dx.doi.org/10.1016/0020-0190(76)90095-8.
http://dx.doi.org/10.1016/0020-0190(76)90095-8
Eduardo Sany Laber and Loana Tito Nogueira. On the hardness of the minimum height decision tree problem. Discrete Applied Mathematics, 144(1-2):209-212, 2004. URL: http://dx.doi.org/10.1016/j.dam.2004.06.002.
http://dx.doi.org/10.1016/j.dam.2004.06.002
Mikhail Ju. Moshkov. On conditional tests. Problemy Kibernetiki. and Sov. Phys. Dokl., 27(7):528-530, 1982.
Mikhail Ju. Moshkov. On conditional tests. Problems of Cybernetics, Nauka, Moscow, 40:131-170, 1982.
Mikhail Ju. Moshkov. Greedy Algorithm of Decision Tree Construction for Real Data Tables. Transactions on Rough Sets I, Lecture Notes in Computer Science 3100, Springer-Verlag, Heidelberg., pages 161-168, 2004. URL: http://dx.doi.org/10.1007/978-3-540-27794-1_7.
http://dx.doi.org/10.1007/978-3-540-27794-1_7
Ayumi Shinohara. Teachability in Computational Learning. New Generation Comput., 8(4):337-347, 1991. URL: http://dx.doi.org/10.1007/BF03037091.
http://dx.doi.org/10.1007/BF03037091
Nader H. Bshouty and Waseem Makhoul
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Improved Algorithms for the Shortest Vector Problem and the Closest Vector Problem in the Infinity Norm
Ajtai, Kumar and Sivakumar [Ajtai et al., 2001] gave the first 2^O(n) algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. The algorithm starts with N in 2^O(n) randomly chosen vectors in the lattice and employs a sieving procedure to iteratively obtain shorter vectors in the lattice, and eventually obtaining the shortest non-zero vector. The running time of the sieving procedure is quadratic in N. Subsequent works [Arvind and Joglekar, 2008; Blömer and Naewe, 2009] generalized the algorithm to other norms.
We study this problem for the special but important case of the l_infty norm. We give a new sieving procedure that runs in time linear in N, thereby improving the running time of the algorithm for SVP in the l_infty norm. As in [Ajtai et al., 2002; Blömer and Naewe, 2009], we also extend this algorithm to obtain significantly faster algorithms for approximate versions of the shortest vector problem and the closest vector problem (CVP) in the l_infty norm.
We also show that the heuristic sieving algorithms of Nguyen and Vidick [Nguyen and Vidick, 2008] and Wang et al. [Wang et al., 2011] can also be analyzed in the l_infty norm. The main technical contribution in this part is to calculate the expected volume of intersection of a unit ball centred at origin and another ball of a different radius centred at a uniformly random point on the boundary of the unit ball. This might be of independent interest.
Lattice
Shortest Vector Problem
Closest Vector Problem
l_infty norm
Theory of computation~Randomness, geometry and discrete structures
35:1-35:13
Regular Paper
https://arxiv.org/abs/1801.02358
Divesh
Aggarwal
Divesh Aggarwal
Centre for Quantum Technologies and School of Computing, National University of Singapore
This research was partially funded by the Singapore Ministry of Education and the National Research Foundation, also through the Tier 3 Grant "Random numbers from quantum processes",MOE2012-T3-1-009.
Priyanka
Mukhopadhyay
Priyanka Mukhopadhyay
Centre for Quantum Technologies, National University of Singapore
This research was funded by the National Research Foundation,Prime Minister’s Office, Singapore and the Ministry of Education, Singapore.
10.4230/LIPIcs.ISAAC.2018.35
Divesh Aggarwal, Daniel Dadush, Oded Regev, and Noah Stephens-Davidowitz. Solving the Shortest Vector Problem in 2ⁿ time via Discrete Gaussian Sampling. In STOC, 2015. Full version available at URL: https://arxiv.org/abs/1412.7994.
https://arxiv.org/abs/1412.7994
Divesh Aggarwal, Daniel Dadush, and Noah Stephens-Davidowitz. Solving the Closest Vector Problem in 2^n Time-The Discrete Gaussian Strikes Again! In Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pages 563-582. IEEE, 2015.
Divesh Aggarwal and Priyanka Mukhopadhyay. Faster algorithms for SVP and CVP in the 𝓁_∞ norm. arXiv preprint, 2018. URL: http://arxiv.org/abs/1801.02358.
http://arxiv.org/abs/1801.02358
Divesh Aggarwal and Noah Stephens-Davidowitz. Just Take the Average! An Embarrassingly Simple 2ⁿ-Time Algorithm for SVP (and CVP). arXiv preprint, 2017. URL: http://arxiv.org/abs/1709.01535.
http://arxiv.org/abs/1709.01535
Miklós Ajtai, Ravi Kumar, and D. Sivakumar. A Sieve Algorithm for the Shortest Lattice Vector Problem. In STOC, pages 601-610, 2001. URL: http://dx.doi.org/10.1145/380752.380857.
http://dx.doi.org/10.1145/380752.380857
Miklós Ajtai, Ravi Kumar, and D. Sivakumar. Sampling short lattice vectors and the closest lattice vector problem. In CCC, pages 41-45, 2002.
Vikraman Arvind and Pushkar S Joglekar. Some sieving algorithms for lattice problems. In LIPIcs-Leibniz International Proceedings in Informatics, volume 2. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2008.
L. Babai. On Lovász' lattice reduction and the nearest lattice point problem. Combinatorica, 6(1):1-13, 1986. URL: http://dx.doi.org/10.1007/BF02579403.
http://dx.doi.org/10.1007/BF02579403
Anja Becker, Léo Ducas, Nicolas Gama, and Thijs Laarhoven. New directions in nearest neighbor searching with applications to lattice sieving. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 10-24. Society for Industrial and Applied Mathematics, 2016.
Huck Bennett, Alexander Golovnev, and Noah Stephens-Davidowitz. On the Quantitative Hardness of CVP. arXiv preprint, 2017. URL: http://arxiv.org/abs/1704.03928.
http://arxiv.org/abs/1704.03928
Johannes Blömer and Stefanie Naewe. Sampling methods for shortest vectors, closest vectors and successive minima. Theoretical Computer Science, 410(18):1648-1665, 2009.
Matthijs J Coster, Antoine Joux, Brian A LaMacchia, Andrew M Odlyzko, Claus-Peter Schnorr, and Jacques Stern. Improved low-density subset sum algorithms. computational complexity, 2(2):111-128, 1992.
Léo Ducas, Tancrède Lepoint, Vadim Lyubashevsky, Peter Schwabe, Gregor Seiler, and Damien Stehlé. CRYSTALS-Dilithium: Digital Signatures from Module Lattices. Technical report, IACR Cryptology ePrint Archive, 2017: 633, 2017.
Friedrich Eisenbrand, Nicolai Hähnle, and Martin Niemeier. Covering cubes and the closest vector problem. In Proceedings of the twenty-seventh annual symposium on Computational geometry, pages 417-423. ACM, 2011.
O. Goldreich, D. Micciancio, S. Safra, and J.-P. Seifert. Approximating shortest lattice vectors is not harder than approximating closest lattice vectors. Information Processing Letters, 71(2):55-61, 1999. URL: http://dx.doi.org/10.1016/S0020-0190(99)00083-6.
http://dx.doi.org/10.1016/S0020-0190(99)00083-6
Ravi Kannan. Minkowski’s Convex Body Theorem and Integer Programming. Mathematics of Operations Research, 12(3):pp. 415-440, 1987. URL: http://www.jstor.org/stable/3689974.
http://www.jstor.org/stable/3689974
Susan Landau and Gary Lee Miller. Solvability by radicals is in polynomial time. In Proceedings of the fifteenth annual ACM symposium on Theory of computing, pages 140-151. ACM, 1983.
A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Math. Ann., 261(4):515-534, 1982. URL: http://dx.doi.org/10.1007/BF01457454.
http://dx.doi.org/10.1007/BF01457454
Hendrik W Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of operations research, 8(4):538-548, 1983.
Mingjie Liu, Xiaoyun Wang, Guangwu Xu, and Xuexin Zheng. Shortest lattice vectors in the presence of gaps. IACR Cryptology ePrint Archive, 2011:139, 2011.
Daniele Micciancio and Panagiotis Voulgaris. A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. SIAM Journal on Computing, 42(3):1364-1391, 2013.
Phong Q Nguyen and Jacques Stern. The two faces of lattices in cryptology. In Cryptography and lattices, pages 146-180. Springer, 2001.
Phong Q Nguyen and Thomas Vidick. Sieve algorithms for the shortest vector problem are practical. Journal of Mathematical Cryptology, 2(2):181-207, 2008.
Chris Peikert et al. A decade of lattice cryptography. Foundations and Trendsregistered in Theoretical Computer Science, 10(4):283-424, 2016.
Xavier Pujol and Damien Stehlé. Solving the Shortest Lattice Vector Problem in Time 2^2.465n. IACR Cryptology ePrint Archive, 2009:605, 2009.
Oded Regev. Lecture notes on lattices in computer science, 2009.
Xiaoyun Wang, Mingjie Liu, Chengliang Tian, and Jingguo Bi. Improved Nguyen-Vidick heuristic sieve algorithm for shortest vector problem. In Proceedings of the 6th ACM Symposium on Information, Computer and Communications Security, pages 1-9. ACM, 2011.
Wei Wei, Mingjie Liu, and Xiaoyun Wang. Finding Shortest Lattice Vectors in the Presence of Gaps. In Topics in Cryptology - CT-RSA 2015, The Cryptographer’s Track at the RSA Conference 2015, San Francisco, CA, USA, April 20-24, 2015. Proceedings, pages 239-257, 2015. URL: http://dx.doi.org/10.1007/978-3-319-16715-2_13.
http://dx.doi.org/10.1007/978-3-319-16715-2_13
Divesh Aggarwal and Priyanka Mukhopadhyay
Creative Commons Attribution 3.0 Unported license
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An Adaptive Version of Brandes' Algorithm for Betweenness Centrality
Betweenness centrality - measuring how many shortest paths pass through a vertex - is one of the most important network analysis concepts for assessing the relative importance of a vertex. The well-known algorithm of Brandes [2001] computes, on an n-vertex and m-edge graph, the betweenness centrality of all vertices in O(nm) worst-case time. In follow-up work, significant empirical speedups were achieved by preprocessing degree-one vertices and by graph partitioning based on cut vertices. We further contribute an algorithmic treatment of degree-two vertices, which turns out to be much richer in mathematical structure than the case of degree-one vertices. Based on these three algorithmic ingredients, we provide a strengthened worst-case running time analysis for betweenness centrality algorithms. More specifically, we prove an adaptive running time bound O(kn), where k < m is the size of a minimum feedback edge set of the input graph.
network science
social network analysis
centrality measures
shortest paths
tree-like graphs
efficient pre- and postprocessing
FPT in P
Theory of computation~Graph algorithms analysis
36:1-36:13
Regular Paper
https://arxiv.org/abs/1802.06701
Matthias
Bentert
Matthias Bentert
Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Germany
Alexander
Dittmann
Alexander Dittmann
Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Germany
Leon
Kellerhals
Leon Kellerhals
Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Germany
Supported by DFG project FPTinP, NI 369/16.
André
Nichterlein
André Nichterlein
Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Germany
Rolf
Niedermeier
Rolf Niedermeier
Algorithmics and Computational Complexity, Faculty IV, TU Berlin, Germany
10.4230/LIPIcs.ISAAC.2018.36
Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic Equivalences Between Graph Centrality Problems, APSP and Diameter. In Proc. of 26th SODA, pages 1681-1697. SIAM, 2015.
Miriam Baglioni, Filippo Geraci, Marco Pellegrini, and Ernesto Lastres. Fast exact computation of betweenness centrality in social networks. In Proc. of 4th ASONAM, pages 450-456. IEEE Computer Society, 2012.
Ulrik Brandes. A faster algorithm for betweenness centrality. J Math Sociol, 25(2):163-177, 2001.
Wouter De Nooy, Andrej Mrvar, and Vladimir Batagelj. Exploratory Social Network Analysis with Pajek. Cambridge University Press, 2011.
Vladimir Estivill-Castro and Derick Wood. A Survey of Adaptive Sorting Algorithms. ACM Comput Surv, 24(4):441-476, 1992.
Linton Freeman. A set of measures of centrality based on betweenness. Sociometry, 40:35-41, 1977.
Archontia C. Giannopoulou, George B. Mertzios, and Rolf Niedermeier. Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs. Theor Comput Sci, 689:67-95, 2017.
David S. Johnson. The genealogy of theoretical computer science: A preliminary report. ACM SIGACT News, 16(2):36-49, 1984.
Mark E. J. Newman. Who Is the Best Connected Scientist? A Study of Scientific Coauthorship Networks. In Proc. of 23rd CNLS, pages 337-370. Springer, 2004.
André Nichterlein, Rolf Niedermeier, Johannes Uhlmann, and Mathias Weller. On tractable cases of Target Set Selection. SNAM, 3(2):233-256, 2013.
Kim Norlen, Gabriel Lucas, Michael Gebbie, and John Chuang. EVA: Extraction, visualization and analysis of the telecommunications and media ownership network. In Proc. of 14th ITS, 2002.
Rami Puzis, Yuval Elovici, Polina Zilberman, Shlomi Dolev, and Ulrik Brandes. Topology manipulations for speeding betweenness centrality computation. J Comp Net, 3(1):84-112, 2015.
Ahmet Erdem Sariyüce, Kamer Kaya, Erik Saule, and Ümit V. Çatalyürek. Graph Manipulations for Fast Centrality Computation. ACM Trans Knowl Discov Data, 11(3):26:1-26:25, 2017.
Flavio Vella, Massimo Bernaschi, and Giancarlo Carbone. Dynamic Merging of Frontiers for Accelerating the Evaluation of Betweenness Centrality. ACM JEA, 23(1):1.4:1-1.4:19, 2018.
Wei Wang and Choon Yik Tang. Distributed computation of node and edge betweenness on tree graphs. In Proc. of 52nd CDC, pages 43-48. IEEE, 2013.
Matthias Bentert, Alexander Dittmann, Leon Kellerhals, André Nichterlein, and Rolf Niedermeier
Creative Commons Attribution 3.0 Unported license
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Algorithms for Coloring Reconfiguration Under Recolorability Constraints
Coloring reconfiguration is one of the most well-studied reconfiguration problems. In the problem, we are given two (vertex-)colorings of a graph using at most k colors, and asked to determine whether there exists a transformation between them by recoloring only a single vertex at a time, while maintaining a k-coloring throughout. It is known that this problem is solvable in linear time for any graph if k <=3, while is PSPACE-complete for a fixed k >= 4. In this paper, we further investigate the problem from the viewpoint of recolorability constraints, which forbid some pairs of colors to be recolored directly. More specifically, the recolorability constraint is given in terms of an undirected graph R such that each node in R corresponds to a color, and each edge in R represents a pair of colors that can be recolored directly. In this paper, we give a linear-time algorithm to solve the problem under such a recolorability constraint if R is of maximum degree at most two. In addition, we show that the minimum number of recoloring steps required for a desired transformation can be computed in linear time for a yes-instance. We note that our results generalize the known positive ones for coloring reconfiguration.
combinatorial reconfiguration
graph algorithm
graph coloring
Mathematics of computing~Graph algorithms
37:1-37:13
Regular Paper
Hiroki
Osawa
Hiroki Osawa
Graduate School of Information Sciences, Tohoku University, Japan
Akira
Suzuki
Akira Suzuki
Graduate School of Information Sciences, Tohoku University, Japan
https://orcid.org/0000-0002-5212-0202
Partially supported by JST CREST Grant Number JPMJCR1402, and JSPS KAKENHI Grant Numbers JP17K12636 and JP18H04091, Japan.
Takehiro
Ito
Takehiro Ito
Graduate School of Information Sciences, Tohoku University, Japan
https://orcid.org/0000-0002-9912-6898
Partially supported by JST CREST Grant Number JPMJCR1402, and JSPS KAKENHI Grant Numbers JP16K00004 and JP18H04091, Japan.
Xiao
Zhou
Xiao Zhou
Graduate School of Information Sciences, Tohoku University, Japan
Partially supported by JSPS KAKENHI Grant Number JP16K00003, Japan.
10.4230/LIPIcs.ISAAC.2018.37
Marthe Bonamy and Nicolas Bousquet. Recoloring graphs via tree decompositions. Eur. J. Comb., 69:200-213, 2018. URL: http://dx.doi.org/10.1016/j.ejc.2017.10.010.
http://dx.doi.org/10.1016/j.ejc.2017.10.010
Marthe Bonamy, Matthew Johnson, Ioannis Lignos, Viresh Patel, and Daniël Paulusma. Reconfiguration graphs for vertex colourings of chordal and chordal bipartite graphs. J. Comb. Optim., 27(1):132-143, 2014. URL: http://dx.doi.org/10.1007/s10878-012-9490-y.
http://dx.doi.org/10.1007/s10878-012-9490-y
Paul S. Bonsma and Luis Cereceda. Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theor. Comput. Sci., 410(50):5215-5226, 2009. URL: http://dx.doi.org/10.1016/j.tcs.2009.08.023.
http://dx.doi.org/10.1016/j.tcs.2009.08.023
Paul S. Bonsma, Amer E. Mouawad, Naomi Nishimura, and Venkatesh Raman. The Complexity of Bounded Length Graph Recoloring and CSP Reconfiguration. In Marek Cygan and Pinar Heggernes, editors, Parameterized and Exact Computation - 9th International Symposium, IPEC 2014, Wroclaw, Poland, September 10-12, 2014. Revised Selected Papers, volume 8894 of Lecture Notes in Computer Science, pages 110-121. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-13524-3_10.
http://dx.doi.org/10.1007/978-3-319-13524-3_10
Paul S. Bonsma and Daniël Paulusma. Using Contracted Solution Graphs for Solving Reconfiguration Problems. In Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, editors, 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, volume 58 of LIPIcs, pages 20:1-20:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.20.
http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.20
Richard C. Brewster, Sean McGuinness, Benjamin Moore, and Jonathan A. Noel. A dichotomy theorem for circular colouring reconfiguration. Theor. Comput. Sci., 639:1-13, 2016. URL: http://dx.doi.org/10.1016/j.tcs.2016.05.015.
http://dx.doi.org/10.1016/j.tcs.2016.05.015
Luis Cereceda, Jan van den Heuvel, and Matthew Johnson. Finding paths between 3-colorings. Journal of Graph Theory, 67(1):69-82, 2011. URL: http://dx.doi.org/10.1002/jgt.20514.
http://dx.doi.org/10.1002/jgt.20514
Tatsuhiko Hatanaka, Takehiro Ito, and Xiao Zhou. The List Coloring Reconfiguration Problem for Bounded Pathwidth Graphs. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 98-A(6):1168-1178, 2015. URL: http://search.ieice.org/bin/summary.php?id=e98-a_6_1168, URL: http://dx.doi.org/10.1587/transfun.E98.A.1168.
http://dx.doi.org/10.1587/transfun.E98.A.1168
Tatsuhiko Hatanaka, Takehiro Ito, and Xiao Zhou. Parameterized complexity of the list coloring reconfiguration problem with graph parameters. Theor. Comput. Sci., 739:65-79, 2018. URL: http://dx.doi.org/10.1016/j.tcs.2018.05.005.
http://dx.doi.org/10.1016/j.tcs.2018.05.005
Jan van den Heuvel. The complexity of change. In Simon R. Blackburn, Stefanie Gerke, and Mark Wildon, editors, Surveys in Combinatorics 2013, volume 409 of London Mathematical Society Lecture Note Series, pages 127-160. Cambridge University Press, 2013. URL: http://dx.doi.org/10.1017/CBO9781139506748.005.
http://dx.doi.org/10.1017/CBO9781139506748.005
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. URL: http://dx.doi.org/10.1016/j.tcs.2010.12.005.
http://dx.doi.org/10.1016/j.tcs.2010.12.005
Matthew Johnson, Dieter Kratsch, Stefan Kratsch, Viresh Patel, and Daniël Paulusma. Finding Shortest Paths Between Graph Colourings. Algorithmica, 75(2):295-321, 2016. URL: http://dx.doi.org/10.1007/s00453-015-0009-7.
http://dx.doi.org/10.1007/s00453-015-0009-7
Naomi Nishimura. Introduction to Reconfiguration. Algorithms, 11(4):52, 2018. URL: http://dx.doi.org/10.3390/a11040052.
http://dx.doi.org/10.3390/a11040052
Hiroki Osawa, Akira Suzuki, Takehiro Ito, and Xiao Zhou. Complexity of Coloring Reconfiguration under Recolorability Constraints. In Yoshio Okamoto and Takeshi Tokuyama, editors, 28th International Symposium on Algorithms and Computation, ISAAC 2017, December 9-12, 2017, Phuket, Thailand, volume 92 of LIPIcs, pages 62:1-62:12. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ISAAC.2017.62.
http://dx.doi.org/10.4230/LIPIcs.ISAAC.2017.62
Marcin Wrochna. Homomorphism Reconfiguration via Homotopy. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 730-742. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.730.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.730
Marcin Wrochna. Reconfiguration in bounded bandwidth and tree-depth. J. Comput. Syst. Sci., 93:1-10, 2018. URL: http://dx.doi.org/10.1016/j.jcss.2017.11.003.
http://dx.doi.org/10.1016/j.jcss.2017.11.003
Hiroki Osawa, Akira Suzuki, Takehiro Ito, and Xiao Zhou
Creative Commons Attribution 3.0 Unported license
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A Cut Tree Representation for Pendant Pairs
Two vertices v and w of a graph G are called a pendant pair if the maximal number of edge-disjoint paths in G between them is precisely min{d(v),d(w)}, where d denotes the degree function. The importance of pendant pairs stems from the fact that they are the key ingredient in one of the simplest and most widely used algorithms for the minimum cut problem today.
Mader showed 1974 that every simple graph with minimum degree delta contains Omega(delta^2) pendant pairs; this is the best bound known so far. We improve this result by showing that every simple graph G with minimum degree delta >= 5 or with edge-connectivity lambda >= 4 or with vertex-connectivity kappa >= 3 contains in fact Omega(delta |V|) pendant pairs. We prove that this bound is tight from several perspectives, and that Omega(delta |V|) pendant pairs can be computed efficiently, namely in linear time when a Gomory-Hu tree is given. Our method utilizes a new cut tree representation of graphs.
Pendant Pairs
Pendant Tree
Maximal Adjacency Ordering
Maximum Cardinality Search
Testing Edge-Connectivity
Gomory-Hu Tree
Mathematics of computing~Graph theory
Mathematics of computing~Graph algorithms
38:1-38:9
Regular Paper
On-Hei S.
Lo
On-Hei S. Lo
Institut für Mathematik, Technische Universität Ilmenau, Weimarer Strasse 25, D-98693 Ilmenau, Germany
This research is supported by the grant SCHM 3186/1-1 (270450205) from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).
Jens M.
Schmidt
Jens M. Schmidt
Institut für Mathematik, Technische Universität Ilmenau, Weimarer Strasse 25, D-98693 Ilmenau, Germany
This research is supported by the grant SCHM 3186/1-1 (270450205) from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).
10.4230/LIPIcs.ISAAC.2018.38
A. Bhalgat, R. Hariharan, T. Kavitha, and D. Panigrahi. An Õ(mn) Gomory-Hu Tree Construction Algorithm for Unweighted Graphs. In Proceedings of the 39th Annual Symposium on Theory of Computing (STOC'07), pages 605-614, 2007. URL: http://dx.doi.org/10.1145/1250790.1250879.
http://dx.doi.org/10.1145/1250790.1250879
E. A. Dinic. Algorithm for Solution of a Problem of Maximum Flow in a Network with Power Estimation. Soviet Math Doklady, 11:1277-1280, 1970.
A. Frank. On the edge-connectivity algorithm of Nagamochi and Ibaraki. Laboratoire Artemis, IMAG, Université J. Fourier, Grenoble, March 1994.
M. Henzinger, S. Rao, and D. Wang. Local Flow Partitioning for Faster Edge Connectivity. In Proceedings of the 28th Annual Symposium on Discrete Algorithms (SODA'17), pages 1919-1938, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.125.
http://dx.doi.org/10.1137/1.9781611974782.125
A. V. Karzanov. On finding a maximum flow in a network with special structure and some applications. Matematicheskie Voprosy Upravleniya Proizvodstvom (in Russian), pages 81-94, 1973.
K. Kawarabayashi and M. Thorup. Deterministic Global Minimum Cut of a Simple Graph in Near-Linear Time. In Proceedings of the 47th Annual Symposium on Theory of Computing (STOC'15), pages 665-674, 2015. URL: http://dx.doi.org/10.1145/2746539.2746588.
http://dx.doi.org/10.1145/2746539.2746588
W. Mader. Existenz gewisser Konfigurationen in n-gesättigten Graphen und in Graphen genügend großer Kantendichte. Mathematische Annalen, 194:295-312, 1971.
W. Mader. Grad und lokaler Zusammenhang in endlichen Graphen. Mathematische Annalen, 205:9-11, 1973.
W. Mader. Kantendisjunkte Wege in Graphen. Monatshefte für Mathematik, 78(5):395-404, 1974.
W. Mader. On vertices of degree n in minimally n-connected graphs and digraphs. Bolyai Society Mathematical Studies (Combinatorics, Paul Erdős is Eighty, Keszthely, 1993), 2:423-449, 1996.
H. Nagamochi and T. Ibaraki. Computing Edge-Connectivity in Multigraphs and Capacitated Graphs. SIAM Journal on Discrete Mathematics, 5(1):54-66, 1992.
M. Stoer and F. Wagner. A simple min-cut algorithm. Journal of the ACM, 44(4):585-591, 1997.
R. E. Tarjan and M. Yannakakis. Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal on Computing, 13(3):566-579, 1984.
