Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016), APPROX/RANDOM 2016, September 7-9, 2016, Paris, France
APPROX/RANDOM 2016
September 7-9, 2016
Paris, France
International Conference on Randomization and Computation
RANDOM
https://randomconference.com/
https://dblp.org/db/conf/random
International Conference on Approximation Algorithms for Combinatorial Optimization Problems
APPROX
https://approxconference.wordpress.com/
https://dblp.org/db/conf/approx
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Klaus
Jansen
Klaus Jansen
Claire
Mathieu
Claire Mathieu
José D. P.
Rolim
José D. P. Rolim
Chris
Umans
Chris Umans
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
60
2016
978-3-95977-018-7
https://www.dagstuhl.de/dagpub/978-3-95977-018-7
Front Matter, Table of Contents, Preface, Program Committees, External Reviewers, List of Authors
Front Matter, Table of Contents, Preface, Program Committees, External Reviewers, List of Authors
Front Matter
Table of Contents
Preface
Program Committees
External Reviewers
List of Authors
0:i-0:xvi
Front Matter
Klaus
Jansen
Klaus Jansen
Claire
Mathieu
Claire Mathieu
José D. P.
Rolim
José D. P. Rolim
Chris
Umans
Chris Umans
10.4230/LIPIcs.APPROX-RANDOM.2016.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces
We show that the Hausdorff metric over constant-size pointsets in constant-dimensional Euclidean space admits an embedding into constant-dimensional l_{infinity} space with constant distortion. More specifically for any s,d>=1, we obtain an embedding of the Hausdorff metric over pointsets of size s in d-dimensional Euclidean space, into l_{\infinity}^{s^{O(s+d)}} with distortion s^{O(s+d)}. We remark that any metric space M admits an isometric embedding into l_{infinity} with dimension proportional to the size of M. In contrast, we obtain an embedding of a space of infinite size into constant-dimensional l_{infinity}.
We further improve the distortion and dimension trade-offs by considering probabilistic embeddings of the snowflake version of the Hausdorff metric. For the case of pointsets of size s in the real line of bounded resolution, we obtain a probabilistic embedding into l_1^{O(s*log(s()} with distortion O(s).
metric embeddings
Hausdorff metric
distortion
dimension
1:1-1:15
Regular Paper
Arturs
Backurs
Arturs Backurs
Anastasios
Sidiropoulos
Anastasios Sidiropoulos
10.4230/LIPIcs.APPROX-RANDOM.2016.1
Artūrs Bačkurs and Piotr Indyk. Better embeddings for planar earth-mover distance over sparse sets. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG'14, pages 280:280-280:289, New York, NY, USA, 2014. ACM. URL: http://dx.doi.org/10.1145/2582112.2582120.
http://dx.doi.org/10.1145/2582112.2582120
Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural computation, 15(6):1373-1396, 2003.
Radu Berinde, Anna C. Gilbert, Piotr Indyk, H. Karloff, and Martin J. Strauss. Combining geometry and combinatorics: A unified approach to sparse signal recovery. In Communication, Control, and Computing, 2008 46th Annual Allerton Conference on, pages 798-805. IEEE, 2008.
J. Bourgain. On lipschitz embedding of finite metric spaces in hilbert space. Israel Journal of Mathematics, 52(1-2):46-52, 1985. URL: http://dx.doi.org/10.1007/BF02776078.
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Manuel Costa, Miguel Castro, Antony Rowstron, and Peter Key. Pic: Practical internet coordinates for distance estimation. In Distributed Computing Systems, 2004. Proceedings. 24th International Conference on, pages 178-187. IEEE, 2004.
Aryeh Dvoretzky. Some results on convex bodies and Banach spaces. Hebrew University, 1960.
Martin Farach-Colton and Piotr Indyk. Approximate nearest neighbor algorithms for hausdorff metrics via embeddings. In Foundations of Computer Science, 1999. 40th Annual Symposium on, pages 171-179. IEEE, 1999.
Sariel Har-Peled and Manor Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM Journal on Computing, 35(5):1148-1184, 2006.
P. Indyk and N. Thaper. Fast color image retrieval via embeddings. Workshop on Statistical and Computational Theories of Vision (at ICCV), 2003.
Piotr Indyk. Algorithmic applications of low-distortion geometric embeddings. In focs, page 10. IEEE, 2001.
Piotr Indyk. High-dimensional Computational Geometry. PhD thesis, Stanford University, 2001.
Piotr Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation. Journal of the ACM (JACM), 53(3):307-323, 2006.
Piotr Indyk, Avner Magen, Anastasios Sidiropoulos, and Anastasios Zouzias. On-line embeddings. In Proc. of APPROX, 2010.
William B Johnson and Joram Lindenstrauss. Extensions of lipschitz mappings into a hilbert space. Contemporary mathematics, 26(189-206):1, 1984.
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Computing Approximate PSD Factorizations
We give an algorithm for computing approximate PSD factorizations of nonnegative matrices. The running time of the algorithm is polynomial in the dimensions of the input matrix, but exponential in the PSD rank and the approximation error. The main ingredient is an exact factorization algorithm when the rows and columns of the factors are constrained to lie in a general polyhedron. This strictly generalizes nonnegative matrix factorizations which can be captured by letting this polyhedron to be the nonnegative orthant.