On-Hei S. Lo and Jens M. Schmidt
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Polyline Drawings with Topological Constraints
Let G be a simple topological graph and let Gamma be a polyline drawing of G. We say that Gamma partially preserves the topology of G if it has the same external boundary, the same rotation system, and the same set of crossings as G. Drawing Gamma fully preserves the topology of G if the planarization of G and the planarization of Gamma have the same planar embedding. We show that if the set of crossing-free edges of G forms a connected spanning subgraph, then G admits a polyline drawing that partially preserves its topology and that has curve complexity at most three (i.e., at most three bends per edge). If, however, the set of crossing-free edges of G is not a connected spanning subgraph, the curve complexity may be Omega(sqrt{n}). Concerning drawings that fully preserve the topology, we show that if G has skewness k, it admits one such drawing with curve complexity at most 2k; for skewness-1 graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal 2-plane graphs and discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology.
Topological graphs
graph drawing
curve complexity
skewness-k graphs
k-planar graphs
Mathematics of computing~Combinatorics
Mathematics of computing~Graph theory
Theory of computation~Design and analysis of algorithms
39:1-39:13
Regular Paper
https://arxiv.org/abs/1809.08111
Emilio
Di Giacomo
Emilio Di Giacomo
Università degli Studi di Perugia, Perugia, Italy
https://orcid.org/0000-0002-9794-1928
Peter
Eades
Peter Eades
University of Sydney, Sydney, Australia
Giuseppe
Liotta
Giuseppe Liotta
Università degli Studi di Perugia, Perugia, Italy
https://orcid.org/0000-0002-2886-9694
Henk
Meijer
Henk Meijer
University College Roosevelt, Middelburg, The Netherlands
Fabrizio
Montecchiani
Fabrizio Montecchiani
Università degli Studi di Perugia, Perugia, Italy
https://orcid.org/0000-0002-0543-8912
10.4230/LIPIcs.ISAAC.2018.39
Bernardo M. Ábrego, Oswin Aichholzer, Silvia Fernández-Merchant, Thomas Hackl, Jürgen Pammer, Alexander Pilz, Pedro Ramos, Gelasio Salazar, and Birgit Vogtenhuber. All Good Drawings of Small Complete Graphs. In EuroCG 2015, pages 57-60, 2015.
Oswin Aichholzer, Thomas Hackl, Alexander Pilz, Gelasio Salazar, and Birgit Vogtenhuber. Deciding monotonicity of good drawings of the complete graph. In EGC 2015, pages 33-36, 2015.
David W. Barnette. 2-Connected Spanning Subgraphs of Planar 3-Connected Graphs. J. Combin. Theory Ser. B, 61(2):210-216, 1994.
Michael A. Bekos, Michael Kaufmann, and Fabrizio Montecchiani. Guest Editors' Foreword and Overview. J. Graph Algorithms Appl., 22(1):1-10, 2018.
Michael A. Bekos, Michael Kaufmann, and Chrysanthi N. Raftopoulou. On Optimal 2- and 3-Planar Graphs. In SOCG 2017, volume 77 of LIPIcs, pages 16:1-16:16. LZI, 2017.
Steven Chaplick, Fabian Lipp, Alexander Wolff, and Johannes Zink. 1-Bend RAC Drawings of NIC-Planar Graphs in Quadratic Area. In GD 2018. Springer, To appear.
Norishige Chiba, Kazunori Onoguchi, and Takao Nishizeki. Drawing plane graphs nicely. Acta Inform., 22(2):187-201, 1985.
Walter Didimo, Giuseppe Liotta, and Fabrizio Montecchiani. A Survey on Graph Drawing Beyond Planarity. CoRR, abs/1804.07257, 2018. URL: http://arxiv.org/abs/1804.07257.
http://arxiv.org/abs/1804.07257
Stephane Durocher and Debajyoti Mondal. Relating Graph Thickness to Planar Layers and Bend Complexity. In ICALP 2016, volume 55 of LIPIcs, pages 10:1-10:13. LZI, 2016.
Peter Eades, Seok-Hee Hong, Giuseppe Liotta, Naoki Katoh, and Sheung-Hung Poon. Straight-Line Drawability of a Planar Graph Plus an Edge. In WADS 2015, pages 301-313. Springer, 2015.
David Eppstein, Mereke van Garderen, Bettina Speckmann, and Torsten Ueckerdt. Convex-Arc Drawings of Pseudolines. CoRR, abs/1601.06865, 2016. URL: http://arxiv.org/abs/1601.06865.
http://arxiv.org/abs/1601.06865
István Fáry. On straight line representations of planar graphs. Acta Univ. Szeged. Sect. Sci. Math., 11:229-233, 1948.
Emilio Di Giacomo, Petere Eades, Giuseppe Liotta, Henk Meijer, and Fabrizio Montecchiani. Polyline Drawings with Topological Constraints. CoRR, abs/1809.08111, 2018. URL: http://arxiv.org/abs/1809.08111.
http://arxiv.org/abs/1809.08111
Jan Kratochvíl, Anna Lubiw, and Jaroslav Nešetřil. Noncrossing Subgraphs in Topological Layouts. SIAM J. Discrete Math., 4(2):223-244, 1991.
Jan Kynčl. Simple Realizability of Complete Abstract Topological Graphs in P. Discrete Comput. Geom., 45(3):383-399, 2011.
Jan Kynčl. Enumeration of simple complete topological graphs. European Journal of Combinatorics, 30(7):1676-1685, 2009.
Sherman K. Stein. Convex maps. Proc. Am. Math. Soc., 2(3):464-466, 1951.
Klaus Wagner. Bemerkungen zum Vierfarbenproblem. Jahresber. Dtsch. Math. Ver., 46:26-32, 1936.
Emilio Di Giacomo, Peter Eades, Giuseppe Liotta, Henk Meijer, and Fabrizio Montecchiani
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Almost Optimal Algorithms for Diameter-Optimally Augmenting Trees
We consider the problem of augmenting an n-vertex tree with one shortcut in order to minimize the diameter of the resulting graph. The tree is embedded in an unknown space and we have access to an oracle that, when queried on a pair of vertices u and v, reports the weight of the shortcut (u,v) in constant time. Previously, the problem was solved in O(n^2 log^3 n) time for general weights [Oh and Ahn, ISAAC 2016], in O(n^2 log n) time for trees embedded in a metric space [Große et al., https://arxiv.org/abs/1607.05547], and in O(n log n) time for paths embedded in a metric space [Wang, WADS 2017]. Furthermore, a (1+epsilon)-approximation algorithm running in O(n+1/epsilon^3) has been designed for paths embedded in R^d, for constant values of d [Große et al., ICALP 2015].
The contribution of this paper is twofold: we address the problem for trees (not only paths) and we also improve upon all known results. More precisely, we design a time-optimal O(n^2) time algorithm for general weights. Moreover, for trees embedded in a metric space, we design (i) an exact O(n log n) time algorithm and (ii) a (1+epsilon)-approximation algorithm that runs in O(n+ epsilon^{-1}log epsilon^{-1}) time.
Graph diameter
augmentation problem
trees
time-efficient algorithms
Theory of computation~Graph algorithms analysis
Theory of computation~Approximation algorithms analysis
40:1-40:13
Regular Paper
A full version of this paper can be found at https://arxiv.org/abs/1809.08822.
Davide
Bilò
Davide Bilò
Department of Humanities and Social Sciences, University of Sassari, Via Roma 151, 07100 Sassari (SS), Italy
https://orcid.org/0000-0003-3169-4300
10.4230/LIPIcs.ISAAC.2018.40
Noga Alon, András Gyárfás, and Miklós Ruszinkó. Decreasing the diameter of bounded degree graphs. Journal of Graph Theory, 35(3):161-172, 2000. URL: http://dx.doi.org/10.1002/1097-0118(200011)35:3%3C161::AID-JGT1%3E3.0.CO;2-Y.
http://dx.doi.org/10.1002/1097-0118(200011)35:3%3C161::AID-JGT1%3E3.0.CO;2-Y
Davide Bilò, Luciano Gualà, and Guido Proietti. Improved approximability and non-approximability results for graph diameter decreasing problems. Theor. Comput. Sci., 417:12-22, 2012. URL: http://dx.doi.org/10.1016/j.tcs.2011.05.014.
http://dx.doi.org/10.1016/j.tcs.2011.05.014
Jean-Lou De Carufel, Carsten Grimm, Anil Maheshwari, and Michiel H. M. Smid. Minimizing the Continuous Diameter when Augmenting Paths and Cycles with Shortcuts. In Rasmus Pagh, editor, 15th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2016, volume 53 of LIPIcs, pages 27:1-27:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.SWAT.2016.27.
http://dx.doi.org/10.4230/LIPIcs.SWAT.2016.27
Jean-Lou De Carufel, Carsten Grimm, Stefan Schirra, and Michiel H. M. Smid. Minimizing the Continuous Diameter When Augmenting a Tree with a Shortcut. In Faith Ellen, Antonina Kolokolova, and Jörg-Rüdiger Sack, editors, Algorithms and Data Structures - 15th International Symposium, WADS 2017, volume 10389 of Lecture Notes in Computer Science, pages 301-312. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-62127-2_26.
http://dx.doi.org/10.1007/978-3-319-62127-2_26
Victor Chepoi and Yann Vaxès. Augmenting Trees to Meet Biconnectivity and Diameter Constraints. Algorithmica, 33(2):243-262, 2002. URL: http://dx.doi.org/10.1007/s00453-001-0113-8.
http://dx.doi.org/10.1007/s00453-001-0113-8
F. R. K. Chung and M. R. Garey. Diameter bounds for altered graphs. Journal of Graph Theory, 8(4):511-534, 1984. URL: http://dx.doi.org/10.1002/jgt.3190080408.
http://dx.doi.org/10.1002/jgt.3190080408
Erik D. Demaine and Morteza Zadimoghaddam. Minimizing the Diameter of a Network Using Shortcut Edges. In Haim Kaplan, editor, Algorithm Theory - SWAT 2010, 12th Scandinavian Symposium and Workshops on Algorithm Theory, volume 6139 of Lecture Notes in Computer Science, pages 420-431. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-13731-0_39.
http://dx.doi.org/10.1007/978-3-642-13731-0_39
Fabrizio Frati, Serge Gaspers, Joachim Gudmundsson, and Luke Mathieson. Augmenting Graphs to Minimize the Diameter. Algorithmica, 72(4):995-1010, 2015. URL: http://dx.doi.org/10.1007/s00453-014-9886-4.
http://dx.doi.org/10.1007/s00453-014-9886-4
Yong Gao, Donovan R. Hare, and James Nastos. The parametric complexity of graph diameter augmentation. Discrete Applied Mathematics, 161(10-11):1626-1631, 2013. URL: http://dx.doi.org/10.1016/j.dam.2013.01.016.
http://dx.doi.org/10.1016/j.dam.2013.01.016
Ulrike Große, Joachim Gudmundsson, Christian Knauer, Michiel H. M. Smid, and Fabian Stehn. Fast Algorithms for Diameter-Optimally Augmenting Paths. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, volume 9134 of Lecture Notes in Computer Science, pages 678-688. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_55.
http://dx.doi.org/10.1007/978-3-662-47672-7_55
Ulrike Große, Joachim Gudmundsson, Christian Knauer, Michiel H. M. Smid, and Fabian Stehn. Fast Algorithms for Diameter-Optimally Augmenting Paths and Trees. CoRR, abs/1607.05547, 2016. URL: http://arxiv.org/abs/1607.05547.
http://arxiv.org/abs/1607.05547
John Hershberger. Finding the Upper Envelope of n Line Segments in it O(n log n) Time. Inf. Process. Lett., 33(4):169-174, 1989. URL: http://dx.doi.org/10.1016/0020-0190(89)90136-1.
http://dx.doi.org/10.1016/0020-0190(89)90136-1
Toshimasa Ishii. Augmenting Outerplanar Graphs to Meet Diameter Requirements. Journal of Graph Theory, 74(4):392-416, 2013. URL: http://dx.doi.org/10.1002/jgt.21719.
http://dx.doi.org/10.1002/jgt.21719
Chung-Lun Li, S.Thomas McCormick, and David Simchi-Levi. On the minimum-cardinality-bounded-diameter and the bounded-cardinality-minimum diameter edge addition problems. Operations Research Letters, 11(5):303-308, 1992.
Nimrod Megiddo. Applying Parallel Computation Algorithms in the Design of Serial Algorithms. J. ACM, 30(4):852-865, 1983. URL: http://dx.doi.org/10.1145/2157.322410.
http://dx.doi.org/10.1145/2157.322410
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http://dx.doi.org/10.4230/LIPIcs.ISAAC.2016.59
Anneke A. Schoone, Hans L. Bodlaender, and Jan van Leeuwen. Diameter increase caused by edge deletion. Journal of Graph Theory, 11(3):409-427, 1987. URL: http://dx.doi.org/10.1002/jgt.3190110315.
http://dx.doi.org/10.1002/jgt.3190110315
Haitao Wang. An Improved Algorithm for Diameter-Optimally Augmenting Paths in a Metric Space. In Faith Ellen, Antonina Kolokolova, and Jörg-Rüdiger Sack, editors, Algorithms and Data Structures - 15th International Symposium, WADS 2017, volume 10389 of Lecture Notes in Computer Science, pages 545-556. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-62127-2_46.
http://dx.doi.org/10.1007/978-3-319-62127-2_46
Davide Bilò
Creative Commons Attribution 3.0 Unported license
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Approximation Algorithms for Facial Cycles in Planar Embeddings
Consider the following combinatorial problem: Given a planar graph G and a set of simple cycles C in G, find a planar embedding E of G such that the number of cycles in C that bound a face in E is maximized. This problem, called Max Facial C-Cycles, was first studied by Mutzel and Weiskircher [IPCO '99, http://dx.doi.org/10.1007/3-540-48777-8_27) and then proved NP-hard by Woeginger [Oper. Res. Lett., 2002, http://dx.doi.org/10.1016/S0167-6377(02)00119-0].
We establish a tight border of tractability for Max Facial C-Cycles in biconnected planar graphs by giving conditions under which the problem is NP-hard and showing that strengthening any of these conditions makes the problem polynomial-time solvable. Our main results are approximation algorithms for Max Facial C-Cycles. Namely, we give a 2-approximation for series-parallel graphs and a (4+epsilon)-approximation for biconnected planar graphs. Remarkably, this provides one of the first approximation algorithms for constrained embedding problems.
Planar Embeddings
Facial Cycles
Complexity
Approximation Algorithms
Mathematics of computing~Graph theory
41:1-41:13
Regular Paper
https://arxiv.org/abs/1607.02347
Giordano
Da Lozzo
Giordano Da Lozzo
Computer Science Department, Roma Tre University, Italy
https://orcid.org/0000-0003-2396-5174
Ignaz
Rutter
Ignaz Rutter
Department of Computer Science and Mathematics, University of Passau, Germany
https://orcid.org/0000-0002-3794-4406
10.4230/LIPIcs.ISAAC.2018.41
Patrizio Angelini, Giuseppe Di Battista, Fabrizio Frati, Vít Jelínek, Jan Kratochvíl, Maurizio Patrignani, and Ignaz Rutter. Testing Planarity of Partially Embedded Graphs. ACM Trans. Alg., 11(4):32:1-32:42, 2015. URL: http://dx.doi.org/10.1145/2629341.
http://dx.doi.org/10.1145/2629341
Patrizio Angelini, Giuseppe Di Battista, and Maurizio Patrignani. Finding a Minimum-Depth Embedding of a Planar Graph in O(n⁴) Time. Algorithmica, 60:890-937, 2011. URL: http://dx.doi.org/10.1007/s00453-009-9380-6.
http://dx.doi.org/10.1007/s00453-009-9380-6
Brenda S. Baker. Approximation Algorithms for NP-complete Problems on Planar Graphs. J. ACM, 41(1):153-180, January 1994. URL: http://dx.doi.org/10.1145/174644.174650.
http://dx.doi.org/10.1145/174644.174650
Thomas Bläsius, Sebastian Lehmann, and Ignaz Rutter. Orthogonal graph drawing with inflexible edges. Comput. Geom., 55:26-40, 2016. URL: http://dx.doi.org/10.1016/j.comgeo.2016.03.001.
http://dx.doi.org/10.1016/j.comgeo.2016.03.001
Thomas Bläsius, Ignaz Rutter, and Dorothea Wagner. Optimal Orthogonal Graph Drawing with Convex Bend Costs. ACM Trans. Algorithms, 12(3):33:1-33:32, 2016.
Giordano Da Lozzo, Vít Jelínek, Jan Kratochvíl, and Ignaz Rutter. Planar Embeddings with Small and Uniform Faces. In ISAAC'14, volume 8889 of LNCS, pages 633-645. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-13075-0.
http://dx.doi.org/10.1007/978-3-319-13075-0
Giordano Da Lozzo and Ignaz Rutter. On the complexity of realizing facial cycles. CoRR, abs/1607.02347, 2016. URL: http://arxiv.org/abs/1607.02347.
http://arxiv.org/abs/1607.02347
Giuseppe Di Battista and Fabrizio Frati. Drawing Trees, Outerplanar Graphs, Series-Parallel Graphs, and Planar Graphs in a Small Area. In Thirty Essays on Geometric Graph Theory, pages 121-165. Springer New York, 2013. URL: http://dx.doi.org/10.1007/978-1-4614-0110-0_9.
http://dx.doi.org/10.1007/978-1-4614-0110-0_9
Giuseppe Di Battista and Roberto Tamassia. On-Line Graph Algorithms with SPQR-Trees. In Michael S. Paterson, editor, ICALP'90, volume 443 of LNCS, pages 598-611. Springer, 1990. URL: http://dx.doi.org/10.1007/BFb0032061.
http://dx.doi.org/10.1007/BFb0032061
Christoph Dornheim. Planar Graphs with Topological Constraints. JGAA, 6(1):27-66, 2002. URL: http://dx.doi.org/10.7155/jgaa.00044.
http://dx.doi.org/10.7155/jgaa.00044
M. R. Garey, David S. Johnson, and Robert Endre Tarjan. The Planar Hamiltonian Circuit Problem is NP-Complete. SIAM J. Comput., 5(4):704-714, 1976. URL: http://dx.doi.org/10.1137/0205049.
http://dx.doi.org/10.1137/0205049
Ashim Garg and Roberto Tamassia. On the Computational Complexity of Upward and Rectilinear Planarity Testing. SIAM J. on Comput., 31(2):601-625, 2001. URL: http://dx.doi.org/10.1137/S0097539794277123.
http://dx.doi.org/10.1137/S0097539794277123
Carsten Gutwenger and Petra Mutzel. A Linear Time Implementation of SPQR-trees. In Joe Marks, editor, GD'00, volume 1984 of LNCS, pages 77-90. Springer, 2001. URL: http://dx.doi.org/10.1007/3-540-44541-2_8.
http://dx.doi.org/10.1007/3-540-44541-2_8
Joseph Douglas Horton and Kyriakos Kilakos. Minimum Edge Dominating Sets. SIAM J. Discrete Math., 6(3):375-387, 1993. URL: http://dx.doi.org/10.1137/0406030.
http://dx.doi.org/10.1137/0406030
Frank Kammer. Determining the Smallest k Such That G Is k-Outerplanar. In ESA'07, volume 4698 of LNCS, pages 359-370. Springer, 2007. URL: http://dx.doi.org/10.1007/978-3-540-75520-3_33.
http://dx.doi.org/10.1007/978-3-540-75520-3_33
Petra Mutzel and René Weiskircher. Optimizing over All Combinatorial Embeddings of a Planar Graph. In IPCO'99, volume 1610 of LNCS, pages 361-376. Springer, 1999. URL: http://dx.doi.org/10.1007/3-540-48777-8_27.
http://dx.doi.org/10.1007/3-540-48777-8_27
Roberto Tamassia. On Embedding a Graph in the Grid with the Minimum Number of Bends. SIAM J. Comput., 16(3):421-444, 1987. URL: http://dx.doi.org/10.1137/0216030.
http://dx.doi.org/10.1137/0216030
Hassler Whitney. Congruent Graphs and the Connectivity of Graphs. American Journal of Mathematics, 54(1):150-168, 1932. URL: http://www.jstor.org/stable/2371086.
http://www.jstor.org/stable/2371086
Gerhard J. Woeginger. Embeddings of planar graphs that minimize the number of long-face cycles. Oper. Res. Lett., 30(3):167-168, 2002. URL: http://dx.doi.org/10.1016/S0167-6377(02)00119-0.
http://dx.doi.org/10.1016/S0167-6377(02)00119-0
Giordano Da Lozzo and Ignaz Rutter
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
An Algorithm for the Maximum Weight Strongly Stable Matching Problem
An instance of the maximum weight strongly stable matching problem with incomplete lists and ties is an undirected bipartite graph G = (A cup B, E), with an adjacency list being a linearly ordered list of ties, which are vertices equally good for a given vertex. We are also given a weight function w on the set E. An edge (x, y) in E setminus M is a blocking edge for M if by getting matched to each other neither of the vertices x and y would become worse off and at least one of them would become better off. A matching is strongly stable if there is no blocking edge with respect to it. The goal is to compute a strongly stable matching of maximum weight with respect to w.
We give a polyhedral characterisation of the problem and prove that the strongly stable matching polytope is integral. This result implies that the maximum weight strongly stable matching problem can be solved in polynomial time. Thereby answering an open question by Gusfield and Irving [Dan Gusfield and Robert W. Irving, 1989]. The main result of this paper is an efficient O(nm log{(Wn)}) time algorithm for computing a maximum weight strongly stable matching, where we denote n = |V|, m = |E| and W is a maximum weight of an edge in G. For small edge weights we show that the problem can be solved in O(nm) time. Note that the fastest known algorithm for the unweighted version of the problem has O(nm) runtime [Telikepalli Kavitha et al., 2007]. Our algorithm is based on the rotation structure which was constructed for strongly stable matchings in [Adam Kunysz et al., 2016].
Stable marriage
Strongly stable matching
Weighted matching
Rotation
Theory of computation~Graph algorithms analysis
42:1-42:13
Regular Paper
Partly supported by Polish National Science Center grant UMO-2013/11/B/ST6/01748.
Adam
Kunysz
Adam Kunysz
Institute of Computer Science, University of Wrocław, Poland
10.4230/LIPIcs.ISAAC.2018.42
Tomás Feder. A New Fixed Point Approach for Stable Networks and Stable Marriages. J. Comput. Syst. Sci., 45(2):233-284, 1992. URL: http://dx.doi.org/10.1016/0022-0000(92)90048-N.
http://dx.doi.org/10.1016/0022-0000(92)90048-N
Tomás Feder. Network Flow and 2-Satisfiability. Algorithmica, 11(3):291-319, 1994. URL: http://dx.doi.org/10.1007/BF01240738.
http://dx.doi.org/10.1007/BF01240738
Tamás Fleiner, Robert W. Irving, and David F. Manlove. Efficient algorithms for generalized Stable Marriage and Roommates problems. Theor. Comput. Sci., 381(1-3):162-176, 2007. URL: http://dx.doi.org/10.1016/j.tcs.2007.04.029.
http://dx.doi.org/10.1016/j.tcs.2007.04.029
Harold N. Gabow and Robert Endre Tarjan. Faster Scaling Algorithms for Network Problems. SIAM J. Comput., 18(5):1013-1036, 1989. URL: http://dx.doi.org/10.1137/0218069.
http://dx.doi.org/10.1137/0218069
David Gale and Lloyd S Shapley. College Admissions and the Stability of Marriage. The American Mathematical Monthly, 69(1):9-15, 1962. URL: http://dx.doi.org/10.2307/2312726.
http://dx.doi.org/10.2307/2312726
Dan Gusfield and Robert W. Irving. The Stable marriage problem - structure and algorithms. Foundations of computing series. MIT Press, 1989.
Robert W. Irving. Stable Marriage and Indifference. Discrete Applied Mathematics, 48(3):261-272, 1994. URL: http://dx.doi.org/10.1016/0166-218X(92)00179-P.
http://dx.doi.org/10.1016/0166-218X(92)00179-P
Kazuo Iwama, David Manlove, Shuichi Miyazaki, and Yasufumi Morita. Stable Marriage with Incomplete Lists and Ties. ICALP'99, Prague, Czech Republic, July 11-15, 1999, pages 443-452, 1999. URL: http://dx.doi.org/10.1007/3-540-48523-6_41.
http://dx.doi.org/10.1007/3-540-48523-6_41
Telikepalli Kavitha, Kurt Mehlhorn, Dimitrios Michail, and Katarzyna E. Paluch. Strongly stable matchings in time O(nm) and extension to the hospitals-residents problem. ACM Trans. Algorithms, 3(2), 2007. URL: http://dx.doi.org/10.1145/1240233.1240238.
http://dx.doi.org/10.1145/1240233.1240238
Zoltán Király. Better and Simpler Approximation Algorithms for the Stable Marriage Problem. Algorithmica, 60(1):3-20, 2011. URL: http://dx.doi.org/10.1007/s00453-009-9371-7.
http://dx.doi.org/10.1007/s00453-009-9371-7
Adam Kunysz. The Strongly Stable Roommates Problem. In Piotr Sankowski and Christos D. Zaroliagis, editors, 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, volume 57 of LIPIcs, pages 60:1-60:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ESA.2016.60.
http://dx.doi.org/10.4230/LIPIcs.ESA.2016.60
Adam Kunysz, Katarzyna E. Paluch, and Pratik Ghosal. Characterisation of Strongly Stable Matchings. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 107-119, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch8.
http://dx.doi.org/10.1137/1.9781611974331.ch8
David Manlove, Robert W. Irving, Kazuo Iwama, Shuichi Miyazaki, and Yasufumi Morita. Hard variants of stable marriage. Theor. Comput. Sci., 276(1-2):261-279, 2002. URL: http://dx.doi.org/10.1016/S0304-3975(01)00206-7.
http://dx.doi.org/10.1016/S0304-3975(01)00206-7
David F. Manlove. Stable marriage with ties and unacceptable partners. Technical report, University of Glasgow, 1999.
David F. Manlove. The structure of stable marriage with indifference. Discrete Applied Mathematics, 122(1-3):167-181, 2002. URL: http://dx.doi.org/10.1016/S0166-218X(01)00322-5.
http://dx.doi.org/10.1016/S0166-218X(01)00322-5
David F. Manlove. Algorithmics of Matching Under Preferences, volume 2 of Series on Theoretical Computer Science. WorldScientific, 2013. URL: http://dx.doi.org/10.1142/8591.
http://dx.doi.org/10.1142/8591
Eric McDermid. A 3/2-Approximation Algorithm for General Stable Marriage. ICALP 2009, Rhodes, Greece, July 5-12, 2009, pages 689-700, 2009. URL: http://dx.doi.org/10.1007/978-3-642-02927-1_57.
http://dx.doi.org/10.1007/978-3-642-02927-1_57
Katarzyna E. Paluch. Faster and Simpler Approximation of Stable Matchings. Algorithms, 7(2):189-202, 2014. URL: http://dx.doi.org/10.3390/a7020189.
http://dx.doi.org/10.3390/a7020189
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Alvin E. Roth, Uriel G. Rothblum, and John H. Vande Vate. Stable Matchings, Optimal Assignments, and Linear Programming. Math. Oper. Res., 18(4):803-828, 1993. URL: http://dx.doi.org/10.1287/moor.18.4.803.
http://dx.doi.org/10.1287/moor.18.4.803
Uriel G. Rothblum. Characterization of stable matchings as extreme points of a polytope. Math. Program., 54:57-67, 1992. URL: http://dx.doi.org/10.1007/BF01586041.
http://dx.doi.org/10.1007/BF01586041
A. Schrijver. Combinatorial Optimization - Polyhedra and Efficiency. Springer, 2003.
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
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http://dx.doi.org/10.1287/moor.23.4.874
J.H. Vande Vate. Linear programming brings marital bliss. Operations Research Letters, 8:147–153, 1989. URL: http://dx.doi.org/10.1016/0167-6377(89)90041-2.
http://dx.doi.org/10.1016/0167-6377(89)90041-2
Adam Kunysz
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Approximation Algorithm for Vertex Cover with Multiple Covering Constraints
We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph G=(V,E) with a maximum edge size f, a cost function w: V - > Z^+, and edge subsets P_1,P_2,...,P_r of E along with covering requirements k_1,k_2,...,k_r for each subset. The objective is to find a minimum cost subset S of V such that, for each edge subset P_i, at least k_i edges of it are covered by S. This problem is a basic yet general form of classical vertex cover problem and a generalization of the edge-partitioned vertex cover problem considered by Bera et al.
We present a primal-dual algorithm yielding an (f * H_r + H_r)-approximation for this problem, where H_r is the r^{th} harmonic number. This improves over the previous ratio of (3cf log r), where c is a large constant used to ensure a low failure probability for Monte-Carlo randomized algorithms. Compared to previous result, our algorithm is deterministic and pure combinatorial, meaning that no Ellipsoid solver is required for this basic problem. Our result can be seen as a novel reinterpretation of a few classical tight results using the language of LP primal-duality.
Vertex cover
multiple cover constraints
Approximation algorithm
Mathematics of computing~Approximation algorithms
43:1-43:11
Regular Paper
This work is supported in part by Ministry of Science and Technology (MOST), Taiwan, under Grants MOST107-2218-E-194-015-MY3 and MOST106-2221-E-001-006-MY3.
Eunpyeong
Hong
Eunpyeong Hong
Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan
Mong-Jen
Kao
Mong-Jen Kao
Department of Computer Science and Information Engineering, National Chung-Cheng University, Chiayi, Taiwan
10.4230/LIPIcs.ISAAC.2018.43
Reuven Bar-Yehuda. Using Homogeneous Weights for Approximating the Partial Cover Problem. J. Algorithms, 39(2):137-144, May 2001.
Suman K. Bera, Shalmoli Gupta, Amit Kumar, and Sambuddha Roy. Approximation algorithms for the partition vertex cover problem. Theoretical Computer Science, 555:2-8, 2014. Special Issue on Algorithms and Computation.
Nader H. Bshouty and Lynn Burroughs. Massaging a Linear Programming Solution to Give a 2-Approximation For a Generalization of the Vertex Cover Problem. In In Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science, pages 298-308. Springer, 1998.
Julia Chuzhoy. Covering Problems with Hard Capacities. SIAM J. Comput., 36(2):498-515, August 2006. URL: http://dx.doi.org/10.1137/S0097539703422479.
http://dx.doi.org/10.1137/S0097539703422479
Toshihiro Fujito. On approximation of the submodular set cover problem. Oper. Res. Lett., 25 (4):169-174, November 1999.
Rajiv Gandhi, Samir Khuller, and Aravind Srinivasan. Approximation Algorithms for Partial Covering Problems. J. Algorithms, 53(1):55-84, October 2004.
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Eunpyeong Hong and Mong-Jen Kao
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Correlation Clustering Generalized
We present new results for LambdaCC and MotifCC, two recently introduced variants of the well-studied correlation clustering problem. Both variants are motivated by applications to network analysis and community detection, and have non-trivial approximation algorithms.
We first show that the standard linear programming relaxation of LambdaCC has a Theta(log n) integrality gap for a certain choice of the parameter lambda. This sheds light on previous challenges encountered in obtaining parameter-independent approximation results for LambdaCC. We generalize a previous constant-factor algorithm to provide the best results, from the LP-rounding approach, for an extended range of lambda.
MotifCC generalizes correlation clustering to the hypergraph setting. In the case of hyperedges of degree 3 with weights satisfying probability constraints, we improve the best approximation factor from 9 to 8. We show that in general our algorithm gives a 4(k-1) approximation when hyperedges have maximum degree k and probability weights. We additionally present approximation results for LambdaCC and MotifCC where we restrict to forming only two clusters.
Correlation Clustering
Approximation Algorithms
Mathematics of computing~Approximation algorithms
44:1-44:13
Regular Paper
https://arxiv.org/abs/1809.09493
David F.
Gleich
David F. Gleich
Department of Computer Science, Purdue University, West Lafayette, Indiana, USA
D.F.G. is supported by the DARPA Simplex program, the Sloan Foundation, and NSF awards IIS-154648, CCF-1149756 and CCF-093937.
Nate
Veldt
Nate Veldt
Department of Mathematics, Purdue University, West Lafayette, Indiana, USA
N.V. supported by NSF award CCF-1149756.
Anthony
Wirth
Anthony Wirth
School of Computing and Information Systems, The University of Melbourne, Parkville, Victoria, Australia
https://orcid.org/0000-0003-3746-6704
A.W. is supported by the Melbourne School of Engineering.
10.4230/LIPIcs.ISAAC.2018.44
Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. O(√ log n) Approximation Algorithms for Min UnCut, Min 2CNF Deletion, and Directed Cut Problems. In STOC 05, pages 573-581. ACM, 2005. URL: http://dx.doi.org/10.1145/1060590.1060675.
http://dx.doi.org/10.1145/1060590.1060675
Nir Ailon, Moses Charikar, and Alantha Newman. Aggregating inconsistent information: ranking and clustering. Journal of the ACM (JACM), 55(5):23, 2008.
Nikhil Bansal, Avrim Blum, and Shuchi Chawla. Correlation Clustering. Machine Learning, 56:89-113, 2004.
Austin R. Benson, David F. Gleich, and Jure Leskovec. Higher-order organization of complex networks. Science, 353(6295):163-166, 2016. URL: http://dx.doi.org/10.1126/science.aad9029.
http://dx.doi.org/10.1126/science.aad9029
Moses Charikar, Venkatesan Guruswami, and Anthony Wirth. Clustering with qualitative information. Journal of Computer and System Sciences, 71(3):360-383, 2005.
Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, and D. Sivakumar. On the Hardness of Approximating Multicut and Sparsest-cut. Computational Complexity, 15(2):94-114, June 2006. URL: http://dx.doi.org/10.1007/s00037-006-0210-9.
http://dx.doi.org/10.1007/s00037-006-0210-9
Shuchi Chawla, Konstantin Makarychev, Tselil Schramm, and Grigory Yaroslavtsev. Near optimal LP rounding algorithm for correlation clustering on complete and complete k-partite graphs. In STOC 15, pages 219-228. ACM, 2015.
Tom Coleman, James Saunderson, and Anthony Wirth. A Local-Search 2-Approximation for 2-Correlation-Clustering. In ESA 08, pages 308-319, 2008. URL: http://dx.doi.org/10.1007/978-3-540-87744-8_26.
http://dx.doi.org/10.1007/978-3-540-87744-8_26
Bhaskar DasGupta, German A. Enciso, Eduardo Sontag, and Yi Zhang. Algorithmic and Complexity Results for Decompositions of Biological Networks into Monotone Subsystems. In WEA 06, pages 253-264, 2006.
Erik D Demaine, Dotan Emanuel, Amos Fiat, and Nicole Immorlica. Correlation clustering in general weighted graphs. Theoretical Computer Science, 361(2-3):172-187, 2006.
Takuro Fukunaga. LP-based pivoting algorithm for higher-order correlation clustering. In Computing and Combinatorics, pages 51-62, Cham, 2018. Springer International Publishing.
Ioannis Giotis and Venkatesan Guruswami. Correlation Clustering with a Fixed Number of Clusters. Theory OF Computing, 2:249-266, 2006.
David F. Gleich, Nate Veldt, and Anthony Wirth. Correlation Clustering Generalized. arXiv, cs.DS, 2018. URL: http://arxiv.org/abs/1809.09493.
http://arxiv.org/abs/1809.09493
S. A. Khot and N. K. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into 𝓁₁. In FOCS 05, pages 53-62, October 2005. URL: http://dx.doi.org/10.1109/SFCS.2005.74.
http://dx.doi.org/10.1109/SFCS.2005.74
Subhash Khot. On the Power of Unique 2-prover 1-round Games. In STOC 02, pages 767-775, 2002. URL: http://dx.doi.org/10.1145/509907.510017.
http://dx.doi.org/10.1145/509907.510017
Sungwoong Kim, Sebastian Nowozin, Pushmeet Kohli, and Chang D. Yoo. Higher-Order Correlation Clustering for Image Segmentation. In NIPS 11, pages 1530-1538, 2011. URL: http://papers.nips.cc/paper/4406-higher-order-correlation-clustering-for-image-segmentation.pdf.
http://papers.nips.cc/paper/4406-higher-order-correlation-clustering-for-image-segmentation.pdf
P. Li, H. Dau, G. Puleo, and O. Milenkovic. Motif clustering and overlapping clustering for social network analysis. In INFOCOM 17, pages 1-9, May 2017. URL: http://dx.doi.org/10.1109/INFOCOM.2017.8056956.
http://dx.doi.org/10.1109/INFOCOM.2017.8056956
Mark EJ Newman and Michelle Girvan. Finding and evaluating community structure in networks. Physical review E, 69(026113), 2004.
G. Puleo and O. Milenkovic. Correlation Clustering with Constrained Cluster Sizes and Extended Weights Bounds. SIAM Journal on Optimization, 25(3):1857-1872, 2015. URL: http://dx.doi.org/10.1137/140994198.
http://dx.doi.org/10.1137/140994198
Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In FOCS 00, pages 3-13, 2000.
Anke van Zuylen and David P. Williamson. Deterministic Pivoting Algorithms for Constrained Ranking and Clustering Problems. Mathematics of Operations Research, 34(3):594-620, 2009. URL: http://dx.doi.org/10.1287/moor.1090.0385.
http://dx.doi.org/10.1287/moor.1090.0385
Nate Veldt, David F Gleich, and Anthony Wirth. A Correlation Clustering Framework for Community Detection. In WWW 18, pages 439-448. International World Wide Web Conferences Steering Committee, 2018.
Anthony Ian Wirth. Approximation Algorithms for Clustering. PhD thesis, Princeton University, November 2004. Computer Science Technical Report 716.
David F. Gleich, Nate Veldt, and Anthony Wirth
Creative Commons Attribution 3.0 Unported license
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Partitioning Vectors into Quadruples: Worst-Case Analysis of a Matching-Based Algorithm
Consider a problem where 4k given vectors need to be partitioned into k clusters of four vectors each. A cluster of four vectors is called a quad, and the cost of a quad is the sum of the component-wise maxima of the four vectors in the quad. The problem is to partition the given 4k vectors into k quads with minimum total cost. We analyze a straightforward matching-based algorithm and prove that this algorithm is a 3/2-approximation algorithm for this problem. We further analyze the performance of this algorithm on a hierarchy of special cases of the problem and prove that, in one particular case, the algorithm is a 5/4-approximation algorithm. Our analysis is tight in all cases except one.
approximation algorithm
matching
clustering problem
Mathematics of computing~Approximation algorithms
45:1-45:12
Regular Paper
https://arxiv.org/abs/1807.01962
Annette M. C.
Ficker
Annette M. C. Ficker
Faculty of Economics and Business, KU Leuven, Leuven, Belgium
Thomas
Erlebach
Thomas Erlebach
Department of Informatics, University of Leicester, Leicester, United Kingdom
https://orcid.org/0000-0002-4470-5868
Supported by a study leave granted by University of Leicester.
Matús
Mihalák
Matús Mihalák
Department of Data Science and Knowledge Engineering, Maastricht University, Maastricht, The Netherlands
https://orcid.org/0000-0002-1898-607X
Frits C. R.
Spieksma
Frits C. R. Spieksma
Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands
https://orcid.org/0000-0002-2547-3782
10.4230/LIPIcs.ISAAC.2018.45
J. Barát and D. Gerbner. Edge-Decomposition of Graphs into Copies of a Tree with Four Edges. The Electronic Journal of Combinatorics, 21(1):1-55, 2014.
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T. Dokka, Y. Crama, and F.C.R. Spieksma. Multi-dimensional vector assignment problems. Discrete Optimization, 14:111-125, 2014.
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Annette M. C. Ficker, Thomas Erlebach, Matús Mihalák, and Frits C. R. Spieksma. Partitioning Vectors into Quadruples: Worst-Case Analysis of a Matching-Based Algorithm. CoRR, abs/1807.01962, 2018. URL: http://arxiv.org/abs/1807.01962.
http://arxiv.org/abs/1807.01962
A. Figueroa, A. Goldstein, T. Jiang, M. Kurowski, A. Lingas, and M. Persson. Approximate clustering of fingerprint vectors with missing values. In Proceedings of the 2005 Australasian Symposium on Theory of Computing (CATS 2005), volume 41 of CRPIT, pages 57-60. Australian Computer Society, 2005.
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D.P. Williamson and D.B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, New York, USA, 1st edition, 2011.
Annette M. C. Ficker, Thomas Erlebach, Matúš Mihalák, and Frits C. R. Spieksma
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Coresets for Fuzzy K-Means with Applications
The fuzzy K-means problem is a popular generalization of the well-known K-means problem to soft clusterings. We present the first coresets for fuzzy K-means with size linear in the dimension, polynomial in the number of clusters, and poly-logarithmic in the number of points. We show that these coresets can be employed in the computation of a (1+epsilon)-approximation for fuzzy K-means, improving previously presented results. We further show that our coresets can be maintained in an insertion-only streaming setting, where data points arrive one-by-one.
clustering
fuzzy k-means
coresets
approximation algorithms
streaming
Theory of computation~Unsupervised learning and clustering
46:1-46:12
Regular Paper
This work was partially supported by the German Research Foundation (DFG) under grant BL 314/8-1.
https://arxiv.org/abs/1612.07516
Johannes
Blömer
Johannes Blömer
Department of Computer Science, Paderborn University, Paderborn, Germany
Sascha
Brauer
Sascha Brauer
Department of Computer Science, Paderborn University, Paderborn, Germany
Kathrin
Bujna
Kathrin Bujna
Department of Computer Science, Paderborn University, Paderborn, Germany
10.4230/LIPIcs.ISAAC.2018.46
N. Ailon, R. Jaiswal, and C. Monteleoni. Streaming k-means approximation. In Advances in Neural Information Processing Systems 22, pages 10-18. Curran Associates, Inc., 2009.
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http://dx.doi.org/10.4230/LIPIcs.SOCG.2015.754
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http://dx.doi.org/10.1080/01969727308546046
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http://dx.doi.org/10.1145/1993636.1993712
D. Feldman, M. Monemizadeh, and C. Sohler. A PTAS for K-means Clustering Based on Weak Coresets. In Proceedings of the Twenty-third Annual Symposium on Computational Geometry, pages 11-18, 2007. URL: http://dx.doi.org/10.1145/1247069.1247072.
http://dx.doi.org/10.1145/1247069.1247072
D. Feldman, M. Schmidt, and C. 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, 2013.
S. Har-Peled and A. Kushal. Smaller Coresets for K-median and K-means Clustering. In Proceedings of the Twenty-first Annual Symposium on Computational Geometry, pages 126-134, 2005. 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 Thirty-sixth Annual ACM Symposium on Theory of Computing, pages 291-300, 2004. URL: http://dx.doi.org/10.1145/1007352.1007400.
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M. R. Rezaee, P. M. J. van der Zwet, B. P. F. Lelieveldt, R. J. van der Geest, and J. H. C. Reiber. A multiresolution image segmentation technique based on pyramidal segmentation and fuzzy clustering. IEEE Transactions on Image Processing, 9(7):1238-1248, 2000. URL: http://dx.doi.org/10.1109/83.847836.
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M. Shindler, A. Wong, and A. Meyerson. Fast and accurate k-means for large datasets. In Advances in neural information processing systems, pages 2375-2383, 2011.
Johannes Blömer, Sascha Brauer, and Kathrin Bujna
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Streaming Algorithms for Planar Convex Hulls
Many classical algorithms are known for computing the convex hull of a set of n point in R^2 using O(n) space. For large point sets, whose size exceeds the size of the working space, these algorithms cannot be directly used. The current best streaming algorithm for computing the convex hull is computationally expensive, because it needs to solve a set of linear programs.
In this paper, we propose simpler and faster streaming and W-stream algorithms for computing the convex hull. Our streaming algorithm has small pass complexity, which is roughly a square root of the current best bound, and it is simpler in the sense that our algorithm mainly relies on computing the convex hulls of smaller point sets. Our W-stream algorithms, one of which is deterministic and the other of which is randomized, have nearly-optimal tradeoff between the pass complexity and space usage, as we established by a new unconditional lower bound.
Convex Hulls
Streaming Algorithms
Lower Bounds
Theory of computation~Computational geometry
47:1-47:13
Regular Paper
https://arxiv.org/abs/1810.00455
Martín
Farach-Colton
Martín Farach-Colton
Department of Computer Science, Rutgers University, Piscataway, USA
This research was supported in part by NSF CCF 1637458, NIH 1 U01 CA198952-01, a NetAPP Faculty Fellowship and a gift from Dell/EMC.
Meng
Li
Meng Li
Department of Computer Science, Rutgers University, Piscataway, USA
Meng-Tsung
Tsai
Meng-Tsung Tsai
Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan
This research was supported in part by the Ministry of Science and Technology of Taiwan under contract MOST grant 107-2218-E-009- 026-MY3, and the Higher Education Sprout Project of National Chiao Tung University and Ministry of Education (MOE), Taiwan.
10.4230/LIPIcs.ISAAC.2018.47
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S. Guha and A. McGregor. Tight Lower Bounds for Multi-pass Stream Computation Via Pass Elimination. In ICALP, pages 760-772, 2008.
N. Gupta and S. Sen. Optimal, output-sensitive algorithms for constructing planar hulls in parallel. Computational Geometry, 8(3):151-166, 1997.
J. Hershberger and S. Suri. Adaptive sampling for geometric problems over data streams. Computational Geometry, 39(3):191-208, 2008.
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M. Kallay. The complexity of incremental convex hull algorithms in R^d. Information Processing Letters, 19(4):197, 1984.
B. Kalyanasundaram and G. Schintger. The Probabilistic Communication Complexity of Set Intersection. SIAM J. Discret. Math., 5(4):545-557, 1992.
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M. H. Overmars and J. van Leeuwen. Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23(2):166-204, 1981.
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R. A. Rufai and D. S. Richards. A Streaming Algorithm for the Convex Hull. In the 27th Canadian Conference on Computational Geometry (CCCG), pages 165-172, 2015.
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M. I. Shamos. Computational Geometry. PhD thesis, Yale University, 1978.
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Martín Farach-Colton, Meng Li, and Meng-Tsung Tsai
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Deterministic Treasure Hunt in the Plane with Angular Hints
A mobile agent equipped with a compass and a measure of length has to find an inert treasure in the Euclidean plane. Both the agent and the treasure are modeled as points. In the beginning, the agent is at a distance at most D>0 from the treasure, but knows neither the distance nor any bound on it. Finding the treasure means getting at distance at most 1 from it. The agent makes a series of moves. Each of them consists in moving straight in a chosen direction at a chosen distance. In the beginning and after each move the agent gets a hint consisting of a positive angle smaller than 2 pi whose vertex is at the current position of the agent and within which the treasure is contained. We investigate the problem of how these hints permit the agent to lower the cost of finding the treasure, using a deterministic algorithm, where the cost is the worst-case total length of the agent's trajectory. It is well known that without any hint the optimal (worst case) cost is Theta(D^2). We show that if all angles given as hints are at most pi, then the cost can be lowered to O(D), which is optimal. If all angles are at most beta, where beta<2 pi is a constant unknown to the agent, then the cost is at most O(D^{2-epsilon}), for some epsilon>0. For both these positive results we present deterministic algorithms achieving the above costs. Finally, if angles given as hints can be arbitrary, smaller than 2 pi, then we show that cost Theta(D^2) cannot be beaten.
treasure hunt
deterministic algorithm
mobile agent
hint
plane
Theory of computation~Design and analysis of algorithms
Computing methodologies~Mobile agents
48:1-48:13
Regular Paper
Sébastien
Bouchard
Sébastien Bouchard
Sorbonne Université, CNRS, INRIA, LIP6, F-75005 Paris, France
Yoann
Dieudonné
Yoann Dieudonné
Laboratoire MIS, Université de Picardie Jules Verne, Amiens, France
Andrzej
Pelc
Andrzej Pelc
Département d'informatique, Université du Québec en Outaouais, Gatineau, Canada
This work was supported in part by NSERC discovery grant 8136 - 2013 and by the Research Chair in Distributed Computing of the Université du Québec en Outaouais.
Franck
Petit
Franck Petit
Sorbonne Université, CNRS, INRIA, LIP6, F-75005 Paris, France
This work was performed within Project ESTATE (Ref. ANR-16-CE25-0009-03), supported by French state funds managed by the ANR (Agence Nationale de la Recherche).
10.4230/LIPIcs.ISAAC.2018.48
Oswin Aichholzer, Franz Aurenhammer, Christian Icking, Rolf Klein, Elmar Langetepe, and Günter Rote. Generalized self-approaching curves. Discrete Applied Mathematics, 109(1-2):3-24, 2001.
Steve Alpern and Shmuel Gal. The Theory of Search Games and Rendezvous. Kluwer Academic Publications, 2003.
Ricardo A. Baeza-Yates, Joseph C. Culberson, and Gregory J. E. Rawlins. Searching in the Plane. Inf. Comput., 106(2):234-252, 1993.
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Lucas Boczkowski, Amos Korman, and Yoav Rodeh. Searching a Tree with Permanently Noisy Advice. In 26th Annual European Symposium on Algorithms, ESA 2018, August 20-22, 2018, Helsinki, Finland, pages 54:1-54:13, 2018.
Anthony Bonato and Richard Nowakowski. The Game of Cops and Robbers on Graphs. American Mathematical Society, 2011.
Timothy H. Chung, Geoffrey A. Hollinger, and Volkan Isler. Search and pursuit-evasion in mobile robotics - A survey. Auton. Robots, 31(4):299-316, 2011.
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G. Matthew Fricke, Joshua P. Hecker, Antonio D. Griego, Linh T. Tran, and Melanie E. Moses. A distributed deterministic spiral search algorithm for swarms. In 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2016, Daejeon, South Korea, October 9-14, 2016, pages 4430-4436, 2016.
Branko Grünbaum. Partitions of mass-distributions and convex bodies by hyperplanes. Pacific J. Math., 10:1257-1261, 1960.
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Ming-Yang Kao, John H. Reif, and Stephen R. Tate. Searching in an Unknown Environment: An Optimal Randomized Algorithm for the Cow-Path Problem. Inf. Comput., 131(1):63-79, 1996.
Elmar Langetepe. On the Optimality of Spiral Search. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 1-12, 2010.
Elmar Langetepe. Searching for an axis-parallel shoreline. Theor. Comput. Sci., 447:85-99, 2012.
Tobias Langner, Barbara Keller, Jara Uitto, and Roger Wattenhofer. Overcoming Obstacles with Ants. In 19th International Conference on Principles of Distributed Systems, OPODIS 2015, December 14-17, 2015, Rennes, France, pages 9:1-9:17, 2015.
Avery Miller and Andrzej Pelc. Tradeoffs between cost and information for rendezvous and treasure hunt. J. Parallel Distrib. Comput., 83:159-167, 2015.
Kevin Spieser and Emilio Frazzoli. The Cow-Path Game: A competitive vehicle routing problem. In Proceedings of the 51th IEEE Conference on Decision and Control, CDC 2012, December 10-13, 2012, Maui, HI, USA, pages 6513-6520, 2012.
Amnon Ta-Shma and Uri Zwick. Deterministic Rendezvous, Treasure Hunts, and Strongly Universal Exploration Sequences. ACM Trans. Algorithms, 10(3):12:1-12:15, 2014.
Sébastien Bouchard, Yoann Dieudonné, Andrzej Pelc, and Franck Petit
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Competitive Searching for a Line on a Line Arrangement
We discuss the problem of searching for an unknown line on a known or unknown line arrangement by a searcher S, and show that a search strategy exists that finds the line competitively, that is, with detour factor at most a constant when compared to the situation where S has all knowledge. In the case where S knows all lines but not which one is sought, the strategy is 79-competitive. We also show that it may be necessary to travel on Omega(n) lines to realize a constant competitive ratio. In the case where initially, S does not know any line, but learns about the ones it encounters during the search, we give a 414.2-competitive search strategy.
Competitive searching
line arrangement
detour factor
search strategy
Theory of computation~Design and analysis of algorithms
49:1-49:12
Regular Paper
Quirijn
Bouts
Quirijn Bouts
ASML Veldhoven, the Netherlands
Thom
Castermans
Thom Castermans
TU Eindhoven, the Netherlands
The Netherlands Organisation for Scientific Research (NWO) supports T.C. under project no. 314.99.117.
Arthur
van Goethem
Arthur van Goethem
TU Eindhoven, the Netherlands
Marc
van Kreveld
Marc van Kreveld
Utrecht University, the Netherlands
Supported by the Netherlands Organisation for Scientific Research on grant no. 612.001.651.