PSD rank
PSD factorizations
2:1-2:12
Regular Paper
Amitabh
Basu
Amitabh Basu
Michael
Dinitz
Michael Dinitz
Xin
Li
Xin Li
10.4230/LIPIcs.APPROX-RANDOM.2016.2
Sanjeev Arora, Rong Ge, Ravindran Kannan, and Ankur Moitra. Computing a nonnegative matrix factorization-provably. In Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of computing, STOC'12, pages 145-162. ACM, 2012.
Saugata Basu, Richard Pollack, and Marie-Françoise Roy. On the combinatorial and algebraic complexity of quantifier elimination. Journal of the ACM (JACM), 43(6):1002-1045, 1996.
Michael W Berry, Murray Browne, Amy N Langville, V Paul Pauca, and Robert J Plemmons. Algorithms and applications for approximate nonnegative matrix factorization. Computational statistics &data analysis, 52(1):155-173, 2007.
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Gábor Braun, Jonah Brown-Cohen, Arefin Huq, Sebastian Pokutta, Prasad Raghavendra, Aurko Roy, Benjamin Weitz, and Daniel Zink. The matching problem has no small symmetric sdp. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'16, pages 1067-1078, Philadelphia, PA, USA, 2016. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2884435.2884510.
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Jop Briët, Daniel Dadush, and Sebastian Pokutta. On the existence of 0/1 polytopes with high semidefinite extension complexity. Mathematical Programming, pages 1-21, 2014.
Michele Conforti, Gérard Cornuéjols, and Giacomo Zambelli. Extended formulations in combinatorial optimization. 4OR, 8(1):1-48, 2010.
Hamza Fawzi, Jo ao Gouveia, Pablo A. Parrilo, Richard Z. Robinson, and Rekha R. Thomas. Positive semidefinite rank. http://arxiv.org/abs/1407.4095, 2015.
Samuel Fiorini, Volker Kaibel, Kanstantsin Pashkovich, and Dirk Oliver Theis. Combinatorial bounds on nonnegative rank and extended formulations. Discrete mathematics, 313(1):67-83, 2013.
Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary, and Ronald de Wolf. Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pages 95-106. ACM, 2012.
Joao Gouveia, Pablo A Parrilo, and Rekha R Thomas. Lifts of convex sets and cone factorizations. Mathematics of Operations Research, 38(2):248-264, 2013.
João Gouveia, Pablo A Parrilo, and Rekha R Thomas. Approximate cone factorizations and lifts of polytopes. Mathematical Programming, 151(2):613-637, 2015.
D Yu Grigor'ev and NN Vorobjov. Solving systems of polynomial inequalities in subexponential time. Journal of symbolic computation, 5(1):37-64, 1988.
Didier Henrion and Jérôme Malick. Projection methods in conic optimization. In Handbook on Semidefinite, Conic and Polynomial Optimization, pages 565-600. Springer, 2012.
Volker Kaibel. Extended formulations in combinatorial optimization. arXiv preprint arXiv:1104.1023, 2011.
J Lee, Prasad Raghavendra, David Steurer, and Ning Tan. On the power of symmetric lp and sdp relaxations. In Proceedings of the 29th Conference on Computational Complexity (CCC), pages 13-21. IEEE, 2014.
James R. Lee, Prasad Raghavendra, and David Steurer. Lower bounds on the size of semidefinite programming relaxations. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, STOC'15, pages 567-576, New York, NY, USA, 2015. ACM. URL: http://dx.doi.org/10.1145/2746539.2746599.
http://dx.doi.org/10.1145/2746539.2746599
Ankur Moitra. An almost optimal algorithm for computing nonnegative rank. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'13, pages 1454-1464. SIAM, 2013.
Victor Y Pan and Zhao Q Chen. The complexity of the matrix eigenproblem. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of computing, pages 507-516. ACM, 1999.
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Thomas Rothvoß. Some 0/1 polytopes need exponential size extended formulations. Mathematical Programming, 142(1-2):255-268, 2013.
Thomas Rothvoß. The matching polytope has exponential extension complexity. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 263-272. ACM, 2014.
Stephen A Vavasis. On the complexity of nonnegative matrix factorization. SIAM Journal on Optimization, 20(3):1364-1377, 2009.
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G. M. Ziegler. Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. URL: http://dx.doi.org/10.1007/978-1-4613-8431-1.
http://dx.doi.org/10.1007/978-1-4613-8431-1
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Hardness of Approximation for H-Free Edge Modification Problems
The H-free Edge Deletion problem asks, for a given graph G and integer k, whether it is possible to delete at most k edges from G to make it H-free, that is, not containing H as an induced subgraph. The H-free Edge Completion problem is defined similarly, but we add edges instead of deleting them. The study of these two problem families has recently been the subject of intensive studies from the point of view of parameterized complexity and kernelization. In particular, it was shown that the problems do not admit polynomial kernels (under plausible complexity assumptions) for almost all graphs H, with several important exceptions occurring when the class of H-free graphs exhibits some structural properties.