Wouter
Meulemans
Wouter Meulemans
TU Eindhoven, the Netherlands
Supported by the Netherlands eScience Center (NLeSC) on project 027.015.G02.
10.4230/LIPIcs.ISAAC.2018.49
Steve Alpern and Shmuel Gal. The Theory of Search Games and Rendezvous, volume 55. Springer Science &Business Media, 2006.
Ricardo Baeza-Yates, Joseph Culberson, and Gregory Rawlins. Searching in the Plane. Information &Computation, 106(2):234-252, 1993.
Ricardo Baeza-Yates and René Schott. Parallel searching in the plane. Computational Geometry, 5(3):143-154, 1995.
Anatole Beck and D.J. Newman. Yet more on the linear search problem. Israel Journal of Mathematics, 8(4):419-429, 1970.
Richard Bellman. A minimization problem. Bulletin of the American Mathematical Society, 62(3):270, 1956.
Avrim Blum, Prabhakar Raghavan, and Baruch Schieber. Navigating in unfamiliar geometric terrain. SIAM Journal on Computing, 26(1):110-137, 1997.
Prosenjit Bose, Andrej Brodnik, Svante Carlsson, Erik D. Demaine, Rudolf Fleischer, Alejandro López-Ortiz, Pat Morin, and J. Ian Munro. Online routing in convex subdivisions. International Journal of Computational Geometry and Applications, 12(4):283-295, 2002.
Prosenjit Bose, Jean-Lou De Carufel, and Stephane Durocher. Searching on a line: A complete characterization of the optimal solution. Theoretical Computer Science, 569:24-42, 2015.
Prosenjit Bose, Jean-Lou De Carufel, Stephane Durocher, and Perouz Taslakian. Competitive online routing on Delaunay triangulations. International Journal of Computational Geometry &Applications, 27(04):241-253, 2017.
Prosenjit Bose, Rolf Fagerberg, André van Renssen, and Sander Verdonschot. Optimal local routing on Delaunay triangulations defined by empty equilateral triangles. SIAM Journal on Computing, 44(6):1626-1649, 2015.
Prosenjit Bose and Pat Morin. Competitive online routing in geometric graphs. Theoretical Computer Science, 324(2-3):273-288, 2004.
Prosenjit Bose and Pat Morin. Online routing in triangulations. SIAM Journal on Computing, 33(4):937-951, 2004.
Erik D. Demaine, Sándor P. Fekete, and Shmuel Gal. Online searching with turn cost. Theoretical Computer Science, 361(2-3):342-355, 2006.
Sándor P. Fekete, Rolf Klein, and Andreas Nüchter. Online searching with an autonomous robot. Computational Geometry, 34(2):102-115, 2006.
Subir Kumar Ghosh and Rolf Klein. Online algorithms for searching and exploration in the plane. Computer Science Review, 4(4):189-201, 2010.
Mikael Hammar, Bengt J. Nilsson, and Sven Schuierer. Parallel searching on m rays. Computational Geometry, 18(3):125-139, 2001.
Christoph A. Hipke, Christian Icking, Rolf Klein, and Elmar Langetepe. How to Find a Point on a Line Within a Fixed Distance. Discrete Applied Mathematics, 93(1):67-73, 1999.
Frank Hoffmann, Christian Icking, Rolf Klein, and Klaus Kriegel. The Polygon Exploration Problem. SIAM Journal on Computing, 31(2):577-600, 2001.
Christian Icking, Rolf Klein, Elmar Langetepe, Sven Schuierer, and Ines Semrau. An optimal competitive strategy for walking in streets. SIAM Journal on Computing, 33(2):462-486, 2004.
J.R. Isbell. An optimal search pattern. Naval Research Logistics Quarterly, 4(4):357-359, 1957.
Bala Kalyanasundaram and Kirk Pruhs. A Competitive Analysis of Algorithms for Searching Unknown Scenes. Computational Geometry, 3:139-155, 1993.
Ming-Yang Kao, John H. Reif, and Stephen R. Tate. Searching in an unknown environment: An optimal randomized algorithm for the cow-path problem. Information and Computation, 131(1):63-79, 1996.
Alejandro López-Ortiz and Sven Schuierer. The ultimate strategy to search on m rays? Theoretical Computer Science, 261(2):267-295, 2001.
Daniel D. Sleator and Robert E. Tarjan. Amortized efficiency of list update and paging rules. Communications of the ACM, 28(2):202-208, 1985.
Marc van Kreveld. Competitive Analysis of the Pokémon Go Search Problem. In Abstracts of the 33rd European Workshop on Computational Geometry, pages 25-28, 2017. http://csconferences.mah.se/eurocg2017/proceedings.pdf.
Quirijn Bouts, Thom Castermans, Arthur van Goethem, Marc van Kreveld, and Wouter Meulemans
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Stabbing Pairwise Intersecting Disks by Five Points
Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab D. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points.
This provides a simple - albeit slightly weaker - algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points.
Disk graph
piercing set
LP-type problem
Mathematics of computing~Combinatorics
50:1-50:12
Regular Paper
Work on this paper was supported in part by grant 1367/2016 from the German-Israeli Science Foundation (GIF).
https://arxiv.org/abs/1801.03158
Sariel
Har-Peled
Sariel Har-Peled
Department of Computer Science, University of Illinois, Urbana, IL 61801, USA
https://orcid.org/0000-0003-2638-9635
Partially supported by a NSF AF awards CCF-1421231, and CCF-1217462.
Haim
Kaplan
Haim Kaplan
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel
Wolfgang
Mulzer
Wolfgang Mulzer
Institut für Informatik, Freie Universität Berlin, 14195 Berlin, Germany
https://orcid.org/0000-0002-1948-5840
Partially supported by DFG grant MU/3501/1 and ERC STG 757609.
Liam
Roditty
Liam Roditty
Department of Computer Science, Bar Ilan University, Ramat Gan 5290002, Israel
Paul
Seiferth
Paul Seiferth
Institut für Informatik, Freie Universität Berlin, 14195 Berlin, Germany
Partially supported by DFG grant MU/3501/1.
Micha
Sharir
Micha Sharir
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel
Partially supported by ISF Grant 892/13, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), by the Blavatnik Research Fund in Computer Science at Tel Aviv University, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University.
Max
Willert
Max Willert
Institut für Informatik, Freie Universität Berlin, 14195 Berlin, Germany
10.4230/LIPIcs.ISAAC.2018.50
Noga Alon and Daniel J. Kleitman. Piercing convex sets and the Hadwiger-Debrunner (p,q)-problem. Adv. Math., 96(1):103-112, 1992.
Timothy M. Chan. An optimal randomized algorithm for maximum Tukey depth. In Proc. 15th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 430-436, 2004.
Bernard Chazelle. The Discrepancy Method - Randomness and Complexity. Cambridge University Press, Cambridge, 2001.
Bernard Chazelle and Jiřı Matoušek. On linear-time deterministic algorithms for optimization problems in fixed dimension. J. Algorithms, 21(3):579-597, 1996.
Ludwig Danzer. Zur Lösung des Gallaischen Problems über Kreisscheiben in der Euklidischen Ebene. Studia Sci. Math. Hungar., 21(1-2):111-134, 1986.
Adrian Dumitrescu and Minghui Jiang. Piercing translates and homothets of a convex body. Algorithmica, 61(1):94-115, 2011.
Branko Grünbaum. On intersections of similar sets. Portugal. Math., 18:155-164, 1959.
Hugo Hadwiger and Hans Debrunner. Ausgewählte Einzelprobleme der kombinatorischen Geometrie in der Ebene. Enseignement Math. (2), 1:56-89, 1955.
Eduard Helly. Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresbericht der Deutschen Mathematiker-Vereinigung, 32:175-176, 1923.
Eduard Helly. Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten. Monatshefte für Mathematik, 37(1):281-302, 1930.
Johann Radon. Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Mathematische Annalen, 83(1):113-115, 1921.
Raimund Seidel. Small-Dimensional Linear Programming and Convex Hulls Made Easy. Discrete Comput. Geom., 6:423-434, 1991.
Micha Sharir and Emo Welzl. A combinatorial bound for linear programming and related problems. Proc. 9th Sympos. Theoret. Aspects Comput. Sci. (STACS), pages 567-579, 1992.
Lajos Stachó. Über ein Problem für Kreisscheibenfamilien. Acta Sci. Math. (Szeged), 26:273-282, 1965.
Lajos Stachó. A solution of Gallai’s problem on pinning down circles. Mat. Lapok, 32(1-3):19-47, 1981/84.
Sariel Har-Peled, Haim Kaplan, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, Micha Sharir, and Max Willert
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Point Location in Incremental Planar Subdivisions
We study the point location problem in incremental (possibly disconnected) planar subdivisions, that is, dynamic subdivisions allowing insertions of edges and vertices only. Specifically, we present an O(n log n)-space data structure for this problem that supports queries in O(log^2 n) time and updates in O(log n log log n) amortized time. This is the first result that achieves polylogarithmic query and update times simultaneously in incremental planar subdivisions. Its update time is significantly faster than the update time of the best known data structure for fully-dynamic (possibly disconnected) planar subdivisions.
Dynamic point location
general incremental planar subdivisions
Theory of computation~Computational geometry
51:1-51:12
Regular Paper
http://arxiv.org/abs/1809.10495
Eunjin
Oh
Eunjin Oh
Max Planck Institute for Informatics, Saarbrücken, Germany
10.4230/LIPIcs.ISAAC.2018.51
Lars Arge, Gerth Stølting Brodal, and Loukas Georgiadis. Improved Dynamic Planar Point Location. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), pages 305-314, 2006.
Hanna Baumgarten, Hermann Jung, and Kurt Mehlhorn. Dynamic Point Location in General Subdivisions. Journal of Algorithms, 17(3):342-380, 1994.
Jon Louis Bentley and James B. Saxe. Decomposable Searching Problems 1: Static-to-Dynamic Transformations. Journal of Algorithms, 1(4):297-396, 1980.
Timothy M. Chan and Yakov Nekrich. Towards an Optimal Method for Dynamic Planar Point Location. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015), pages 390-409, 2015.
Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete &Computational Geometry, 6(3):485-524, 1991.
Siu-Wing Cheng and Ravi Janardan. New Results on Dynamic Planar Point Location. SIAM Journal on Computing, 21(5):972-999, 1992.
Yi-Jen Chiang, Franco P. Preparata, and Roberto Tamassia. A Unified Approach to Dynamic Point Location, Ray shooting, and Shortest Paths in Planar Maps. SIAM Journal on Computing, 25(1):207-233, 1996.
Yi-Jen Chiang and Roberto Tamassia. Dynamization of the trapezoid method for planar point location in monotone subdivisions. International Journal of Computational Geometry & Applications, 2(3):311-333, 1992.
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag TELOS, 2008.
Michael T. Goodrich and Roberto Tamassia. Dynamic Trees and Dynamic Point Location. SIAM Journal on Computing, 28(2):612-636, 1998.
Hiroshi Imai and Takao Asano. Dynamic orthogonal segment intersection search. Journal of Algorithms, 8(1):1-18, 1987.
Kurt Mehlhorn and Stefan Näher. Dynamic fractional cascading. Algorithmica, 5(1):215-241, 1990.
Eunjin Oh and Hee-Kap Ahn. Point Location in Dynamic Planar Subdivision. In Proceedings of the 34th International Symposium on Computational Geometry (SOCG 2018), volume 99 of Leibniz International Proceedings in Informatics (LIPIcs), pages 63:1-63:14, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2018.63.
http://dx.doi.org/10.4230/LIPIcs.SoCG.2018.63
Franco P. Preparata and Roberto Tamassia. Fully Dynamic Point Location in a Monotone Subdivision. SIAM Journal on Computing, 18(4):811-830, 1989.
Jack Snoeyink. Point Location. In Handbook of Discrete and Computational Geometry, Third Edition, pages 1005-1023. Chapman and Hall/CRC, 2017.
Eunjin Oh
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Convex Partial Transversals of Planar Regions
We consider the problem of testing, for a given set of planar regions R and an integer k, whether there exists a convex shape whose boundary intersects at least k regions of R. We provide polynomial-time algorithms for the case where the regions are disjoint axis-aligned rectangles or disjoint line segments with a constant number of orientations. On the other hand, we show that the problem is NP-hard when the regions are intersecting axis-aligned rectangles or 3-oriented line segments. For several natural intermediate classes of shapes (arbitrary disjoint segments, intersecting 2-oriented segments) the problem remains open.
computational geometry
algorithms
NP-hardness
convex transversals
Theory of computation~Computational geometry
52:1-52:12
Regular Paper
https://arxiv.org/abs/1809.10078
Vahideh
Keikha
Vahideh Keikha
Dept. of Mathematics and Computer Science, University of Sistan and Baluchestan, Zahedan, Iran
Mees
van de Kerkhof
Mees van de Kerkhof
Dept. of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands
M.v.d.K. supported by the Netherlands Organisation for Scientific Research under proj. 628.011.005.
Marc
van Kreveld
Marc van Kreveld
Dept. of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands
M.v.K. supported by the Netherlands Organisation for Scientific Research under proj. 612.001.651.
Irina
Kostitsyna
Irina Kostitsyna
Dept. of Mathematics and Computer Science, TU Eindhoven, Eindhoven, The Netherlands
Maarten
Löffler
Maarten Löffler
Dept. of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands
M.L. supported by the Netherlands Organisation for Scientific Research under proj. 614.001.504.
Frank
Staals
Frank Staals
Dept. of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands
F.S. supported by the Netherlands Organisation for Scientific Research under proj. 612.001.651.
Jérôme
Urhausen
Jérôme Urhausen
Dept. of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands
J.U. supported by the Netherlands Organisation for Scientific Research under proj. 612.001.651.
Jordi L.
Vermeulen
Jordi L. Vermeulen
Dept. of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands
J.V. supported by the Netherlands Organisation for Scientific Research under proj. 612.001.651.
Lionov
Wiratma
Lionov Wiratma
Dept. of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands, Dept. of Informatics, Parahyangan Catholic University, Bandung, Indonesia
L.W. supported by the Mnst. of Research, Techn. and High. Ed. of Indonesia (No. 138.41/E4.4/2015).
10.4230/LIPIcs.ISAAC.2018.52
E. M. Arkin, A. Banik, P. Carmi, G. Citovsky, M. J. Katz, J. S. B. Mitchell, and M. Simakov. Conflict-free Covering. In Proc. 27th Canadian Conference on Computational Geometry (CCCG), pages 17-23, 2015.
E. M. Arkin, C. Dieckmann, C. Knauer, J. S. B. Mitchell, V. Polishchuk, L. Schlipf, and S. Yang. Convex transversals. Computational Geometry, 47(2, Part B):224-239, 2014. URL: http://dx.doi.org/10.1016/j.comgeo.2012.10.009.
http://dx.doi.org/10.1016/j.comgeo.2012.10.009
M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, 3rd edition, 2008.
D. Eppstein, M. Overmars, G. Rote, and G. Woeginger. Finding Minimum Area k-gons. Discrete &Computational Geometry, 7(1):45-58, 1992. URL: http://dx.doi.org/10.1007/BF02187823.
http://dx.doi.org/10.1007/BF02187823
G. Even, Z. Lotker, D. Ron, and S. Smorodinsky. Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM Journal on Computing, 33(1):94-136, 2003.
M. T. Goodrich and J. S. Snoeyink. Stabbing parallel segments with a convex polygon. Computer Vision, Graphics, and Image Processing, 49(2):152-170, 1990.
S. Har-Peled and S. Smorodinsky. Conflict-free coloring of points and simple regions in the plane. Discrete &Computational Geometry, 34(1):47-70, 2005.
M. J. Katz, N. Lev-Tov, and G. Morgenstern. Conflict-free coloring of points on a line with respect to a set of intervals. Computational Geometry, 45(9):508-514, 2012.
V. Keikha, M. van de Kerkhof, M. van Kreveld, I. Kostitsyna, M. Löffler, F. Staals, J. Urhausen, J. L. Vermeulen, and L. Wiratma. Convex partial transversals of planar regions. arXiv preprint, 2018. URL: http://arxiv.org/abs/1809.10078.
http://arxiv.org/abs/1809.10078
J. S. B. Mitchell, G. Rote, G. Sundaram, and G. Woeginger. Counting convex polygons in planar point sets. Information Processing Letters, 56(1):45-49, 1995. URL: http://dx.doi.org/10.1016/0020-0190(95)00130-5.
http://dx.doi.org/10.1016/0020-0190(95)00130-5
M. H. Overmars and E. Welzl. New Methods for Computing Visibility Graphs. In Proc. 4th Annual Symposium on Computational Geometry (SCG), pages 164-171, 1988. URL: http://dx.doi.org/10.1145/73393.73410.
http://dx.doi.org/10.1145/73393.73410
G. Rote, G. Woeginger, B. Zhu, and Z. Wang. Counting k-subsets and convex k-gons in the plane. Information Processing Letters, 38(3):149-151, 1991.
L. Schlipf. Notes on Convex Transversals. arXiv preprint, 2012. URL: http://arxiv.org/abs/1211.5107.
http://arxiv.org/abs/1211.5107
M. Tompa. An optimal solution to a wire-routing problem. Journal of Computer and System Sciences, 23(2):127-150, 1981. URL: http://dx.doi.org/10.1016/0022-0000(81)90010-6.
http://dx.doi.org/10.1016/0022-0000(81)90010-6
Vahideh Keikha, Mees van de Kerkhof, Marc van Kreveld, Irina Kostitsyna, Maarten Löffler, Frank Staals, Jérôme Urhausen, Jordi L. Vermeulen, and Lionov Wiratma
Creative Commons Attribution 3.0 Unported license
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Extending the Centerpoint Theorem to Multiple Points
The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set P of n points in R^d, there is a point c, not necessarily from P, such that each halfspace containing c contains at least n/(d+1) points of P. Such a point c is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for the point set P. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median.
We present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set Q of (few) points such that every halfspace that contains one point of Q contains a large fraction of the points of P and every halfspace that contains more of Q contains an even larger fraction of P. This setting is comparable to the well-studied concepts of weak epsilon-nets and weak epsilon-approximations, where it is stronger than the former but weaker than the latter. We show that for any point set of size n in R^d and for any positive alpha_1,...,alpha_k where alpha_1 <= alpha_2 <= ... <= alpha_k and for every i,j with i+j <= k+1 we have that (d-1)alpha_k+alpha_i+alpha_j <= 1, we can find Q of size k such that each halfspace containing j points of Q contains least alpha_j n points of P. For two-dimensional point sets we further show that for every alpha and beta with alpha <= beta and alpha+beta <= 2/3 we can find Q with |Q|=3 such that each halfplane containing one point of Q contains at least alpha n of the points of P and each halfplane containing all of Q contains at least beta n points of P. All these results generalize to the setting where P is any mass distribution. For the case where P is a point set in R^2 and |Q|=2, we provide algorithms to find such points in time O(n log^3 n).
centerpoint
point sets
Tukey depth
Theory of computation~Design and analysis of algorithms
53:1-53:13
Regular Paper
https://arxiv.org/abs/1810.10231
Alexander
Pilz
Alexander Pilz
Institute of Software Technology, Graz University of Technology, Austria
https://orcid.org/0000-0002-6059-1821
Supported by a Schrödinger fellowship of the Austrian Science Fund (FWF): J-3847-N35.
Patrick
Schnider
Patrick Schnider
Department of Computer Science, ETH Zurich, Switzerland
10.4230/LIPIcs.ISAAC.2018.53
Greg Aloupis. Geometric measures of data depth. In Regina Y. Liu, Robert Serfling, and Diane L. Souvaine, editors, Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications, pages 147-158. DIMACS/AMS, 2003.
Greg Aloupis, Carmen Cortés, Francisco Gómez, Michael Soss, and Godfried Toussaint. Lower bounds for computing statistical depth. Comput. Statist. Data Anal., 40(2):223-229, 2002. URL: http://dx.doi.org/10.1016/S0167-9473(02)00032-4.
http://dx.doi.org/10.1016/S0167-9473(02)00032-4
Boris Aronov, Franz Aurenhammer, Ferran Hurtado, Stefan Langerman, David Rappaport, Carlos Seara, and Shakhar Smorodinsky. Small weak epsilon-nets. Comput. Geom., 42(5):455-462, 2009. URL: http://dx.doi.org/10.1016/j.comgeo.2008.02.005.
http://dx.doi.org/10.1016/j.comgeo.2008.02.005
Maryam Babazadeh and Hamid Zarrabi-Zadeh. Small Weak Epsilon-Nets in Three Dimensions. In Proc. 18th Canadian Conference on Computational Geometry (CCCG), 2006. URL: http://www.cs.queensu.ca/cccg/papers/cccg13.pdf.
http://www.cs.queensu.ca/cccg/papers/cccg13.pdf
Boris Bukh and Gabriel Nivasch. One-Sided Epsilon-Approximants. In Martin Loebl, Jaroslav Nešetřil, and Robin Thomas, editors, A Journey Through Discrete Mathematics: A Tribute to Jiří Matoušek, pages 343-356. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-44479-6_12.
http://dx.doi.org/10.1007/978-3-319-44479-6_12
Probal Chaudhuri. On a geometric notion of quantiles for multivariate data. J. American Statist. Assoc., 91(434):862-872, 1996.
David Haussler and Emo Welzl. ε-nets and simplex range queries. Discrete Comput. Geom., 2:127-151, 1987. URL: http://dx.doi.org/10.1007/BF02187876.
http://dx.doi.org/10.1007/BF02187876
Shreesh Jadhav and Asish Mukhopadhyay. Computing a Centerpoint of a Finite Planar Set of Points in Linear Time. Discrete Comput. Geom., 12:291-312, 1994. URL: http://dx.doi.org/10.1007/BF02574382.
http://dx.doi.org/10.1007/BF02574382
Stefan Langerman and William L. Steiger. Optimization in Arrangements. In 20th Symp. on Theoretical Aspects of Computer Science (STACS), volume 2607 of LNCS, pages 50-61, 2003. URL: http://dx.doi.org/10.1007/3-540-36494-3_6.
http://dx.doi.org/10.1007/3-540-36494-3_6
Jiří Matoušek. Computing the Center of Planar Point Sets. In Jacob E. Goodman, Richard Pollack, and William Steiger, editors, Discrete and Computational Geometry: Papers from the DIMACS Special Year, volume 6 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 221-230. DIMACS/AMS, 1990.
Jiří Matoušek. Construction of epsilon-Nets. Discrete Comput. Geom., 5:427-448, 1990. URL: http://dx.doi.org/10.1007/BF02187804.
http://dx.doi.org/10.1007/BF02187804
Jiří Matoušek. Tight upper bounds for the discrepancy of half-spaces. Discrete Comput. Geom., 13(3):593-601, 1995. URL: http://dx.doi.org/10.1007/BF02574066.
http://dx.doi.org/10.1007/BF02574066
Jiří Matoušek, Emo Welzl, and Lorenz Wernisch. Discrepancy and approximations for bounded VC-dimension. Combinatorica, 13(4):455-466, 1993. URL: http://dx.doi.org/10.1007/BF01303517.
http://dx.doi.org/10.1007/BF01303517
Nabil Mustafa and Kasturi Varadarajan. Epsilon-approximations and epsilon-nets. In Handbook of Discrete and Computational Geometry. HAL, 2017. URL: https://hal.archives-ouvertes.fr/hal-01468664.
https://hal.archives-ouvertes.fr/hal-01468664
Nabil H. Mustafa and Saurabh Ray. An optimal extension of the centerpoint theorem. Comput. Geom., 42(6):505-510, 2009. URL: http://dx.doi.org/10.1016/j.comgeo.2007.10.004.
http://dx.doi.org/10.1016/j.comgeo.2007.10.004
Sambuddha Roy and William Steiger. Some Combinatorial and Algorithmic Applications of the Borsuk-Ulam Theorem. Graphs Combin., 23:331-341, 2007. URL: http://dx.doi.org/10.1007/s00373-007-0716-1.
http://dx.doi.org/10.1007/s00373-007-0716-1
Mudassir Shabbir. Some results in computational and combinatorial geometry. PhD thesis, Rutgers The State University of New Jersey, 2014.
John W. Tukey. Mathematics and the picturing of data. In Proc. International Congress of Mathematicians, pages 523-531, 1975.
Alexander Pilz and Patrick Schnider
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Approximate Query Processing over Static Sets and Sliding Windows
Indexing of static and dynamic sets is fundamental to a large set of applications such as information retrieval and caching. Denoting the characteristic vector of the set by B, we consider the problem of encoding sets and multisets to support approximate versions of the operations rank(i) (i.e., computing sum_{j <= i} B[j]) and select(i) (i.e., finding min{p|rank(p) >= i}) queries. We study multiple types of approximations (allowing an error in the query or the result) and present lower bounds and succinct data structures for several variants of the problem. We also extend our model to sliding windows, in which we process a stream of elements and compute suffix sums. This is a generalization of the window summation problem that allows the user to specify the window size at query time. Here, we provide an algorithm that supports updates and queries in constant time while requiring just (1+o(1)) factor more space than the fixed-window summation algorithms.
Streaming
Algorithms
Sliding window
Lower bounds
Theory of computation~Data compression
54:1-54:12
Regular Paper
https://arxiv.org/abs/1809.05419
Ran
Ben Basat
Ran Ben Basat
Harvard University, Cambridge, USA
Supported by the Zuckerman Foundation, the Technion Hiroshi Fujiwara cyber security research center, and the Israel Cyber Directorate.
Seungbum
Jo
Seungbum Jo
University of Siegen, Germany
https://orcid.org/0000-0002-8644-3691
The author of this paper is supported by the DFG research project LO748/11-1.
Srinivasa Rao
Satti
Srinivasa Rao Satti
Seoul National University, South Korea
https://orcid.org/0000-0003-0636-9880
Shubham
Ugare
Shubham Ugare
IIT Guwahati, Guwahati, India
10.4230/LIPIcs.ISAAC.2018.54
R. Ben Basat, S. Jo, S. Rao Satti, and S. Ugare. Approximate Query Processing over Static Sets and Sliding Windows. ArXiv e-prints, September 2018. URL: http://arxiv.org/abs/1809.05419.
http://arxiv.org/abs/1809.05419
Ran Ben-Basat, Gil Einziger, and Roy Friedman. Give Me Some Slack: Efficient Network Measurements. In MFCS, pages 34:1-34:16, 2018.
Ran Ben-Basat, Gil Einziger, Roy Friedman, and Yaron Kassner. Efficient Summing over Sliding Windows. In SWAT, pages 11:1-11:14, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.SWAT.2016.11.
http://dx.doi.org/10.4230/LIPIcs.SWAT.2016.11
Ran Ben-Basat, Roy Friedman, and Rana Shahout. Heavy Hitters over Interval Queries. CoRR, abs/1804.10740, 2018. URL: http://arxiv.org/abs/1804.10740.
http://arxiv.org/abs/1804.10740
David R. Clark and J. Ian Munro. Efficient Suffix Trees on Secondary Storage. In SODA, pages 383-391, 1996.
Mayur Datar, Aristides Gionis, Piotr Indyk, and Rajeev Motwani. Maintaining Stream Statistics over Sliding Windows. SIAM J. Comput., 31(6):1794-1813, 2002. URL: http://dx.doi.org/10.1137/S0097539701398363.
http://dx.doi.org/10.1137/S0097539701398363
Hicham El-Zein, J. Ian Munro, and Yakov Nekrich. Succinct Color Searching in One Dimension. In ISAAC, pages 30:1-30:11, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ISAAC.2017.30.
http://dx.doi.org/10.4230/LIPIcs.ISAAC.2017.30
Phillip B. Gibbons and Srikanta Tirthapura. Distributed streams algorithms for sliding windows. In SPAA, pages 63-72, 2002. URL: http://dx.doi.org/10.1145/564870.564880.
http://dx.doi.org/10.1145/564870.564880
Alexander Golynski, J. Ian Munro, and S. Srinivasa Rao. Rank/Select Operations on Large Alphabets: A Tool for Text Indexing. In SODA, pages 368-373, 2006.
Alexander Golynski, Alessio Orlandi, Rajeev Raman, and S. Srinivasa Rao. Optimal Indexes for Sparse Bit Vectors. Algorithmica, 69(4):906-924, 2014. URL: http://dx.doi.org/10.1007/s00453-013-9767-2.
http://dx.doi.org/10.1007/s00453-013-9767-2
Wing-Kai Hon, Kunihiko Sadakane, and Wing-Kin Sung. Succinct data structures for Searchable Partial Sums with optimal worst-case performance. Theor. Comput. Sci., 412(39):5176-5186, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2011.05.023.
http://dx.doi.org/10.1016/j.tcs.2011.05.023
Guy Joseph Jacobson. Succinct Static Data Structures. PhD thesis, Carnegie Mellon University, Pittsburgh, PA, USA, 1988. AAI8918056.
Seungbum Jo, Stelios Joannou, Daisuke Okanohara, Rajeev Raman, and Srinivasa Rao Satti. Compressed Bit vectors Based on Variable-to-Fixed Encodings. Comput. J., 60(5):761-775, 2017.
P. B. Miltersen. Cell probe complexity - a survey. FSTTCS, 1999.
J.Ian Munro, Venkatesh Raman, and S.Srinivasa Rao. Space Efficient Suffix Trees. J. Algorithms, 39(2):205-222, 2001.
Gonzalo Navarro and Eliana Providel. Fast, Small, Simple Rank/Select on Bitmaps. In SEA, pages 295-306, 2012. URL: http://dx.doi.org/10.1007/978-3-642-30850-5_26.
http://dx.doi.org/10.1007/978-3-642-30850-5_26
Daisuke Okanohara and Kunihiko Sadakane. Practical Entropy-Compressed Rank/Select Dictionary. In ALENEX, pages 60-70, 2007. URL: http://dx.doi.org/10.1137/1.9781611972870.6.
http://dx.doi.org/10.1137/1.9781611972870.6
Rajeev Raman, Venkatesh Raman, and S. Srinivasa Rao. Succinct Dynamic Data Structures. In WADS, pages 426-437, 2001. URL: http://dx.doi.org/10.1007/3-540-44634-6_39.
http://dx.doi.org/10.1007/3-540-44634-6_39
Rajeev Raman, Venkatesh Raman, and Srinivasa Rao Satti. Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets. ACM Trans. Algorithms, 3(4):43, 2007. URL: http://dx.doi.org/10.1145/1290672.1290680.
http://dx.doi.org/10.1145/1290672.1290680
Ran Ben-Basat, Seungbum Jo, Srinivasa Rao Satti, and Shubham Ugare
Creative Commons Attribution 3.0 Unported license
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Multi-Finger Binary Search Trees
We study multi-finger binary search trees (BSTs), a far-reaching extension of the classical BST model, with connections to the well-studied k-server problem. Finger search is a popular technique for speeding up BST operations when a query sequence has locality of reference. BSTs with multiple fingers can exploit more general regularities in the input. In this paper we consider the cost of serving a sequence of queries in an optimal (offline) BST with k fingers, a powerful benchmark against which other algorithms can be measured.
We show that the k-finger optimum can be matched by a standard dynamic BST (having a single root-finger) with an O(log{k}) factor overhead. This result is tight for all k, improving the O(k) factor implicit in earlier work. Furthermore, we describe new online BSTs that match this bound up to a (log{k})^{O(1)} factor. Previously only the "one-finger" special case was known to hold for an online BST (Iacono, Langerman, 2016; Cole et al., 2000). Splay trees, assuming their conjectured optimality (Sleator and Tarjan, 1983), would have to match our bounds for all k.
Our online algorithms are randomized and combine techniques developed for the k-server problem with a multiplicative-weights scheme for learning tree metrics. To our knowledge, this is the first time when tools developed for the k-server problem are used in BSTs. As an application of our k-finger results, we show that BSTs can efficiently serve queries that are close to some recently accessed item. This is a (restricted) form of the unified property (Iacono, 2001) that was previously not known to hold for any BST algorithm, online or offline.
binary search trees
dynamic optimality
finger search
k-server
Theory of computation~Design and analysis of algorithms
Theory of computation~Data structures design and analysis
55:1-55:26
Regular Paper
Parinya
Chalermsook
Parinya Chalermsook
Aalto University, Finland
Parinya Chalermsook is supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 759557) and by Academy of Finland Research Fellows, under grant No. 310415.
Mayank
Goswami
Mayank Goswami
Queens College, City University of New York, USA
László
Kozma
László Kozma
TU Eindhoven, The Netherlands
László Kozma is supperted through ERC consolidator grant No. 617951.
Kurt
Mehlhorn
Kurt Mehlhorn
MPI für Informatik, Saarbrücken, Germany
Thatchaphol
Saranurak
Thatchaphol Saranurak
KTH Royal Institute of Technology, Sweden
Thatchaphol Saranurak is supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 715672, and by the Swedish Research Council (Reg. No. 2015-04659).
10.4230/LIPIcs.ISAAC.2018.55
Sanjeev Arora, Elad Hazan, and Satyen Kale. The Multiplicative Weights Update Method: a Meta-Algorithm and Applications. Theory of Computing, 8(1):121-164, 2012. URL: http://dx.doi.org/10.4086/toc.2012.v008a006.
http://dx.doi.org/10.4086/toc.2012.v008a006
Nikhil Bansal, Niv Buchbinder, Aleksander Madry, and Joseph Naor. A Polylogarithmic-Competitive Algorithm for the k-Server Problem. J. ACM, 62(5):40:1-40:49, 2015. URL: http://dx.doi.org/10.1145/2783434.
http://dx.doi.org/10.1145/2783434
Yair Bartal and Eddie Grove. The harmonic k-server algorithm is competitive. J. ACM, 47(1):1-15, 2000. URL: http://dx.doi.org/10.1145/331605.331606.
http://dx.doi.org/10.1145/331605.331606
Avrim Blum and Carl Burch. On-line Learning and the Metrical Task System Problem. In Proceedings of the Tenth Annual Conference on Computational Learning Theory, COLT 1997, Nashville, Tennessee, USA, July 6-9, 1997., pages 45-53, 1997. URL: http://dx.doi.org/10.1145/267460.267475.
http://dx.doi.org/10.1145/267460.267475
Avrim Blum, Shuchi Chawla, and Adam Kalai. Static Optimality and Dynamic Search-Optimality in Lists and Trees. Algorithmica, 36(3):249-260, 2003. URL: http://dx.doi.org/10.1007/s00453-003-1015-8.
http://dx.doi.org/10.1007/s00453-003-1015-8
Allan Borodin and Ran El-Yaniv. Online computation and competitive analysis. Cambridge University Press, 1998.
Prosenjit Bose, Jonathan F Buss, and Anna Lubiw. Pattern matching for permutations. Information Processing Letters, 65(5):277-283, 1998.
Prosenjit Bose, Karim Douïeb, John Iacono, and Stefan Langerman. The Power and Limitations of Static Binary Search Trees with Lazy Finger. In ISAAC, pages 181-192, 2014. URL: http://dx.doi.org/10.1007/978-3-319-13075-0_15.
http://dx.doi.org/10.1007/978-3-319-13075-0_15
Gerth Stølting Brodal. Finger Search Trees. In Handbook of Data Structures and Applications. Chapman and Hall/CRC, 2004. URL: http://dx.doi.org/10.1201/9781420035179.ch11.
http://dx.doi.org/10.1201/9781420035179.ch11
Mark R. Brown and Robert Endre Tarjan. Design and Analysis of a Data Structure for Representing Sorted Lists. SIAM J. Comput., 9(3):594-614, 1980. URL: http://dx.doi.org/10.1137/0209045.
http://dx.doi.org/10.1137/0209045
Mihai Bădoiu, Richard Cole, Erik D. Demaine, and John Iacono. A Unified Access Bound on Comparison-Based Dynamic Dictionaries. Theoretical Computer Science, 382(2):86-96, August 2007.
Sébastien Bubeck, Michael B. Cohen, James R. Lee, Yin Tat Lee, and Aleksander Madry. k-server via multiscale entropic regularization. In STOC, 2018.
Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn, and Thatchaphol Saranurak. Pattern-Avoiding Access in Binary Search Trees. In FOCS, pages 410-423, 2015.
Marek Chrobak and Lawrence L. Larmore. An Optimal On-Line Algorithm for k-Servers on Trees. SIAM J. Comput., 20(1):144-148, 1991. URL: http://dx.doi.org/10.1137/0220008.
http://dx.doi.org/10.1137/0220008
R. Cole. On the Dynamic Finger Conjecture for Splay Trees. Part II: The Proof. SIAM Journal on Computing, 30(1):44-85, 2000. URL: http://dx.doi.org/10.1137/S009753979732699X.
http://dx.doi.org/10.1137/S009753979732699X
Richard Cole, Bud Mishra, Jeanette Schmidt, and Alan Siegel. On the Dynamic Finger Conjecture for Splay Trees. Part I: Splay Sorting Log n-Block Sequences. SIAM J. Comput., 30(1):1-43, April 2000. URL: http://dx.doi.org/10.1137/S0097539797326988.
http://dx.doi.org/10.1137/S0097539797326988
Erik D. Demaine, Dion Harmon, John Iacono, Daniel M. Kane, and Mihai Pǎtraşcu. The geometry of binary search trees. In SODA 2009, pages 496-505, 2009. URL: http://dl.acm.org/citation.cfm?id=1496770.1496825.
http://dl.acm.org/citation.cfm?id=1496770.1496825
Erik D. Demaine, John Iacono, Stefan Langerman, and Özgür Özkan. Combining Binary Search Trees. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 388-399, 2013. URL: http://dx.doi.org/10.1007/978-3-642-39206-1_33.
http://dx.doi.org/10.1007/978-3-642-39206-1_33
Erik D. Demaine, Stefan Langerman, and Eric Price. Confluently Persistent Tries for Efficient Version Control. Algorithmica, 57(3):462-483, 2010. URL: http://dx.doi.org/10.1007/s00453-008-9274-z.
http://dx.doi.org/10.1007/s00453-008-9274-z
Jonathan Derryberry and Daniel Dominic Sleator. Skip-Splay: Toward Achieving the Unified Bound in the BST Model. In WADS 2009, Banff, Canada, August 21-23, 2009. Proceedings, pages 194-205, 2009.
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http://arxiv.org/abs/1302.6914
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Christos Levcopoulos and Ola Petersson. Sorting Shuffled Monotone Sequences. Inf. Comput., 112(1):37-50, 1994. URL: http://dx.doi.org/10.1006/inco.1994.1050.
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Joan M. Lucas. Canonical forms for competitive binary search tree algorithms. Tech. Rep. DCS-TR-250, Rutgers University, 1988.
Mark S. Manasse, Lyle A. McGeoch, and Daniel Dominic Sleator. Competitive Algorithms for Server Problems. J. Algorithms, 11(2):208-230, 1990. URL: http://dx.doi.org/10.1016/0196-6774(90)90003-W.
http://dx.doi.org/10.1016/0196-6774(90)90003-W
K. Mehlhorn and P. Sanders. Algorithms and Data Structures: The Basic Toolbox. Springer, 2008.
Kurt Mehlhorn. Data Structures and Algorithms 1: Sorting and Searching, volume 1 of EATCS Monographs on Theoretical Computer Science. Springer, 1984. URL: http://dx.doi.org/10.1007/978-3-642-69672-5.
http://dx.doi.org/10.1007/978-3-642-69672-5
Alistair Moffat and Ola Petersson. An Overview of Adaptive Sorting. Australian Computer Journal, 24(2):70-77, 1992.
J.Ian Munro. On the Competitiveness of Linear Search. In Mike S. Paterson, editor, Algorithms - ESA 2000, volume 1879 of Lecture Notes in Computer Science, pages 338-345. Springer Berlin Heidelberg, 2000. URL: http://dx.doi.org/10.1007/3-540-45253-2_31.
http://dx.doi.org/10.1007/3-540-45253-2_31
William Pugh. Skip Lists: A Probabilistic Alternative to Balanced Trees. Commun. ACM, 33(6):668-676, 1990. URL: http://dx.doi.org/10.1145/78973.78977.
http://dx.doi.org/10.1145/78973.78977
Prabhakar Raghavan and Marc Snir. Memory versus randomization in on-line algorithms. IBM Journal of Research and Development, 38(6):683-708, 1994. URL: http://dx.doi.org/10.1147/rd.386.0683.
http://dx.doi.org/10.1147/rd.386.0683
Amitai Regev. Asymptotic values for degrees associated with strips of Young diagrams. Advances in Mathematics, 41(2):115-136, 1981.
Raimund Seidel and Cecilia R. Aragon. Randomized Search Trees. Algorithmica, 16(4/5):464-497, 1996. URL: http://dx.doi.org/10.1007/BF01940876.
http://dx.doi.org/10.1007/BF01940876
Steven S. Seiden. A General Decomposition Theorem for the k-Server Problem. Inf. Comput., 174(2):193-202, 2002. URL: http://dx.doi.org/10.1006/inco.2002.3144.
http://dx.doi.org/10.1006/inco.2002.3144
René Sitters. The Generalized Work Function Algorithm Is Competitive for the Generalized 2-Server Problem. SIAM J. Comput., 43(1):96-125, 2014. URL: http://dx.doi.org/10.1137/120885309.
http://dx.doi.org/10.1137/120885309
Daniel Dominic Sleator and Robert Endre Tarjan. Self-Adjusting Binary Search Trees. J. ACM, 32(3):652-686, 1985. URL: http://dx.doi.org/10.1145/3828.3835.
http://dx.doi.org/10.1145/3828.3835
Robert Endre Tarjan and Christopher J. Van Wyk. An O(n log log n)-Time Algorithm for Triangulating a Simple Polygon. SIAM J. Comput., 17(1):143-178, 1988. URL: http://dx.doi.org/10.1137/0217010.
http://dx.doi.org/10.1137/0217010
Athanasios K. Tsakalidis. AVL-Trees for Localized Search. Information and Control, 67(1-3):173-194, 1985. URL: http://dx.doi.org/10.1016/S0019-9958(85)80034-6.
http://dx.doi.org/10.1016/S0019-9958(85)80034-6
R. Wilber. Lower Bounds for Accessing Binary Search Trees with Rotations. SIAM Journal on Computing, 18(1):56-67, 1989. URL: http://dx.doi.org/10.1137/0218004.
http://dx.doi.org/10.1137/0218004
Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn, and Thatchaphol Saranurak
Creative Commons Attribution 3.0 Unported license
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On Counting Oracles for Path Problems
We initiate the study of counting oracles for various path problems in graphs. Distance oracles have gained a lot of attention in recent years, with studies of the underlying space and time tradeoffs. For a given graph G, a distance oracle is a data structure which can be used to answer distance queries for pairs of vertices s,t in V(G). In this work, we extend the set up to answering counting queries: for a pair of vertices s,t, the oracle needs to provide the number of (shortest or all) paths from s to t. We present O(n^{1.5}) preprocessing time, O(n^{1.5}) space, and O(sqrt{n}) query time algorithms for oracles counting shortest paths in planar graphs and for counting all paths in planar directed acyclic graphs. We extend our results to other graphs which admit small balanced separators and present applications where our oracle improves the currently best known running times.
Counting oracle
Path problems
Shortest paths
Separators
Theory of computation~Graph algorithms analysis
56:1-56:12
Regular Paper
Research supported by NSF grant CCF-1319987 and by an REU (Research Experience for Undergraduates) supplement.
Ivona
Bezáková
Ivona Bezáková
Department of Computer Science, Rochester Institute of Technology, Rochester, NY, USA
Andrew
Searns
Andrew Searns
Rochester Institute of Technology, Rochester, NY, USA
10.4230/LIPIcs.ISAAC.2018.56
Ittai Abraham, Shiri Chechik, Daniel Delling, Andrew V. Goldberg, and Renato F. Werneck. On Dynamic Approximate Shortest Paths for Planar Graphs with Worst-case Costs. SODA, pages 740-753, 2016.
Mohamad Akra and Louay Bazzi. On the Solution of Linear Recurrence Equations. Computational Optimization and Applications, 10(2):195-210, 1998.
Srinivasa Rao Arikati, Danny Z. Chen, L. Paul Chew, Gautam Das, Michiel H. M. Smid, and Christos D. Zaroliagis. Planar Spanners and Approximate Shortest Path Queries among Obstacles in the Plane. In Proceedings of Algorithms - ESA '96, Fourth Annual European Symposium, pages 514-528, 1996.
Michael O. Ball and J. Scott Provan. Calculating Bounds on Reachability and Connectnedness in Stochastic Networks. Networks, 13:253-278, 1983.
Ivona Bezáková and Adam J. Friedlander. Counting and Sampling Minimum (s,t)-Cuts in Weighted Planar Graphs in Polynomial Time. Theor. Comp. Sci., 417:2-11, 2012.
Sergio Cabello. Many Distances in Planar Graphs. Algorithmica, 62(1-2):361-381, 2012.
Erin W. Chambers, Kyle Fox, and Amir Nayyeri. Counting and Sampling Minimum Cuts in Genus g Graphs. Discrete & Computational Geometry, 52(3):450-475, 2014.
Danny Z. Chen and Jinhui Xu. Shortest path queries in planar graphs. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 469-478, 2000.
Vincent Cohen-Addad, Søren Dahlgaard, and Christian Wulff-Nilsen. Fast and Compact Exact Distance Oracle for Planar Graphs. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 962-973, 2017.
Hristo Djidjev. On-Line Algorithms for Shortest Path Problems on Planar Digraphs. In Proceedings of Graph-Theoretic Concepts in Computer Science, 22nd International Workshop, WG, pages 151-165, 1996.
Jittat Fakcharoenphol and Satish Rao. Planar graphs, negative weight edges, shortest paths, and near linear time. J. Comput. Syst. Sci., 72(5):868-889, 2006.
Jörg Flum and Martin Grohe. The Parameterized Complexity of Counting Problems. SIAM J. Comput., 33(4):892-922, 2004.
Pawel Gawrychowski, Shay Mozes, Oren Weimann, and Christian Wulff-Nilsen. Better Tradeoffs for Exact Distance Oracles in Planar Graphs. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 515-529, 2018.
John R. Gilber, Joan P. Hutchinson, and Robert Endre Tarjan. A Separator Theorem for Graphs of Bounded Genus. Journal of Algorithms, 5(3):391-405, 1984.
Monika Rauch Henzinger, Philip N. Klein, Satish Rao, and Sairam Subramanian. Faster Shortest-Path Algorithms for Planar Graphs. J. Comput. Syst. Sci., 55(1):3-23, 1997.
Richard J. Lipton and Robert E. Tarjan. A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 36(2):177-189, 1979. URL: http://dx.doi.org/10.1137/0136016.
http://dx.doi.org/10.1137/0136016
Matúš Mihalák, Rastislav Šrámek, and Peter Widmayer. Approximately Counting Approximately-Shortest Paths in Directed Acyclic Graphs. Theory Comput. Syst., 58(1):45-59, 2016.
Shay Mozes and Christian Sommer. Exact distance oracles for planar graphs. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA,, pages 209-222, 2012.
J. Scott Provan and Michael O. Ball. The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected. SIAM J. Comput., 12(4):777-788, 1983.
Leslie G. Valiant. The Complexity of Enumeration and Reliability Problems. SIAM J. Comput., 8(3):410-421, 1979.
Masaki Yamamoto. Approximately counting paths and cycles in a graph. Discrete Applied Mathematics, 217:381-387, 2017.
Ivona Bezáková and Andrew Searns
Creative Commons Attribution 3.0 Unported license
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Reconstructing Phylogenetic Tree From Multipartite Quartet System
A phylogenetic tree is a graphical representation of an evolutionary history in a set of taxa in which the leaves correspond to taxa and the non-leaves correspond to speciations. One of important problems in phylogenetic analysis is to assemble a global phylogenetic tree from smaller pieces of phylogenetic trees, particularly, quartet trees. Quartet Compatibility is to decide whether there is a phylogenetic tree inducing a given collection of quartet trees, and to construct such a phylogenetic tree if it exists. It is known that Quartet Compatibility is NP-hard but there are only a few results known for polynomial-time solvable subclasses.
In this paper, we introduce two novel classes of quartet systems, called complete multipartite quartet system and full multipartite quartet system, and present polynomial time algorithms for Quartet Compatibility for these systems. We also see that complete/full multipartite quartet systems naturally arise from a limited situation of block-restricted measurement.
phylogenetic tree
quartet system
reconstruction
Mathematics of computing~Combinatorial algorithms
57:1-57:13
Regular Paper
Hiroshi
Hirai
Hiroshi Hirai
Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Japan
Supported by KAKENHI Grant Numbers JP26280004, JP17K00029.
Yuni
Iwamasa
Yuni Iwamasa
Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Japan
Supported by JSPS Research Fellowship for Young Scientists.
10.4230/LIPIcs.ISAAC.2018.57
A. V. Aho, Y. Sagiv, T. G. Szymanski, and J. D. Ullman. Inferring a tree from lowest common ancestors with an application to the optimization of relational expressions. SIAM Journal on Computing, 10(3):405-421, 1981.
H.-J. Bandelt and A. Dress. Reconstructing the shape of a tree from observed dissimilarity data. Advances in Applied Mathematics, 7:309-343, 1986.
V. Berry, D. Bryant, T. Jiang, P. Kearney, M. Li, T. Wareham, and H. Zhang. A practical algorithm for recovering the best supported edges of an evolutionary tree. In Proceedings of the 11th ACM-SIAM Symposium on Discrete Algorithms (SODA'00), pages 287-296, 2000.
V. Berry, T. Jiang, P. Kearney, M. Li, and T. Wareham. Quartet cleaning: Improved algorithms and simulations. In Proceedings of the 7th European Symposium on Algorithm (ESA'99), volume 1643 of Lecture Notes in Computer Science, pages 313-324, Heidelberg, 1999. Springer.
D. Bryant and M. Steel. Extension operations on sets of leaf-labelled trees. Advances in Applied Mathematics, 16:425-453, 1995.
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M.-S Chang, C.-C Lin, and P. Rossmanith. New fixed-parameter algorithms for the minimum quartet inconsistency problem. Theory of Computing Systems, 47(2):342-367, 2010.
H. Colonius and H. H. Schulze. Tree structure from proximity data. British Journal of Mathematical and Statistical Psychology, 34:167-180, 1981.
W. M. Fitch. A non-sequential method for constructing trees and hierarchical classifications. Journal of Molecular Evolution, 18:30-37, 1981.
J. Gramm and R. Niedermeier. A fixed-parameter algorithm for minimum quartet inconsistency. Journal of Computer and System Sciences, 67:723-741, 2003.
H. Hirai, Y. Iwamasa, K. Murota, and S. Živný. A tractable class of binary VCSPs via M-convex intersection. arXiv, 2018. URL: http://arxiv.org/abs/1801.02199v1.
http://arxiv.org/abs/1801.02199v1
T. Jiang, P. Kearney, and M. Li. A polynomial time approximation scheme for inferring evolutionary trees from quartet topologies and its application. SIAM Journal on Computing, 30(6):1942-1961, 2001.
R. Reaz, M. S. Bayzid, and M. S. Rahman. Accurate phylogenetic tree reconstruction from quartets: A heuristic approach. PLoS ONE, 9(8):e104008, 2014.
S. Sattath and A. Tversky. Additive similarity trees. Psychometrika, 42(319-345), 1977.
A. Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg, 2003.
C. Semple and M. Steel. A supertree method for rooted trees. Discrete Applied Mathematics, 105:147-158, 2000.
C. Semple and M. Steel. Phylogenetics. Oxford University Press, Oxford, 2003.
M. Steel. The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification, 9:91-116, 1992.
K. Strimmer and A. Haeseler. Quartet puzzling: A quartet maximum-likelihood method for reconstructing tree topologies. Journal of Molecular Biology and Evolution, 13:964-969, 1996.
Hiroshi Hirai and Yuni Iwamasa
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain
We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of n vertices and h holes. We introduce a graph of oriented distances to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in O(min(n^omega, n^2 + nh log h + chi^2)) time, where omega<2.373 denotes the matrix multiplication exponent and chi in Omega(n) cap O(n^2) is the number of edges of the graph of oriented distances. We also provide an alternative algorithm for computing the diameter that runs in O(n^2 log n) time.
Rectilinear link distance
polygonal domain
diameter
radius
Theory of computation~Computational geometry
58:1-58:13
Regular Paper
https://arxiv.org/abs/1712.05538
Elena
Arseneva
Elena Arseneva
St. Petersburg State University, St. Petersburg, Russia
https://orcid.org/0000-0002-5267-4512
Partially supported by the SNF Early Postdoc Mobility grant P2TIP2-168563, Switzerland, and F.R.S.-FNRS, Belgium.
Man-Kwun
Chiu
Man-Kwun Chiu
Institut für Informatik, Freie Universität Berlin, Berlin, Germany
Supported in part by ERC StG 757609.
Matias
Korman
Matias Korman
Tufts University, Boston, USA
Supported in part by KAKENHI No. 17K12635, Japan and NSF award CCF-1422311.
Aleksandar
Markovic
Aleksandar Markovic
TU Eindhoven, Eindhoven, The Netherlands
Supported by the Netherlands' Organisation for Scientific Research (NWO) under project no. 024.002.003.
Yoshio
Okamoto
Yoshio Okamoto
University of Electro-Communications, Tokyo, Japan, RIKEN Center for Advanced Intelligent Project, Tokyo, Japan
https://orcid.org/0000-0002-9826-7074
Partially supported by JSPS KAKENHI Grant Number 15K00009 and JST CREST Grant Number JPMJCR1402, and Kayamori Foundation of Informational Science Advancement.
Aurélien
Ooms
Aurélien Ooms
Université libre de Bruxelles (ULB), Brussels, Belgium
https://orcid.org/0000-0002-5733-1383
Supported by the Fund for Research Training in Industry and Agriculture (FRIA).
André
van Renssen
André van Renssen
University of Sydney, Sydney, Australia
https://orcid.org/0000-0002-9294-9947
Supported by JST ERATO Grant Number JPMJER1201, Japan.
Marcel
Roeloffzen
Marcel Roeloffzen
TU Eindhoven, Eindhoven, The Netherlands
10.4230/LIPIcs.ISAAC.2018.58
Hee-Kap Ahn, Luis Barba, Prosenjit Bose, Jean-Lou De Carufel, Matias Korman, and Eunjin Oh. A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon. Discrete & Computational Geometry, 56(4):836-859, 2016. URL: http://dx.doi.org/10.1007/s00454-016-9796-0.
http://dx.doi.org/10.1007/s00454-016-9796-0
Sang Won Bae, Matias Korman, Joseph S. B. Mitchell, Yoshio Okamoto, Valentin Polishchuk, and Haitao Wang. Computing the L₁ Geodesic Diameter and Center of a Polygonal Domain. Discrete & Computational Geometry, 57(3):674-701, 2017. URL: http://dx.doi.org/10.1007/s00454-016-9841-z.
http://dx.doi.org/10.1007/s00454-016-9841-z
Sang Won Bae, Matias Korman, and Yoshio Okamoto. The Geodesic Diameter of Polygonal Domains. Discrete & Computational Geometry, 50(2):306-329, 2013. URL: http://dx.doi.org/10.1007/s00454-013-9527-8.
http://dx.doi.org/10.1007/s00454-013-9527-8
Sang Won Bae, Matias Korman, Yoshio Okamoto, and Haitao Wang. Computing the L₁ geodesic diameter and center of a simple polygon in linear time. Computational Geometry: Theory and Applications, 48(6):495-505, 2015. URL: http://dx.doi.org/10.1016/j.comgeo.2015.02.005.
http://dx.doi.org/10.1016/j.comgeo.2015.02.005
Timothy M. Chan and Dimitrios Skrepetos. All-Pairs Shortest Paths in Geometric Intersection Graphs. In Proc. 16th Algorithms and Data Structures Symposium, pages 253-264, 2017.
Hristo Djidjev, Andrzej Lingas, and Jörg-Rüdiger Sack. An O(n log n) Algorithm for Computing the Link Center of a Simple Polygon. Discrete & Computational Geometry, 8:131-152, 1992. URL: http://dx.doi.org/10.1007/BF02293040.
http://dx.doi.org/10.1007/BF02293040
Yoav Giyora and Haim Kaplan. Optimal dynamic vertical ray shooting in rectilinear planar subdivisions. ACM Transactions on Algorithms, 5(3):28:1-28:51, 2009. URL: http://dx.doi.org/10.1145/1541885.1541889.
http://dx.doi.org/10.1145/1541885.1541889
John Hershberger and Subhash Suri. Matrix Searching with the Shortest-Path Metric. SIAM Journal on Computing, 26(6):1612-1634, 1997. URL: http://dx.doi.org/10.1137/S0097539793253577.
http://dx.doi.org/10.1137/S0097539793253577
John Hershberger and Subhash Suri. An Optimal Algorithm for Euclidean Shortest Paths in the Plane. SIAM Journal on Computing, 28(6):2215-2256, 1999. URL: http://dx.doi.org/10.1137/S0097539795289604.
http://dx.doi.org/10.1137/S0097539795289604
François Le Gall. Powers of tensors and fast matrix multiplication. In Katsusuke Nabeshima, Kosaku Nagasaka, Franz Winkler, and Ágnes Szántó, editors, Proc. 25th International Symposium on Symbolic and Algebraic Computation, ISSAC, pages 296-303. ACM, 2014. URL: http://dx.doi.org/10.1145/2608628.2608664.
http://dx.doi.org/10.1145/2608628.2608664
Joseph S. B. Mitchell. An optimal algorithm for shortest rectilinear paths among obstacles. In Proc. 1st Canadian Conference on Computational Geometry, CCCG, 1989.
Joseph S. B. Mitchell. L₁ shortest paths among polygonal obstacles in the plane. Algorithmica, 8(1):55-88, 1992.
Joseph S. B. Mitchell, Valentin Polishchuk, and Mikko Sysikaski. Minimum-link paths revisited. Computational Geometry: Theory and Applications, 47(6):651-667, 2014. URL: http://dx.doi.org/10.1016/j.comgeo.2013.12.005.
http://dx.doi.org/10.1016/j.comgeo.2013.12.005
Joseph S. B. Mitchell, Valentin Polishchuk, Mikko Sysikaski, and Haitao Wang. An Optimal Algorithm for Minimum-Link Rectilinear Paths in Triangulated Rectilinear Domains. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Proc. Automata, Languages, and Programming - 42nd International Colloquium, ICALP, volume 9134 of Lecture Notes in Computer Science, pages 947-959. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_77.
http://dx.doi.org/10.1007/978-3-662-47672-7_77
Bengt J. Nilsson and Sven Schuierer. Computing the Rectilinear Link Diameter of a Polygon. In Proc. Computational Geometry - Methods, Algorithms and Applications, International Workshop on Computational Geometry CG'91, pages 203-215, 1991. URL: http://dx.doi.org/10.1007/3-540-54891-2_15.
http://dx.doi.org/10.1007/3-540-54891-2_15
Bengt J. Nilsson and Sven Schuierer. An Optimal Algorithm for the Rectilinear Link Center of a Rectilinear Polygon. Computational Geometry: Theory and Applications, 6:169-194, 1996. URL: http://dx.doi.org/10.1016/0925-7721(95)00026-7.
http://dx.doi.org/10.1016/0925-7721(95)00026-7
Subhash Suri. On some link distance problems in a simple polygon. IEEE Transactions on Robotics and Automation, 6(1):108-113, 1990. URL: http://dx.doi.org/10.1109/70.88124.
http://dx.doi.org/10.1109/70.88124
Haitao Wang. On the geodesic centers of polygonal domains. Journal of Computational Geometry, 9(1):131-190, 2018.
Elena Arseneva, Man-Kwun Chiu, Matias Korman, Aleksandar Markovic, Yoshio Okamoto, Aurélien Ooms, André van Renssen, and Marcel Roeloffzen
Creative Commons Attribution 3.0 Unported license
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Minimizing Distance-to-Sight in Polygonal Domains
In this paper, we consider the quickest pair-visibility problem in polygonal domains. Given two points in a polygonal domain with h holes of total complexity n, we want to minimize the maximum distance that the two points travel in order to see each other in the polygonal domain. We present an O(n log^2 n+h^2 log^4 h)-time algorithm for this problem. We show that this running time is almost optimal unless the 3sum problem can be solved in O(n^{2-epsilon}) time for some epsilon>0.
Visibility in polygonal domains
shortest path in polygonal domains
Theory of computation~Computational geometry
59:1-59:12
Regular Paper
Eunjin
Oh
Eunjin Oh
Max Planck Institute for Informatics Saarbrücken, Germany
10.4230/LIPIcs.ISAAC.2018.59
Hee-Kap Ahn, Eunjin Oh, Lena Schlipf, Fabian Stehn, and Darren Strash. On Romeo and Juliet Problems: Minimizing Distance-to-Sight. In Proceedings of the 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018), 2018.
Esther M. Arkin, Alon Efrat, Christian Knauer, Joseph S. B. Mitchell, Valentin Polishchuk, Günter Rote, Lena Schlipf, and Topi Talvitie. Shortest Path to a Segment and Quickest Visibility Queries. Journal of Computational Geometry, 7(2):77-100, 2016.
Danny Z. Chen and Haitao Wang. Computing Shortest Paths Among Curved Obstacles in the Plane. ACM Transactions on Algorithms, 11(4):26:1-26:46, 2015.
Danny Z. Chen and Haitao Wang. Computing the Visibility Polygon of an Island in a Polygonal Domain. Algorithmica, 77(1):40-64, 2017.
Richard Cole. Parallel Merge Sort. SIAM Journal on Computing, 17(4):770-785, 1988.
David P. Dobkin and Diane L. Souvaine. Computational geometry in a curved world. Algorithmica, 5(1):421-457, 1990.
Anka Gajentaan and Mark H. Overmars. On a class of O(n²) problems in computational geometry. Compututational Geometry: Theory and Applications, 45(4):140-152, 2012.
Anurag Ganguli, Jorge Cortes, and Francesco Bullo. Visibility-based multi-agent deployment in orthogonal environments. In Proceedings of American Control Conference, pages 3426-3431, 2007.
Michael T. Goodrich, Steven B. Shauck, and Sumanta Guha. Parallel methods for visibility and shortest-path problems in simple polygons. Algorithmica, 8(1):461-486, 1992.
John Hershberger and Subhash Suri. An optimal algorithm for Euclidean shortest paths in the plane. SIAM Journal on Computing, 28(6):2215-2256, 1999.
S. Kapoor, S. N. Maheshwari, and J. S. B. Mitchell. An Efficient Algorithm for Euclidean Shortest Paths Among Polygonal Obstacles in the Plane. Discrete & Computational Geometry, 18(4):377-383, 1997.
Nimrod Megiddo. Applying parallel computation algorithms in the design of serial algorithms. Journal of the ACM, 30(4):852-865, 1983.
Mark H. Overmars and Jan van Leeuwen. Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23(2):166-204, 1981.
Haitao Wang. Quickest Visibility Queries in Polygonal Domains. In Proceedings of the 33rd International Symposium on Computational Geometry (SoCG 2017), volume 77, pages 61:1-61:16, 2017.
Erik L. Wynters and Joseph S. B. Mitchell. Shortest Paths for a Two-Robot Rendez-Vous. In Proceedings of the 5th Canadian Conference on Computational Geometry (CCCG 1993), pages 216-221, 1993.
Eunjin Oh
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Partially Walking a Polygon
Deciding two-guard walkability of an n-sided polygon is a well-understood problem. We study the following more general question: How far can two guards reach from a given source vertex while staying mutually visible, in the (more realistic) case that the polygon is not entirely walkable? There can be Theta(n) such maximal walks, and we show how to find all of them in O(n log n) time.
Polygon
guard walk
visibility
Theory of computation~Design and analysis of algorithms
60:1-60:9
Regular Paper
This work was supported by Project I 1836-N15, Austria Science Fund (FWF).
Franz
Aurenhammer
Franz Aurenhammer
Institute for Theoretical Computer Science, University of Technology, Graz, Austria
Michael
Steinkogler
Michael Steinkogler
Institute for Theoretical Computer Science, University of Technology, Graz, Austria
Rolf
Klein
Rolf Klein
Universität Bonn, Institut für Informatik, Bonn, Germany
10.4230/LIPIcs.ISAAC.2018.60
Wolfgang Aigner, Franz Aurenhammer, and Bert Jüttler. On triangulation axes of polygons. Information Processing Letters, 115(1):45-51, 2015.
Franz Aurenhammer, Michael Steinkogler, and Rolf Klein. Maximal two-guard walks in a polygon. In 34nd European Workshop on Computational Geometry, EuroCG 2018, pages 17-22, 2018.
Binay Bhattacharya, Asish Mukhopadhyay, and Giri Narasimhan. Optimal algorithms for two-guard walkability of simple polygons. In Workshop on Algorithms and Data Structures, WADS 2001. Lecture Notes in Computer Science, volume 2125, pages 438-449. Springer, 2001.
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Jorge Urrutia. Art gallery and illumination problems. In Handbook of Computational Geometry, pages 973-1027. Elsevier, 2000.
Franz Aurenhammer, Michael Steinkogler, and Rolf Klein
Creative Commons Attribution 3.0 Unported license
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Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity
We initiate the study of the following natural geometric optimization problem. The input is a set of axis-aligned rectangles in the plane. The objective is to find a set of horizontal line segments of minimum total length so that every rectangle is stabbed by some line segment. A line segment stabs a rectangle if it intersects its left and its right boundary. The problem, which we call Stabbing, can be motivated by a resource allocation problem and has applications in geometric network design. To the best of our knowledge, only special cases of this problem have been considered so far.
Stabbing is a weighted geometric set cover problem, which we show to be NP-hard. While for general set cover the best possible approximation ratio is Theta(log n), it is an important field in geometric approximation algorithms to obtain better ratios for geometric set cover problems. Chan et al. [SODA'12] generalize earlier results by Varadarajan [STOC'10] to obtain sub-logarithmic performances for a broad class of weighted geometric set cover instances that are characterized by having low shallow-cell complexity. The shallow-cell complexity of Stabbing instances, however, can be high so that a direct application of the framework of Chan et al. gives only logarithmic bounds. We still achieve a constant-factor approximation by decomposing general instances into what we call laminar instances that have low enough complexity.
Our decomposition technique yields constant-factor approximations also for the variant where rectangles can be stabbed by horizontal and vertical segments and for two further geometric set cover problems.
Geometric optimization
NP-hard
approximation
shallow-cell complexity
line stabbing
Theory of computation~Packing and covering problems
Theory of computation~Computational geometry
61:1-61:13
Regular Paper
https://arxiv.org/abs/1806.02851
Timothy M.
Chan
Timothy M. Chan
University of Illinois at Urbana-Champaign, U.S.A.
Thomas C.
van Dijk
Thomas C. van Dijk
Universität Würzburg, Germany
https://orcid.org/0000-0001-6553-7317
Supported by DFG grant DI 2161/2-1.
Krzysztof
Fleszar
Krzysztof Fleszar
Max-Planck-Institut für Informatik, Saarbrücken, Germany
https://orcid.org/0000-0002-1129-3289
This research was partially supported by Conicyt Grant PII 20150140 and by Millennium Nucleus Information and Coordination in Networks RC130003.
Joachim
Spoerhase
Joachim Spoerhase
Aalto University, Espoo, Finland, Universität Würzburg, Germany
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).
Alexander
Wolff
Alexander Wolff
Universität Würzburg, Germany
https://orcid.org/0000-0001-5872-718X
10.4230/LIPIcs.ISAAC.2018.61
Boris Aronov, Esther Ezra, and Micha Sharir. Small-Size ε-Nets for Axis-Parallel Rectangles and Boxes. SIAM J. Comput., 39(7):3248-3282, 2010. URL: http://dx.doi.org/10.1137/090762968.
http://dx.doi.org/10.1137/090762968
Nikhil Bansal and Kirk Pruhs. The Geometry of Scheduling. SIAM J. Computing, 43(5):1684-1698, 2014. URL: http://dx.doi.org/10.1137/130911317.
http://dx.doi.org/10.1137/130911317
Hervé Brönnimann and Michael T. Goodrich. Almost Optimal Set Covers in Finite VC-Dimension. Discrete Comput. Geom., 14(4):463-479, 1995. URL: http://dx.doi.org/10.1007/BF02570718.
http://dx.doi.org/10.1007/BF02570718
Timothy M. Chan and Elyot Grant. Exact Algorithms and APX-hardness Results for Geometric Packing and Covering Problems. Comput. Geom. Theory Appl., 47(2):112-124, 2014. URL: http://dx.doi.org/10.1016/j.comgeo.2012.04.001.
http://dx.doi.org/10.1016/j.comgeo.2012.04.001
Timothy M. Chan, Elyot Grant, Jochen Könemann, and Malcolm Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Proc. 23th Annu. ACM-SIAM Symp. Discrete Algorithms (SODA'12), pages 1576-1585, 2012. URL: http://portal.acm.org/citation.cfm?id=2095241&CFID=63838676&CFTOKEN=79617016.
http://portal.acm.org/citation.cfm?id=2095241&CFID=63838676&CFTOKEN=79617016
Aparna Das, Krzysztof Fleszar, Stephen G. Kobourov, Joachim Spoerhase, Sankar Veeramoni, and Alexander Wolff. Approximating the Generalized Minimum Manhattan Network Problem. Algorithmica, 80(4):1170-1190, 2018. URL: http://dx.doi.org/10.1007/s00453-017-0298-0.
http://dx.doi.org/10.1007/s00453-017-0298-0
Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Proc. Symp. Theory Comput. (STOC'14), pages 624-633, 2014. URL: http://dx.doi.org/10.1145/2591796.2591884.
http://dx.doi.org/10.1145/2591796.2591884
Guy Even, Retsef Levi, Dror Rawitz, Baruch Schieber, Shimon Shahar, and Maxim Sviridenko. Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs. ACM Trans. Algorithms, 4(3), 2008. URL: http://dx.doi.org/10.1145/1367064.1367074.
http://dx.doi.org/10.1145/1367064.1367074
Uriel Feige. A Threshold of ln n for Approximating Set Cover. J. ACM, 45(4):634-652, 1998. URL: http://dx.doi.org/10.1145/285055.285059.
http://dx.doi.org/10.1145/285055.285059
Gerd Finke, Vincent Jost, Maurice Queyranne, and András Sebö. Batch processing with interval graph compatibilities between tasks. Discrete Appl. Math., 156(5):556-568, 2008. URL: http://dx.doi.org/10.1016/j.dam.2006.03.039.
http://dx.doi.org/10.1016/j.dam.2006.03.039
Michael R. Garey, David S. Johnson, and Larry Stockmeyer. Some simplified NP-complete problems. In Proc. 6th Annu. ACM Symp. Theory Comput. (STOC'74), pages 47-63, 1974. URL: http://dx.doi.org/10.1145/800119.803884.
http://dx.doi.org/10.1145/800119.803884
Daya Ram Gaur, Toshihide Ibaraki, and Ramesh Krishnamurti. Constant Ratio Approximation Algorithms for the Rectangle Stabbing Problem and the Rectilinear Partitioning Problem. J. Algorithms, 43(1):138-152, 2002. URL: http://dx.doi.org/10.1006/jagm.2002.1221.
http://dx.doi.org/10.1006/jagm.2002.1221
Panos Giannopoulos, Christian Knauer, Günter Rote, and Daniel Werner. Fixed-parameter tractability and lower bounds for stabbing problems. Comput. Geom. Theory Appl., 46(7):839-860, 2013. URL: http://dx.doi.org/10.1016/j.comgeo.2011.06.005.
http://dx.doi.org/10.1016/j.comgeo.2011.06.005
Sofia Kovaleva and Frits C. R. Spieksma. Approximation Algorithms for Rectangle Stabbing and Interval Stabbing Problems. SIAM J. Discrete Math., 20(3):748-768, 2006. URL: http://dx.doi.org/10.1137/S089548010444273X.
http://dx.doi.org/10.1137/S089548010444273X
Ching-Chi Lin, Hsueh-I Lu, and I-Fan Sun. Improved Compact Visibility Representation of Planar Graph via Schnyder’s Realizer. SIAM J. Discrete Math., 18(1):19-29, 2004. URL: http://dx.doi.org/10.1137/S0895480103420744.
http://dx.doi.org/10.1137/S0895480103420744
Nabil H. Mustafa, Rajiv Raman, and Saurabh Ray. Quasi-Polynomial Time Approximation Scheme for Weighted Geometric Set Cover on Pseudodisks and Halfspaces. SIAM J. Computing, 44(6):1650-1669, 2015. URL: http://dx.doi.org/10.1137/14099317X.
http://dx.doi.org/10.1137/14099317X
Nabil H. Mustafa and Saurabh Ray. Improved Results on Geometric Hitting Set Problems. Discrete Comput. Geom., 44(4):883-895, 2010. URL: http://dx.doi.org/10.1007/s00454-010-9285-9.
http://dx.doi.org/10.1007/s00454-010-9285-9
Kasturi Varadarajan. Weighted geometric set cover via quasi-uniform sampling. In Proc. 42nd ACM Symp. Theory Comput. (STOC'10), pages 641-648, 2010. URL: http://dx.doi.org/10.1145/1806689.1806777.
http://dx.doi.org/10.1145/1806689.1806777
Timothy M. Chan, Thomas C. van Dijk, Krzysztof Fleszar, Joachim Spoerhase, and Alexander Wolff
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Impatient Online Matching
We investigate the problem of Min-cost Perfect Matching with Delays (MPMD) in which requests are pairwise matched in an online fashion with the objective to minimize the sum of space cost and time cost. Though linear-MPMD (i.e., time cost is linear in delay) has been thoroughly studied in the literature, it does not well model impatient requests that are common in practice. Thus, we propose convex-MPMD where time cost functions are convex, capturing the situation where time cost increases faster and faster. Since the existing algorithms for linear-MPMD are not competitive any more, we devise a new deterministic algorithm for convex-MPMD problems. For a large class of convex time cost functions, our algorithm achieves a competitive ratio of O(k) on any k-point uniform metric space. Moreover, our deterministic algorithm is asymptotically optimal, which uncover a substantial difference between convex-MPMD and linear-MPMD which allows a deterministic algorithm with constant competitive ratio on any uniform metric space.
online algorithm
online matching
convex function
competitive analysis
lower bound
Theory of computation~Online algorithms
62:1-62:12
Regular Paper
Xingwu
Liu
Xingwu Liu
SKL Computer Architecture, ICT, CAS , University of Chinese Academy of Sciences, Beijing, China
Zhida
Pan
Zhida Pan
Institute of Computing Technology, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing, China
Yuyi
Wang
Yuyi Wang
ETH Zurich, Switzerland
Roger
Wattenhofer
Roger Wattenhofer
ETH Zurich, Switzerland
10.4230/LIPIcs.ISAAC.2018.62
Gagan Aggarwal, Gagan Goel, Chinmay Karande, and Aranyak Mehta. Online Vertex-Weighted Bipartite Matching and Single-bid Budgeted Allocations. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, page 1253–1264, 2011.
Itai Ashlagi, Yossi Azar, Moses Charikar, Ashish Chiplunkar, Ofir Geri, Haim Kaplan, Rahul Makhijani, Yuyi Wang, and Roger Wattenhofer. Min-cost Bipartite Perfect Matching with Delays. In 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), Berkeley, California, USA, August 2017.
Yossi Azar, Ashish Chiplunkar, and Haim Kaplan. Polylogarithmic Bounds on the Competitiveness of Min-cost Perfect Matching with Delays. arXiv, 2016. URL: http://arxiv.org/abs/1610.05155.
http://arxiv.org/abs/1610.05155
Yossi Azar, Ashish Chiplunkar, and Haim Kaplan. Polylogarithmic Bounds on the Competitiveness of Min-cost Perfect Matching with Delays. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, page 1051–1061, 2017.
Yossi Azar, Arun Ganesh, Rong Ge, and Debmalya Panigrahi. Online service with delay. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 551-563. ACM, 2017.
Benjamin E. Birnbaum and Claire Mathieu. On-line bipartite matching made simple. SIGACT News, 39(1):80–87, 2008.
Nikhil R. Devanur, Kamal Jain, and Robert D. Kleinberg. Randomized Primal-Dual analysis of RANKING for Online BiPartite Matching. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, page 101–107, 2013.
Jack Edmonds. Maximum matching and a polyhedron with 0, 1-vertices. Journal of Research of the National Bureau of Standards B, 69:125–130, 1965.
Jack Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17:449–467, 1965.
Yuval Emek, Shay Kutten, and Roger Wattenhofer. Online Matching: Haste makes Waste! In 48th Annual Symposium on Theory of Computing (STOC), June 2016.
Yuval Emek, Yaacov Shapiro, and Yuyi Wang. Minimum Cost Perfect Matching with Delays for Two Sources. In 10th International Conference on Algorithms and Complexity (CIAC), Athens, Greece, May 2017.
Gagan Goel and Aranyak Mehta. Online budgeted matching in random input models with applications to Adwords. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, page 982–991, 2008.
Bala Kalyanasundaram and Kirk Pruhs. Online Weighted Matching. J. Algorithms, 14(3):478–488, 1993.
Anna R. Karlin, Mark S. Manasse, Lyle A. McGeoch, and Susan Owicki. Competitive randomized algorithms for nonuniform problems. Algorithmica, 11(6):542-571, 1994.
Richard M. Karp, Umesh V. Vazirani, and Vijay V. Vazirani. An Optimal Algorithm for On-line Bipartite Matching. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, page 352–358, 1990.
Samir Khuller, Stephen G. Mitchell, and Vijay V. Vazirani. On-Line Algorithms for Weighted Bipartite Matching and Stable Marriages. Theor. Comput. Sci., 127(2):255–267, 1994.
Aranyak Mehta. Online Matching and Ad Allocation. Foundations and Trends in Theoretical Computer Science, 8(4):265–368, 2013.
Aranyak Mehta, Amin Saberi, Umesh V. Vazirani, and Vijay V. Vazirani. AdWords and Generalized On-line Matching. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), page 264–273, 2005.
Adam Meyerson, Akash Nanavati, and Laura J. Poplawski. Randomized online algorithms for minimum metric bipartite matching. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, 2006.
Shuichi Miyazaki. On the advice complexity of online bipartite matching and online stable marriage. Inf. Process. Lett., 114(12):714–717, 2014.
Joseph Naor and David Wajc. Near-Optimum Online Ad Allocation for Targeted Advertising. In Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC, page 131–148, 2015.
Xingwu Liu, Zhida Pan, Yuyi Wang, and Roger Wattenhofer
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Extensions of Self-Improving Sorters
Ailon et al. (SICOMP 2011) proposed a self-improving sorter that tunes its performance to the unknown input distribution in a training phase. The distribution of the input numbers x_1,x_2,...,x_n must be of the product type, that is, each x_i is drawn independently from an arbitrary distribution D_i, and the D_i's are independent of each other. We study two extensions that relax this requirement. The first extension models hidden classes in the input. We consider the case that numbers in the same class are governed by linear functions of the same hidden random parameter. The second extension considers a hidden mixture of product distributions.
sorting
self-improving algorithms
entropy
Theory of computation~Design and analysis of algorithms
63:1-63:12
Regular Paper
Siu-Wing
Cheng
Siu-Wing Cheng
HKUST, Hong Kong, China
https://orcid.org/0000-0002-3557-9935
Supported by Research Grants Council, Hong Kong, China (project no. 16200317).
Lie
Yan
Lie Yan
Hangzhou, China
Part of the work was conducted while the author was at HKUST and supported by the Hong Kong PhD Fellowship.
10.4230/LIPIcs.ISAAC.2018.63
N. Ailon, B. Chazelle, K.L. Clarkson, D. Liu, W. Mulzer, and C. Seshadhir. Self-improving algorithms. SIAM Journal on Computing, 40(2):350-375, 2011.
K.L. Clarkson, W. Mulzer, and C. Seshadhri. Self-improving algorithms for coordinatewise maxima and convex hulls. SIAM Journal on Computing, 43(2):617-653, 2014.
T.M. Cover and J.A. Thomas. Elements of Information Theory. Wiley-Interscience, New York, 2nd edition, 2006.
M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag Berlin Heidelberg, 3rd edition, 2008.
J.R. Driscoll, N. Sarnak, D.D. Sleator, and R.E. Tarjan. Making data structures persistent. Journal of Computer and System Sciences, 38:86-124, 1989.
M.L. Fredman. Two applications of a probabilistic search technique: sorting X+Y and building balanced search trees. In Proceedings of the 7th Symposium on Theory of Computing, pages 240-244, 1975.
G.F. Italiano and R. Raman. Topics in Data Structures. In M.J. Atallah and M. Blanton, editors, Algorithms and Theory of Computation Handbook, pages 5:1-29. Chapman &Hall/CRC, 2nd edition, 2009.
K. Mehlhorn. Nearly optimal binary search trees. Acta Informatica, 5:287-295, 1975.
P. van Emde Boas and R. Kaas and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10:99-127, 1977.
Siu-Wing Cheng and Lie Yan
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Online Scheduling of Car-Sharing Requests Between Two Locations with Many Cars and Flexible Advance Bookings
We study an on-line scheduling problem that is motivated by applications such as car-sharing, in which users submit ride requests, and the scheduler aims to accept requests of maximum total profit using k servers (cars). Each ride request specifies the pick-up time and the pick-up location (among two locations, with the other location being the destination). The scheduler has to decide whether or not to accept a request immediately at the time when the request is submitted (booking time). We consider two variants of the problem with respect to constraints on the booking time: In the fixed booking time variant, a request must be submitted a fixed amount of time before the pick-up time. In the variable booking time variant, a request can be submitted at any time during a certain time interval (called the booking horizon) that precedes the pick-up time. We present lower bounds on the competitive ratio for both variants and propose a balanced greedy algorithm (BGA) that achieves the best possible competitive ratio. We prove that, for the fixed booking time variant, BGA is 1.5-competitive if k=3i ( i in N) and the fixed booking length is not less than the travel time between the two locations; for the variable booking time variant, BGA is 1.5-competitive if k=3i ( i in N) and the length of the booking horizon is less than the travel time between the two locations, and BGA is 5/3-competitive if k=5i ( i in N) and the length of the booking horizon is not less than the travel time between the two locations.
Car-sharing system
Competitive analysis
On-line scheduling
Theory of computation~Online algorithms
64:1-64:13
Regular Paper
Kelin
Luo
Kelin Luo
School of Management, Xi'an Jiaotong University, Xi'an, China
https://orcid.org/0000-0003-2006-0601
This work was partially supported by the China Postdoctoral Science Foundation (Grant No. 2016M592811), and the China Scholarship Council (Grant No. 201706280058).
Thomas
Erlebach
Thomas Erlebach
Department of Informatics, University of Leicester, Leicester, United Kingdom
https://orcid.org/0000-0002-4470-5868
Yinfeng
Xu
Yinfeng Xu
School of Management, Xi'an Jiaotong University, Xi'an, China
10.4230/LIPIcs.ISAAC.2018.64
Norbert Ascheuer, Sven Oliver Krumke, and Jörg Rambau. Online Dial-a-Ride Problems: Minimizing the Completion Time. In Proc. of the 17th Annual Symposium on Theoretical Aspects of Computer Science (STACS '00), volume 1770 of LNCS, pages 639-650. Springer, 2000. URL: http://dx.doi.org/10.1007/3-540-46541-3_53.
http://dx.doi.org/10.1007/3-540-46541-3_53
Antje Bjelde, Yann Disser, Jan Hackfeld, Christoph Hansknecht, Maarten Lipmann, Julie Meißner, Kevin Schewior, Miriam Schlöter, and Leen Stougie. Tight Bounds for Online TSP on the Line. In Proc. of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '17), pages 994-1005. SIAM, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.63.
http://dx.doi.org/10.1137/1.9781611974782.63
Katerina Böhmová, Yann Disser, Matús Mihalák, and Rastislav Srámek. Scheduling Transfers of Resources over Time: Towards Car-Sharing with Flexible Drop-Offs. In Proc. of the 12th Latin American Symposium on Theoretical Informatics (LATIN'16), volume 9644 of LNCS, pages 220-234. Springer, 2016. URL: http://dx.doi.org/10.1007/978-3-662-49529-2_17.
http://dx.doi.org/10.1007/978-3-662-49529-2_17
Allan Borodin and Ran El-Yaniv. Online computation and competitive analysis. Cambridge University Press, 1998.
Ananya Christman, William Forcier, and Aayam Poudel. From theory to practice: maximizing revenues for on-line dial-a-ride. J. Comb. Optim., 35(2):512-529, 2018. URL: http://dx.doi.org/10.1007/s10878-017-0188-z.
http://dx.doi.org/10.1007/s10878-017-0188-z
Sven Oliver Krumke, Willem de Paepe, Diana Poensgen, Maarten Lipmann, Alberto Marchetti-Spaccamela, and Leen Stougie. On Minimizing the Maximum Flow Time in the Online Dial-a-Ride Problem. In Proc. of the 3rd International Workshop on Approximation and Online Algorithms (WAOA 2005), Revised Papers, volume 3879 of LNCS, pages 258-269. Springer, 2006. URL: http://dx.doi.org/10.1007/11671411_20.
http://dx.doi.org/10.1007/11671411_20
Kelin Luo, Thomas Erlebach, and Yinfeng Xu. Car-Sharing between Two Locations: Online Scheduling with Flexible Advance Bookings. In Proc. of the 24th International Computing and Combinatorics Conference (COCOON '18), 2018. To appear.
Kelin Luo, Thomas Erlebach, and Yinfeng Xu. Car-Sharing between Two Locations: Online Scheduling with two Servers. In Proc. of 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS '18), 117. LIPIcs, 2018. To appear. URL: http://www.cs.le.ac.uk/~te17/papers/mfcs2018.pdf.
http://www.cs.le.ac.uk/~te17/papers/mfcs2018.pdf
Fanglei Yi and Lei Tian. On the Online Dial-A-Ride Problem with Time-Windows. In Proc. of the 1st International Conference on Algorithmic Applications in Management (AAIM '05), volume 3521 of LNCS, pages 85-94. Springer, 2005. URL: http://dx.doi.org/10.1007/11496199_11.
http://dx.doi.org/10.1007/11496199_11
Kelin Luo, Thomas Erlebach, and Yinfeng Xu
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Packing Returning Secretaries
We study online secretary problems with returns in combinatorial packing domains with n candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n arrivals occur in random order. We propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio at least 0.5721 - o(1) for growing n, and an algorithm with ratio at least 0.5459 for all n >= 1. We extend all algorithms and ratios to k >= 2 arrivals per candidate.
In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed (returned into the pool). We mainly focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Theta(n log n) is always sufficient. For matroids, we show that the expected number can be reduced to O(r log (n/r)), where r <=n/2 is the minimum of the ranks of matroid and dual matroid. For bipartite matching, we show a bound of O(r log n), where r is the size of the optimum matching. For general packing, we show a lower bound of Omega(n log log n), even when the size of the optimum is r = Theta(log n).
Secretary Problem
Coupon Collector Problem
Matroids
Theory of computation~Online algorithms
65:1-65:12
Regular Paper
Supported by a grant from the German-Israeli Foundation for Scientific Research and Development (GIF).
https://arxiv.org/abs/1810.11216
Martin
Hoefer
Martin Hoefer
Goethe University Frankfurt/Main, Germany
Lisa
Wilhelmi
Lisa Wilhelmi
Goethe University Frankfurt/Main, Germany
10.4230/LIPIcs.ISAAC.2018.65
Moshe Babaioff, Nicole Immorlica, David Kempe, and Robert Kleinberg. A Knapsack Secretary Problem with Applications. In Proc. 10th Workshop Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 16-28, 2007.
Moshe Babaioff, Nicole Immorlica, David Kempe, and Robert Kleinberg. Online Auctions and Generalized Secretary Problems. SIGecom Exchanges, 7(2), 2008.
Moshe Babaioff, Nicole Immorlica, and Robert Kleinberg. Matroids, secretary problems, and online mechanisms. In Proc. 18th Symp. Discrete Algorithms (SODA), pages 434-443, 2007.
Arnon Boneh and Micha Hofri. The coupon-collector problem revisited - a survey of engineering problems and computational methods. Comm. Stat. Stoch. Models, 13(1):39-66, 1997.
Ning Chen, Martin Hoefer, Marvin Künnemann, Chengyu Lin, and Peihan Miao. Secretary Markets with Local Information. In Proc. 42nd Intl. Coll. Automata, Languages &Programming (ICALP), volume 2, pages 552-563, 2015.
Michael Dinitz. Recent advances on the matroid secretary problem. SIGACT News, 44(2):126-142, 2013.
Uriel Feige. On Maximizing Welfare When Utility Functions Are Subadditive. SIAM J. Comput., 39(1):122-142, 2009.
Moran Feldman, Ola Svensson, and Rico Zenklusen. A Simple O(log log(rank))-Competitive Algorithm for the Matroid Secretary Problem. Math. Oper. Res., 43(2):638-650, 2018.
Moran Feldman and Rico Zenklusen. The Submodular Secretary Problem Goes Linear. SIAM J. Comput., 47(2):330-366, 2018.
Thomas Ferguson. Who Solved the Secretary Problem? Statistical Science, 4(3):282-289, 1989.
Amos Fiat, Ilia Gorelik, Haim Kaplan, and Slava Novgorodov. The Temp Secretary Problem. In Proc. 23rd European Symp. Algorithms (ESA), pages 631-642, 2015.
Philippe Flajolet, Danièle Gardy, and Loÿs Thimonier. Birthday Paradox, Coupon Collectors, Caching Algorithms and Self-Organizing Search. Disc. Appl. Math., 39(3):207-229, 1992.
Oliver Göbel, Martin Hoefer, Thomas Kesselheim, Thomas Schleiden, and Berthold Vöcking. Online Independent Set Beyond the Worst-Case: Secretaries, Prophets and Periods. In Proc. 41st Intl. Coll. Automata, Languages &Programming (ICALP), volume 2, pages 508-519, 2014.
MohammadTaghi Hajiaghayi, Robert Kleinberg, and David Parkes. Adaptive limited-supply online auctions. In Proc. 5th Conf. Electronic Commerce (EC), pages 71-80, 2004.
Martin Hoefer and Bojana Kodric. Combinatorial Secretary Problems with Ordinal Information. In Proc. 44th Intl. Coll. Automata, Languages &Programming (ICALP), pages 133:1-133:14, 2017.
Thomas Kesselheim, Klaus Radke, Andreas Tönnis, and Berthold Vöcking. An Optimal Online Algorithm for Weighted Bipartite Matching and Extensions to Combinatorial Auctions. In Proc. 21st European Symp. Algorithms (ESA), pages 589-600, 2013.
Thomas Kesselheim, Klaus Radke, Andreas Tönnis, and Berthold Vöcking. Primal Beats Dual on Online Packing LPs in the Random-Order Model. In Proc. 46th Symp. Theory of Comput. (STOC), pages 303-312, 2014.
Thomas Kesselheim and Andreas Tönnis. Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints. In Proc. 20th Workshop Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 16:1-16:22, 2017.
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Shai Vardi. The Returning Secretary. In Proc. 32nd Symp. Theoret. Aspects of Comput. Sci. (STACS), pages 716-729, 2015. Full version CoRR abs/1404.0614. URL: http://arxiv.org/abs/1404.0614.
http://arxiv.org/abs/1404.0614
Martin Hoefer and Lisa Wilhelmi
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Simple 2^f-Color Choice Dictionaries
A c-color choice dictionary of size n in N is a fundamental data structure in the development of space-efficient algorithms that stores the colors of n elements and that supports operations to get and change the color of an element as well as an operation choice that returns an arbitrary element of that color. For an integer f>0 and a constant c=2^f, we present a word-RAM algorithm for a c-color choice dictionary of size n that supports all operations above in constant time and uses only nf+1 bits, which is optimal if all operations have to run in o(n/w) time where w is the word size.
In addition, we extend our choice dictionary by an operation union without using more space.
space efficient
succinct
word RAM
Theory of computation~Data structures design and analysis
66:1-66:12
Regular Paper
Frank
Kammer
Frank Kammer
THM, University of Applied Sciences Mittelhessen, Germany
Andrej
Sajenko
Andrej Sajenko
THM, University of Applied Sciences Mittelhessen, Germany
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 379157101.
10.4230/LIPIcs.ISAAC.2018.66
Tetsuo Asano, Amr Elmasry, and Jyrki Katajainen. Priority Queues and Sorting for Read-Only Data. In Proc. 10th International Conference on Theory and Applications of Models of Computation (TAMC 2013), volume 7876 of LNCS, pages 32-41. Springer, 2013. URL: http://dx.doi.org/10.1007/978-3-642-38236-9_4.
http://dx.doi.org/10.1007/978-3-642-38236-9_4
Tetsuo Asano, Taisuke Izumi, Masashi Kiyomi, Matsuo Konagaya, Hirotaka Ono, Yota Otachi, Pascal Schweitzer, Jun Tarui, and Ryuhei Uehara. Depth-First Search Using O(n) Bits. In Proc. 25th International Symposium on Algorithms and Computation (ISAAC 2014), volume 8889 of LNCS, pages 553-564. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-13075-0_44.
http://dx.doi.org/10.1007/978-3-319-13075-0_44
Niranka Banerjee, Sankardeep Chakraborty, and Venkatesh Raman. Improved Space Efficient Algorithms for BFS, DFS and Applications. In Proc. 22nd International Conference on Computing and Combinatorics (COCOON), volume 9797 of LNCS, pages 119-130. Springer, 2016. URL: http://dx.doi.org/10.1007/978-3-319-42634-1_10.
http://dx.doi.org/10.1007/978-3-319-42634-1_10
Sankardeep Chakraborty, Anish Mukherjee, Venkatesh Raman, and Srinivasa Rao Satti. A Framework for In-place Graph Algorithms. In Proc. 26th Annual European Symposium on Algorithms (ESA 2018), volume 112 of LIPIcs, pages 13:1-13:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.
Omar Darwish and Amr Elmasry. Optimal Time-Space Tradeoff for the 2D Convex-Hull Problem. In Proc. 22nd Annual European Symposium on Algorithms (ESA 2014), volume 8737 of LNCS, pages 284-295. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44777-2_24.
http://dx.doi.org/10.1007/978-3-662-44777-2_24
Samir Datta, Raghav Kulkarni, and Anish Mukherjee. Space-Efficient Approximation Scheme for Maximum Matching in Sparse Graphs. In Proc. 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016), volume 58 of LIPIcs, pages 28:1-28:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.28.
http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.28
Amr Elmasry, Torben Hagerup, and Frank Kammer. Space-efficient Basic Graph Algorithms. In Proc. 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), volume 30 of LIPIcs, pages 288-301. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.288.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.288
Amr Elmasry and Frank Kammer. Space-Efficient Plane-Sweep Algorithms. In Proc. 27th International Symposium on Algorithms and Computation (ISAAC 2016), volume 64 of LIPIcs, pages 30:1-30:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ISAAC.2016.30.
http://dx.doi.org/10.4230/LIPIcs.ISAAC.2016.30
Torben Hagerup. An Optimal Choice Dictionary. CoRR, abs/1711.00808, 2017. URL: http://arxiv.org/abs/1711.00808.
http://arxiv.org/abs/1711.00808
Torben Hagerup. Small Uncolored and Colored Choice Dictionaries. CoRR, abs/1809.07661, 2018. URL: http://arxiv.org/abs/1809.07661.
http://arxiv.org/abs/1809.07661
Torben Hagerup and Frank Kammer. Succinct Choice Dictionaries. CoRR, abs/1604.06058, 2016. URL: http://arxiv.org/abs/1604.06058.
http://arxiv.org/abs/1604.06058
Torben Hagerup, Frank Kammer, and Moritz Laudahn. Space-efficient Euler partition and bipartite edge coloring. Theor. Comput. Sci., 2018. URL: http://dx.doi.org/10.1016/j.tcs.2018.01.008.
http://dx.doi.org/10.1016/j.tcs.2018.01.008
Frank Kammer, Dieter Kratsch, and Moritz Laudahn. Space-Efficient Biconnected Components and Recognition of Outerplanar Graphs. Algorithmica, 2018. URL: http://dx.doi.org/10.1007/s00453-018-0464-z.
http://dx.doi.org/10.1007/s00453-018-0464-z
Frank Kammer and Andrej Sajenko. Space efficient (graph) algorithms. https://github.com/thm-mni-ii/sea, 2018.
https://github.com/thm-mni-ii/sea
Takashi Katoh and Keisuke Goto. In-Place Initializable Arrays. CoRR, abs/1709.08900, 2017. URL: http://arxiv.org/abs/1709.08900.
http://arxiv.org/abs/1709.08900
Frank Kammer and Andrej Sajenko
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Succinct Data Structures for Chordal Graphs
We study the problem of approximate shortest path queries in chordal graphs and give a n log n + o(n log n) bit data structure to answer the approximate distance query to within an additive constant of 1 in O(1) time.
We study the problem of succinctly storing a static chordal graph to answer adjacency, degree, neighbourhood and shortest path queries. Let G be a chordal graph with n vertices. We design a data structure using the information theoretic minimal n^2/4 + o(n^2) bits of space to support the queries:
- whether two vertices u,v are adjacent in time f(n) for any f(n) in omega(1).
- the degree of a vertex in O(1) time.
- the vertices adjacent to u in (f(n))^2 time per neighbour
- the length of the shortest path from u to v in O(nf(n)) time
Succinct Data Structure
Chordal Graph
Theory of computation~Shortest paths
Theory of computation~Data compression
67:1-67:12
Regular Paper
This work was supported by NSERC of Canada and the Canada Research Chairs Programme.
J. Ian
Munro
J. Ian Munro
Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada
https://orcid.org/0000-0002-7165-7988
Kaiyu
Wu
Kaiyu Wu
Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada
https://orcid.org/0000-0001-7562-1336
10.4230/LIPIcs.ISAAC.2018.67
Niranka Banerjee, Venkatesh Raman, and Srinivasa Rao Satti. Maintaining Chordal Graphs Dynamically: Improved Upper and Lower Bounds. In Fedor V. Fomin and Vladimir V. Podolskii, editors, Computer Science - Theory and Applications - 13th International Computer Science Symposium in Russia, CSR 2018, Moscow, Russia, June 6-10, 2018, Proceedings, volume 10846 of Lecture Notes in Computer Science, pages 29-40. Springer, 2018. URL: http://dx.doi.org/10.1007/978-3-319-90530-3_4.
http://dx.doi.org/10.1007/978-3-319-90530-3_4
Daniel K. Blandford, Guy E. Blelloch, and Ian A. Kash. Compact representations of separable graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA., pages 679-688. ACM/SIAM, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644219.
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http://dx.doi.org/10.1007/s00453-012-9712-9
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http://dx.doi.org/10.1016/j.tcs.2013.09.031
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http://dx.doi.org/10.1109/FOCS.2008.83
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http://dx.doi.org/10.1007/BF02582944
J. Ian Munro and Kaiyu Wu
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Tree Path Majority Data Structures
We present the first solution to tau-majorities on tree paths. Given a tree of n nodes, each with a label from [1..sigma], and a fixed threshold 0<tau<1, such a query gives two nodes u and v and asks for all the labels that appear more than tau * |P_{uv}| times in the path P_{uv} from u to v, where |P_{uv}| denotes the number of nodes in P_{uv}. Note that the answer to any query is of size up to 1/tau. On a w-bit RAM, we obtain a linear-space data structure with O((1/tau)lg^* n lg lg_w sigma) query time. For any kappa > 1, we can also build a structure that uses O(n lg^{[kappa]} n) space, where lg^{[kappa]} n denotes the function that applies logarithm kappa times to n, and answers queries in time O((1/tau)lg lg_w sigma). The construction time of both structures is O(n lg n). We also describe two succinct-space solutions with the same query time of the linear-space structure. One uses 2nH + 4n + o(n)(H+1) bits, where H <=lg sigma is the entropy of the label distribution, and can be built in O(n lg n) time. The other uses nH + O(n) + o(nH) bits and is built in O(n lg n) time w.h.p.
Majorities on Trees
Succinct data structures
Theory of computation~Data structures design and analysis
68:1-68:12
Regular Paper
An extended version of the paper is available at https://arxiv.org/abs/1806.01804.
Travis
Gagie
Travis Gagie
CeBiB - Center for Biotechnology and Bioengineering, Chile, School of Computer Science and Telecommunications, Diego Portales University, Chile
Funded by FONDECYT grant 1171058, Chile.
Meng
He
Meng He
Faculty of Computer Science, Dalhousie University, Canada
Funded by NSERC, Canada.
Gonzalo
Navarro
Gonzalo Navarro
CeBiB - Center for Biotechnology and Bioengineering, Chile, IMFD - Millenium Institute for Foundational Research on Data, Chile, Dept. of Computer Science, University of Chile, Chile
Funded with basal funds FB0001, Conicyt, Chile, by Millenium Institute for Foundational Research on Data (IMFD), Chile, and by Fondecyt grant 1170048, Chile.
10.4230/LIPIcs.ISAAC.2018.68
D. Belazzougui, T. Gagie, J. I. Munro, G. Navarro, and Y. Nekrich. Range Majorities and Minorities in Arrays. CoRR, abs/1606.04495, 2016. URL: http://arxiv.org/abs/1606.04495.
http://arxiv.org/abs/1606.04495
D. Belazzougui, T. Gagie, and G. Navarro. Better Space Bounds for Parameterized Range Majority and Minority. In Proc. 12th WADS, LNCS 8037, pages 121-132, 2013.
D. Belazzougui and G. Navarro. Alphabet-Independent Compressed Text Indexing. ACM Trans. Alg., 10(4):article 23, 2014.
D. Belazzougui and G. Navarro. Optimal Lower and Upper Bounds for Representing Sequences. ACM Trans. Alg., 11(4):article 31, 2015.
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T. M. Chan, S. Durocher, K. G. Larsen, J. Morrison, and B. T. Wilkinson. Linear-Space Data Structures for Range Mode Query in Arrays. Theor. Comp. Syst., 55(4):719-741, 2014.
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Travis Gagie, Meng He, and Gonzalo Navarro
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Encoding Two-Dimensional Range Top-k Queries Revisited
We consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering Top-k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For 2 x n arrays, we first give upper and lower bounds on space for answering sorted and unsorted 3-sided Top-k queries. For m x n arrays, with m <=n and k <=mn, we obtain (m lg{(k+1)n choose n}+4nm(m-1)+o(n))-bit encoding for answering sorted 4-sided Top-k queries. This improves the min{(O(mn lg{n}),m^2 lg{(k+1)n choose n} + m lg{m}+o(n))}-bit encoding of Jo et al. [CPM, 2016] when m = o(lg{n}). This is a consequence of a new encoding that encodes a 2 x n array to support sorted 4-sided Top-k queries on it using an additional 4n bits, in addition to the encodings to support the Top-k queries on individual rows. This new encoding is a non-trivial generalization of the encoding of Jo et al. [CPM, 2016] that supports sorted 4-sided Top-2 queries on it using an additional 3n bits. We also give almost optimal space encodings for 3-sided Top-k queries, and show lower bounds on encodings for 3-sided and 4-sided Top-k queries.
Encoding model
top-k query
range minimum query
Theory of computation~Data compression
69:1-69:13
Regular Paper
https://arxiv.org/abs/1809.07067
Seungbum
Jo
Seungbum Jo
University of Siegen, Germany
https://orcid.org/0000-0002-8644-3691
The author of this paper is supported by the DFG research project LO748/11-1.
Srinivasa Rao
Satti
Srinivasa Rao Satti
Seoul National University, South Korea
https://orcid.org/0000-0003-0636-9880
10.4230/LIPIcs.ISAAC.2018.69
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Gerth Stølting Brodal, Andrej Brodnik, and Pooya Davoodi. The Encoding Complexity of Two Dimensional Range Minimum Data Structures. In Algorithms - ESA 2013 - 21st Annual European Symposium, Sophia Antipolis, France, September 2-4, 2013. Proceedings, pages 229-240, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40450-4_20.
http://dx.doi.org/10.1007/978-3-642-40450-4_20
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http://dx.doi.org/10.1007/s00453-011-9499-0
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http://dx.doi.org/10.1007/s11786-009-0007-8
Pawel Gawrychowski and Patrick K. Nicholson. Encodings of Range Maximum-Sum Segment Queries and Applications. In Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Ischia Island, Italy, June 29 - July 1, 2015, Proceedings, pages 196-206, 2015. URL: http://dx.doi.org/10.1007/978-3-319-19929-0_17.
http://dx.doi.org/10.1007/978-3-319-19929-0_17
Pawel Gawrychowski and Patrick K. Nicholson. Optimal Encodings for Range Top-k k , Selection, and Min-Max. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 593-604, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47672-7_48.
http://dx.doi.org/10.1007/978-3-662-47672-7_48
Mordecai J. Golin, John Iacono, Danny Krizanc, Rajeev Raman, Srinivasa Rao Satti, and Sunil M. Shende. Encoding 2D range maximum queries. Theor. Comput. Sci., 609:316-327, 2016. URL: http://dx.doi.org/10.1016/j.tcs.2015.10.012.
http://dx.doi.org/10.1016/j.tcs.2015.10.012
Roberto Grossi, Ankur Gupta, and Jeffrey Scott Vitter. High-order entropy-compressed text indexes. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA., pages 841-850, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644250.
http://dl.acm.org/citation.cfm?id=644108.644250
Roberto Grossi, John Iacono, Gonzalo Navarro, Rajeev Raman, and S. Srinivasa Rao. Asymptotically Optimal Encodings of Range Data Structures for Selection and Top-k Queries. ACM Trans. Algorithms, 13(2):28:1-28:31, 2017. URL: http://dx.doi.org/10.1145/3012939.
http://dx.doi.org/10.1145/3012939
Seungbum Jo, Rahul Lingala, and Srinivasa Rao Satti. Encoding Two-Dimensional Range Top-k Queries. In 27th Annual Symposium on Combinatorial Pattern Matching, CPM 2016, June 27-29, 2016, Tel Aviv, Israel, pages 3:1-3:11, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CPM.2016.3.
http://dx.doi.org/10.4230/LIPIcs.CPM.2016.3
Seungbum Jo and Srinivasa Rao Satti. Encoding two-dimensional range top-k queries revisited. CoRR, abs/1809.07067, 2018. URL: http://arxiv.org/abs/1809.07067.
http://arxiv.org/abs/1809.07067
N.D Kazarinoff. Geometric inequalities. New York: Random House, 1961.
P. B. Miltersen. Cell probe complexity - a survey. FSTTCS, 1999.
J. Ian Munro and Venkatesh Raman. Succinct Representation of Balanced Parentheses and Static Trees. SIAM J. Comput., 31(3):762-776, 2001. URL: http://dx.doi.org/10.1137/S0097539799364092.
http://dx.doi.org/10.1137/S0097539799364092
Seungbum Jo and Srinivasa Rao Satti
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Longest Unbordered Factor in Quasilinear Time
A border u of a word w is a proper factor of w occurring both as a prefix and as a suffix. The maximal unbordered factor of w is the longest factor of w which does not have a border. Here an O(n log n)-time with high probability (or O(n log n log^2 log n)-time deterministic) algorithm to compute the Longest Unbordered Factor Array of w for general alphabets is presented, where n is the length of w. This array specifies the length of the maximal unbordered factor starting at each position of w. This is a major improvement on the running time of the currently best worst-case algorithm working in O(n^{1.5}) time for integer alphabets [Gawrychowski et al., 2015].
longest unbordered factor
factorisation
period
border
strings
Theory of computation~Pattern matching
70:1-70:13
Regular Paper
https://arxiv.org/abs/1805.09924
Tomasz
Kociumaka
Tomasz Kociumaka
Institute of Informatics, University of Warsaw, Warsaw, Poland
https://orcid.org/0000-0002-2477-1702
Ritu
Kundu
Ritu Kundu
Department of Informatics, King’s College London, London, UK
https://orcid.org/0000-0003-1353-4004
Manal
Mohamed
Manal Mohamed
Department of Informatics, King’s College London, London, UK
https://orcid.org/0000-0002-1435-5051
Solon P.
Pissis
Solon P. Pissis
Department of Informatics, King’s College London, London, UK
https://orcid.org/0000-0002-1445-1932
10.4230/LIPIcs.ISAAC.2018.70
Michael A. Bender and Martin Farach-Colton. The LCA Problem Revisited. In Gaston H. Gonnet, Daniel Panario, and Alfredo Viola, editors, LATIN 2000: Theoretical Informatics, 4th Latin American Symposium, Punta del Este, Uruguay, April 10-14, 2000, Proceedings, volume 1776 of Lecture Notes in Computer Science, pages 88-94. Springer, 2000. URL: http://dx.doi.org/10.1007/10719839_9.
http://dx.doi.org/10.1007/10719839_9
Patrick Hagge Cording and Mathias Bæk Tejs Knudsen. Maximal Unbordered Factors of Random Strings. In Shunsuke Inenaga, Kunihiko Sadakane, and Tetsuya Sakai, editors, String Processing and Information Retrieval - 23rd International Symposium, SPIRE 2016, Beppu, Japan, October 18-20, 2016, Proceedings, volume 9954 of Lecture Notes in Computer Science, pages 93-96, 2016. URL: http://dx.doi.org/10.1007/978-3-319-46049-9_9.
http://dx.doi.org/10.1007/978-3-319-46049-9_9
Maxime Crochemore, Christophe Hancart, and Thierry Lecroq. Algorithms on strings. Cambridge University Press, 2007.
Maxime Crochemore and Lucian Ilie. Computing Longest Previous Factor in Linear Time and Applications. Inf. Process. Lett., 106(2):75-80, 2008.
Maxime Crochemore, Lucian Ilie, Costas S. Iliopoulos, Marcin Kubica, Wojciech Rytter, and Tomasz Waleń. Computing the Longest Previous Factor. Eur. J. Comb., 34(1):15-26, 2013.
Maxime Crochemore and Wojciech Rytter. Jewels of stringology. World Scientific, 2002. URL: http://dx.doi.org/10.1142/4838.
http://dx.doi.org/10.1142/4838
Jean-Pierre Duval. Relationship between the period of a finite word and the length of its unbordered segments. Discrete Mathematics, 40(1):31-44, 1982. URL: http://dx.doi.org/10.1016/0012-365X(82)90186-8.
http://dx.doi.org/10.1016/0012-365X(82)90186-8
Jean-Pierre Duval, Thierry Lecroq, and Arnaud Lefebvre. Linear computation of unbordered conjugate on unordered alphabet. Theor. Comput. Sci., 522:77-84, 2014. URL: http://dx.doi.org/10.1016/j.tcs.2013.12.008.
http://dx.doi.org/10.1016/j.tcs.2013.12.008
Andrzej Ehrenfeucht and D. M. Silberger. Periodicity and unbordered segments of words. Discrete Mathematics, 26(2):101-109, 1979. URL: http://dx.doi.org/10.1016/0012-365X(79)90116-X.
http://dx.doi.org/10.1016/0012-365X(79)90116-X
Pawel Gawrychowski, Gregory Kucherov, Benjamin Sach, and Tatiana A. Starikovskaya. Computing the Longest Unbordered Substring. In Costas S. Iliopoulos, Simon J. Puglisi, and Emine Yilmaz, editors, String Processing and Information Retrieval - 22nd International Symposium, SPIRE 2015, London, UK, September 1-4, 2015, Proceedings, volume 9309 of Lecture Notes in Computer Science, pages 246-257. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-23826-5_24.
http://dx.doi.org/10.1007/978-3-319-23826-5_24
Stepan Holub and Dirk Nowotka. The Ehrenfeucht-Silberger problem. J. Comb. Theory, Ser. A, 119(3):668-682, 2012. URL: http://dx.doi.org/10.1016/j.jcta.2011.11.004.
http://dx.doi.org/10.1016/j.jcta.2011.11.004
Donald E. Knuth, James H. Morris Jr., and Vaughan R. Pratt. Fast Pattern Matching in Strings. SIAM J. Comput., 6(2):323-350, 1977. URL: http://dx.doi.org/10.1137/0206024.
http://dx.doi.org/10.1137/0206024
Tomasz Kociumaka, Jakub Radoszewski, Wojciech Rytter, and Tomasz Walen. Efficient Data Structures for the Factor Periodicity Problem. In Liliana Calderón-Benavides, Cristina N. González-Caro, Edgar Chávez, and Nivio Ziviani, editors, String Processing and Information Retrieval - 19th International Symposium, SPIRE 2012, Cartagena de Indias, Colombia, October 21-25, 2012. Proceedings, volume 7608 of Lecture Notes in Computer Science, pages 284-294. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-34109-0_30.
http://dx.doi.org/10.1007/978-3-642-34109-0_30
Tomasz Kociumaka, Jakub Radoszewski, Wojciech Rytter, and Tomasz Walen. Internal Pattern Matching Queries in a Text and Applications. In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 532-551. SIAM, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.36.
http://dx.doi.org/10.1137/1.9781611973730.36
Alexander Loptev, Gregory Kucherov, and Tatiana A. Starikovskaya. On Maximal Unbordered Factors. In Ferdinando Cicalese, Ely Porat, and Ugo Vaccaro, editors, Combinatorial Pattern Matching - 26th Annual Symposium, CPM 2015, Ischia Island, Italy, June 29 - July 1, 2015, Proceedings, volume 9133 of Lecture Notes in Computer Science, pages 343-354. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-19929-0_29.
http://dx.doi.org/10.1007/978-3-319-19929-0_29
Milan Ruzic. Constructing Efficient Dictionaries in Close to Sorting Time. In Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, and Igor Walukiewicz, editors, Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7-11, 2008, Proceedings, Part I: Tack A: Algorithms, Automata, Complexity, and Games, volume 5125 of Lecture Notes in Computer Science, pages 84-95. Springer, 2008. URL: http://dx.doi.org/10.1007/978-3-540-70575-8_8.
http://dx.doi.org/10.1007/978-3-540-70575-8_8
Tomasz Kociumaka, Ritu Kundu, Manal Mohamed, and Solon P. Pissis
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Packing Sporadic Real-Time Tasks on Identical Multiprocessor Systems
In real-time systems, in addition to the functional correctness recurrent tasks must fulfill timing constraints to ensure the correct behavior of the system. Partitioned scheduling is widely used in real-time systems, i.e., the tasks are statically assigned onto processors while ensuring that all timing constraints are met. The decision version of the problem, which is to check whether the deadline constraints of tasks can be satisfied on a given number of identical processors, has been known NP-complete in the strong sense. Several studies on this problem are based on approximations involving resource augmentation, i.e., speeding up individual processors. This paper studies another type of resource augmentation by allocating additional processors, a topic that has not been explored until recently. We provide polynomial-time algorithms and analysis, in which the approximation factors are dependent upon the input instances. Specifically, the factors are related to the maximum ratio of the period to the relative deadline of a task in the given task set. We also show that these algorithms unfortunately cannot achieve a constant approximation factor for general cases. Furthermore, we prove that the problem does not admit any asymptotic polynomial-time approximation scheme (APTAS) unless P=NP when the task set has constrained deadlines, i.e., the relative deadline of a task is no more than the period of the task.
multiprocessor partitioned scheduling
approximation factors
Computer systems organization~Real-time systems
71:1-71:14
Regular Paper
https://arxiv.org/abs/1809.04355
Jian-Jia
Chen
Jian-Jia Chen
Department of Computer Science, TU Dortmund University, Germany
https://orcid.org/0000-0001-8114-9760
Nikhil
Bansal
Nikhil Bansal
Eindhoven University of Technology, The Netherlands
Samarjit
Chakraborty
Samarjit Chakraborty
Technical University of Munich (TUM), Germany
https://orcid.org/0000-0002-0503-6235
Georg
von der Brüggen
Georg von der Brüggen
Department of Computer Science, TU Dortmund University, Germany
https://orcid.org/0000-0002-8137-3612
10.4230/LIPIcs.ISAAC.2018.71
Nikhil Bansal, Cyriel Rutten, Suzanne van der Ster, Tjark Vredeveld, and Ruben van der Zwaan. Approximating Real-Time Scheduling on Identical Machines. In LATIN: Theoretical Informatics - 11th Latin American Symposium, pages 550-561, 2014.
Sanjoy Baruah. The Partitioned EDF Scheduling of Sporadic Task Systems. In Real-Time Systems Symposium (RTSS), pages 116-125, 2011.
Sanjoy K. Baruah and Nathan Fisher. The Partitioned Multiprocessor Scheduling of Sporadic Task Systems. In Real-Time Systems Symposium (RTSS), pages 321-329, 2005.
Sanjoy K. Baruah and Nathan Fisher. The Partitioned Multiprocessor Scheduling of Deadline-Constrained Sporadic Task Systems. IEEE Trans. Computers, 55(7):918-923, 2006.
Sanjoy K. Baruah, Aloysius K. Mok, and Louis E. Rosier. Preemptively scheduling hard-real-time sporadic tasks on one processor. In Real-Time Systems Symposium (RTSS), pages 182-190, 1990.
Almut Burchard, Jörg Liebeherr, Yingfeng Oh, and Sang Hyuk Son. New Strategies for Assigning Real-Time Tasks to Multiprocessor Systems. IEEE Trans. Computers, 44(12):1429-1442, 1995.
Chandra Chekuri and Sanjeev Khanna. On Multidimensional Packing Problems. SIAM J. Comput., 33(4):837-851, 2004. URL: http://dx.doi.org/10.1137/S0097539799356265.
http://dx.doi.org/10.1137/S0097539799356265
Jian-Jia Chen. Partitioned Multiprocessor Fixed-Priority Scheduling of Sporadic Real-Time Tasks. In Euromicro Conference on Real-Time Systems (ECRTS), pages 251-261, 2016.
Jian-Jia Chen, Nikhil Bansal, Samarjit Chakraborty, and Georg von der Brüggen. Packing Sporadic Real-Time Tasks on Identical Multiprocessor Systems. Computing Research Repository (CoRR), 2018. http://arxiv.org/abs/1809.04355.
Jian-Jia Chen and Samarjit Chakraborty. Resource Augmentation Bounds for Approximate Demand Bound Functions. In IEEE Real-Time Systems Symposium, pages 272-281, 2011.
Jian-Jia Chen and Samarjit Chakraborty. Partitioned Packing and Scheduling for Sporadic Real-Time Tasks in Identical Multiprocessor Systems. In ECRTS, pages 24-33, 2012.
Jian-Jia Chen, Georg von der Brüggen, Wen-Hung Huang, and Robert I Davis. On the Pitfalls of Resource Augmentation Factors and Utilization Bounds in Real-Time Scheduling. In Euromicro Conference on Real-Time Systems, ECRTS, pages 9:1-9:25, 2017.
Lin Chen, Nicole Megow, and Kevin Schewior. An O(log m)-Competitive Algorithm for Online Machine Minimization. In Symposium on Discrete Algorithms, SODA, pages 155-163, 2016.
Lin Chen, Nicole Megow, and Kevin Schewior. The Power of Migration in Online Machine Minimization. In Symposium on Parallelism in Algorithms and Architectures, pages 175-184, 2016.
Robert I. Davis and Alan Burns. A survey of hard real-time scheduling for multiprocessor systems. ACM Comput. Surv., 43(4):35, 2011.
Wenceslas Fernandez de la Vega and George S. Lueker. Bin packing can be solved within 1+epsilon in linear time. Combinatorica, 1(4):349-355, 1981. URL: http://dx.doi.org/10.1007/BF02579456.
http://dx.doi.org/10.1007/BF02579456
Friedrich Eisenbrand and Thomas Rothvoß. EDF-schedulability of synchronous periodic task systems is coNP-hard. In Symposium on Discrete Algorithms (SODA), pages 1029-1034, 2010. URL: http://www.siam.org/proceedings/soda/2010/SODA10_083_eisenbrandf.pdf.
http://www.siam.org/proceedings/soda/2010/SODA10_083_eisenbrandf.pdf
Pontus Ekberg and Wang Yi. Uniprocessor Feasibility of Sporadic Tasks Remains coNP-Complete under Bounded Utilization. In IEEE Real-Time Systems Symposium, RTSS, pages 87-95, 2015.
Pontus Ekberg and Wang Yi. Uniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines Is Strongly coNP-Complete. In Euromicro Conference on Real-Time Systems, ECRTS, pages 281-286, 2015.
M. R. Garey and D. S. Johnson. Computers and intractability: A guide to the theory of NP-completeness. W. H. Freeman and Co., 1979.
Ronald L. Graham. Bounds on Multiprocessing Timing Anomalies. SIAM Journal of Applied Mathematics, 17(2):416-429, 1969.
Dorit S. Hochbaum and David B. Shmoys. Using dual approximation algorithms for scheduling problems theoretical and practical results. J. ACM, 34(1):144-162, 1987. URL: http://dx.doi.org/10.1145/7531.7535.
http://dx.doi.org/10.1145/7531.7535
Sungjin Im, Benjamin Moseley, Kirk Pruhs, and Clifford Stein. An O(log log m)-competitive Algorithm for Online Machine Minimization. In Real-Time Systems Symposium, (RTSS), pages 343-350, 2017.
Viggo Kann. Maximum bounded 3-dimensional matching is MAX SNP-complete. Inf. Process. Lett., 37(1):27-35, January 1991.
N. Karmarkar and R. M. Karp. An Efficient Approximation Scheme for the One-Dimensional Bin-Packing Problem. In Symp. on Foundations of Computer Science (FOCS), pages 312-320, 1982.
Andreas Karrenbauer and Thomas Rothvoß. A 3/2-Approximation Algorithm for Rate-Monotonic Multiprocessor Scheduling of Implicit-Deadline Tasks. In International Workshop of Approximation and Online Algorithms WAOA, pages 166-177, 2010. URL: http://dx.doi.org/10.1007/978-3-642-18318-8_15.
http://dx.doi.org/10.1007/978-3-642-18318-8_15
C. L. Liu and James W. Layland. Scheduling Algorithms for Multiprogramming in a Hard-Real-Time Environment. Journal of the ACM, 20(1):46-61, 1973.
A. K. Mok. Fundamental design problems of distributed systems for the hard-real-time environment. Technical report, Massachusetts Institute of Technology, Cambridge, MA, USA, 1983.
K. Pruhs, E. Torng, and J. Sgall. Online scheduling. In Joseph Y. T. Leung, editor, Handbook of Scheduling: Algorithms, Models, and Performance Analysis, chapter 15, pages 15:1-15:41. Chapman and Hall/CRC, 2004.
Vijay V. Vazirani. Approximation Algorithms. Springer, 2001.
Gerhard J. Woeginger. There is no Asymptotic PTAS for Two-Dimensional Vector Packing. Inf. Process. Lett., 64(6):293-297, 1997. URL: http://dx.doi.org/10.1016/S0020-0190(97)00179-8.
http://dx.doi.org/10.1016/S0020-0190(97)00179-8
Jina-Jia Chen, Nikhil Bansal, Samarjit Chakraborty, and Georg von der Brüggen
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Relaxed FPTAS for Chance-Constrained Knapsack
The stochastic knapsack problem is a stochastic version of the well known deterministic knapsack problem, in which some of the input values are random variables. There are several variants of the stochastic problem. In this paper we concentrate on the chance-constrained variant, where item values are deterministic and item sizes are stochastic. The goal is to find a maximum value allocation subject to the constraint that the overflow probability is at most a given value. Previous work showed a PTAS for the problem for various distributions (Poisson, Exponential, Bernoulli and Normal). Some strictly respect the constraint and some relax the constraint by a factor of (1+epsilon). All algorithms use Omega(n^{1/epsilon}) time. A very recent work showed a "almost FPTAS" algorithm for Bernoulli distributions with O(poly(n) * quasipoly(1/epsilon)) time.
In this paper we present a FPTAS for normal distributions with a solution that satisfies the chance constraint in a relaxed sense. The normal distribution is particularly important, because by the Berry-Esseen theorem, an algorithm solving the normal distribution also solves, under mild conditions, arbitrary independent distributions. To the best of our knowledge, this is the first (relaxed or non-relaxed) FPTAS for the problem. In fact, our algorithm runs in poly(n/epsilon) time. We achieve the FPTAS by a delicate combination of previous techniques plus a new alternative solution to the non-heavy elements that is based on a non-convex program with a simple structure and an O(n^2 log {n/epsilon}) running time. We believe this part is also interesting on its own right.
Stochastic knapsack
Chance constraint
Approximation algorithms
Combinatorial optimization
Theory of computation~Approximation algorithms analysis
Theory of computation~Stochastic approximation
Theory of computation~Discrete optimization
Theory of computation~Nonconvex optimization
72:1-72:12
Regular Paper
Galia
Shabtai
Galia Shabtai
School of Electrical Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Danny
Raz
Danny Raz
Faculty of Computer Science, The Technion, Haifa 32000, Israel
Yuval
Shavitt
Yuval Shavitt
School of Electrical Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
10.4230/LIPIcs.ISAAC.2018.72
Anand Bhalgat, Ashish Goel, and Sanjeev Khanna. Improved approximation results for stochastic knapsack problems. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 1647-1665, 2011.
Daniel Blado, Weihong Hu, and Alejandro Toriello. Semi-infinite relaxations for the dynamic knapsack problem with stochastic item sizes. SIAM Journal on Optimization, 26(3):1625-1648, 2016.
Anindya De. Boolean function analysis meets stochastic optimization: An approximation scheme for stochastic knapsack. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1286-1305. SIAM, 2018.
Ashish Goel and Piotr Indyk. Stochastic load balancing and related problems. In IEEE FOCS, pages 579-586, 1999.
Vineet Goyal and R Ravi. A PTAS for the chance-constrained knapsack problem with random item sizes. Operations Research Letters, 38:162-164, 2010.
Jon Kleinberg, Yuval Rabani, and Éva Tardos. Allocating bandwidth for bursty connections. SIAM Journal on Computing, 30(1):191-217, 2000.
Jian Li and Wen Yuan. Stochastic combinatorial optimization via poisson approximation. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 971-980. ACM, 2013.
Evdokia Nikolova. Approximation algorithms for offline risk-averse combinatorial optimization. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 338-351, 2010.
Galia Shabtai, Danny Raz, and Yuval Shavitt. Risk aware stochastic placement of cloud services: the case of two data centers. In International Workshop on Algorithmic Aspects of Cloud Computing, pages 59-88. Springer, 2017.
Galia Shabtai, Danny Raz, and Yuval Shavitt. A relaxed FPTAS for Chance-Constrained Knapsack - Technical Report, 2018. To appear.
Galia Shabtai, Danny Raz, and Yuval Shavitt
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Covering Clients with Types and Budgets
In this paper, we consider a variant of the facility location problem. Imagine the scenario where facilities are categorized into multiple types such as schools, hospitals, post offices, etc. and the cost of connecting a client to a facility is realized by the distance between them. Each client has a total budget on the distance she/he is willing to travel. The goal is to open the minimum number of facilities such that the aggregate distance of each client to multiple types is within her/his budget. This problem closely resembles to the set cover and r-domination problems. Here, we study this problem in different settings. Specifically, we present some positive and negative results in the general setting, where no assumption is made on the distance values. Then we show that better results can be achieved when clients and facilities lie in a metric space.
Facility Location
Geometric Set Cover
Local Search
Theory of computation~Packing and covering problems
Theory of computation~Facility location and clustering
73:1-73:12
Regular Paper
PSL Project MULTIFAC
Dimitris
Fotakis
Dimitris Fotakis
Yahoo Research-New York, USA & National Technical University of Athens, Greece
Laurent
Gourvès
Laurent Gourvès
Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France
Claire
Mathieu
Claire Mathieu
CNRS, France
Abhinav
Srivastav
Abhinav Srivastav
ENS Paris & Université Paris-Dauphine, France
10.4230/LIPIcs.ISAAC.2018.73
A. Ageev, Y. Ye, and J. Zhang. Improved Combinatorial Approximation Algorithms for the k-Level Facility Location Problem. SIAM J. Discrete Math., 18(1):207-2017, 2004.
N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, and J. Naor. The Online Set Cover Problem. SIAM J. Computing, 39(2):361-370, 2009.
S. Arora, N. Gupta, S. Khuller, Y. Sabharwal, and S. Singhal. Facility Location with Red-blue Demands. Operation Research Letter, 42(6):462-465, September 2014.
N. Bansal and K. Pruhs. Weighted geometric set multi-cover via quasi-uniform sampling. J. Comp. Geometry, 7(1):221-236, 2016.
R. Bar-Yehuda, G. Flysher, J. Mestre, and D. Rawitz. Approximation of Partial Capacitated Vertex Cover. SIAM J. Discrete Math., 24(4):1441-1469, 2010.
J. Byrka, S. Li, and B. Rybicki. Improved Approximation Algorithm for k-level Uncapacitated Facility Location Problem (with Penalties). Theory of Computing Systems, 58(1):19-44, 2016.
A. Caprara, G. F. Italiano, G. Mohan, A. Panconesi, and A. Srinivasan. Wavelength rerouting in optical networks, or the Venetian Routing problem. J. Algorithms, 45(2):93-125, 2002.
R.D. Carr, S. Doddi, G. Konjevod, and M.V. Marathe. On the red-blue set cover problem. In Proc. of SODA, pages 345-353, 2000.
I. Dinur and D. Steurer. Analytical approach to parallel repetition. In Proc. of STOC, pages 624-633, 2014.
Y. Dodis and S. Khanna. Designing Networks with Bounded Pairwise Distance. In Proc. of STOC, pages 750-759, 1999.
A. Efrat, F. Hoffmann, C. Knauer, K. Kriegel, G. Rote, and C. Wenk. Covering Shapes by Ellipses. In Proc. of SODA, pages 453-454, 2002.
M. Elkin and D. Peleg. The Hardness of Approximating Spanner Problems. Theory Comp. Systems, 41(4):691-729, 2007.
D. Fotakis. Online and Incremental Algorithms for Facility Location. SIGACT News, 42(1):97-131, 2011.
M.R. Garey and D.S. Johnson. Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman &Co., 1990.
M. Hajiaghayi, R. Khandekar, and G. Kortsarz. Budgeted Red-blue Median and Its Generalizations. In Proc. of ESA, pages 314-325, 2010.
D.S. Hochbaum and W. Maass. Approximation Schemes for Covering and Packing Problems in Image Processing and VLSI. J. ACM, 32(1):130-136, 1985.
D.S. Hochbaum and W. Maass. Fast approximation algorithms for a nonconvex covering problem. J. Algorithms, 8(3):305-323, 1987.
K. Jain and V.V. Vazirani. An approximation algorithm for the fault tolerant metric facility location problem. In Proc. of APPROX, pages 177-183, 2000.
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