In this work we complement the parameterized study of edge modification problems to H-free graphs by considering their approximability. We prove that whenever H is 3-connected and has at least two non-edges, then both H-free Edge Deletion and H-free Edge Completion are very hard to approximate: they do not admit poly(OPT)-approximation in polynomial time, unless P=NP, or even in time subexponential in OPT, unless the Exponential Time Hypothesis fails. The assumption of the existence of two non-edges appears to be important: we show that whenever H is a complete graph without one edge, then H-free Edge Deletion is tightly connected to the \minhorn problem, whose approximability is still open. Finally, in an attempt to extend our hardness results beyond 3-connected graphs, we consider the cases of H being a path or a cycle, and we achieve an almost complete dichotomy there.
hardness of approximation
parameterized complexity
kernelization
edge modification problems
3:1-3:17
Regular Paper
Ivan
Bliznets
Ivan Bliznets
Marek
Cygan
Marek Cygan
Pawel
Komosa
Pawel Komosa
Michal
Pilipczuk
Michal Pilipczuk
10.4230/LIPIcs.APPROX-RANDOM.2016.3
N. R. Aravind, R. B. Sandeep, and Naveen Sivadasan. Parameterized lower bound and NP-completeness of some h-free edge deletion problems. In COCOA 2015, volume 9486 of LNCS, pages 424-438. Springer, 2015.
Ivan Bliznets, Marek Cygan, Paweł Komosa, Lukáš Mach, and Michał Pilipczuk. Lower bounds for the parameterized complexity of Minimum Fill-in and other completion problems. In SODA 2016, pages 1132-1151. SIAM, 2016.
Leizhen Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett., 58(4):171-176, 1996.
Leizhen Cai and Yufei Cai. Incompressibility of H-free edge modification problems. Algorithmica, 71(3):731-757, 2015.
Yufei Cai. Polynomial kernelisation of H-free edge modification problems. Master’s thesis, The Chinese University of Hong Kong, 2012. Available at author’s website.
Marek Cygan, Marcin Pilipczuk, Michał Pilipczuk, Erik Jan van Leeuwen, and Marcin Wrochna. Polynomial kernelization for removing induced claws and diamonds. CoRR, abs/1503.00704, 2015. To appear in the proceedings of WG 2015.
Pål Grønås Drange, Fedor V. Fomin, Michał Pilipczuk, and Yngve Villanger. Exploring the subexponential complexity of completion problems. TOCT, 7(4):14, 2015.
Archontia C. Giannopoulou, Daniel Lokshtanov, Saket Saurabh, and Ondrej Suchý. Tree deletion set has a polynomial kernel (but no OPT^O(1) approximation). In FSTTCS 2014, volume 29 of LIPIcs, pages 85-96. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2014.
Sylvain Guillemot, Frédéric Havet, Christophe Paul, and Anthony Perez. On the (non-)existence of polynomial kernels for P_𝓁-free edge modification problems. Algorithmica, 65(4):900-926, 2013.
Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001.
Sanjeev Khanna, Madhu Sudan, Luca Trevisan, and David P. Williamson. The approximability of constraint satisfaction problems. SIAM Journal on Computing, 30(6):1863-1920, 2001. URL: http://dx.doi.org/10.1137/S0097539799349948.
http://dx.doi.org/10.1137/S0097539799349948
Stefan Kratsch and Magnus Wahlström. Two edge modification problems without polynomial kernels. Discrete Optimization, 10(3):193-199, 2013.
Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bulletin of the EATCS, 105:41-72, 2011.
Assaf Natanzon. Complexity and approximation of some graph modification problems. Master’s thesis, Department of Computer Science, Tel Aviv University, 1999.
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On Approximating Target Set Selection
We study the Target Set Selection (TSS) problem introduced by Kempe, Kleinberg, and Tardos (2003). This problem models the propagation of influence in a network, in a sequence of rounds. A set of nodes is made "active" initially. In each subsequent round, a vertex is activated if at least a certain number of its neighbors are (already) active. In the minimization version, the goal is to activate a small set of vertices initially - a seed, or target, set - so that activation spreads to the entire graph. In the absence of a sublinear-factor algorithm for the general version, we provide a (sublinear) approximation algorithm for the bounded-round version, where the goal is to activate all the vertices in r rounds. Assuming a known conjecture on the hardness of Planted Dense Subgraph, we establish hardness-of-approximation results for the bounded-round version. We show that they translate to general Target Set Selection, leading to a hardness factor of n^(1/2-epsilon) for all epsilon > 0. This is the first polynomial hardness result for Target Set Selection, and the strongest conditional result known for a large class of monotone satisfiability problems. In the maximization version of TSS, the goal is to pick a target set of size k so as to maximize the number of nodes eventually active. We show an n^(1-epsilon) hardness result for the undirected maximization version of the problem, thus establishing that the undirected case is as hard as the directed case. Finally, we demonstrate an SETH lower bound for the exact computation of the optimal seed set.
target set selection
influence propagation
approximation algorithms
hardness of approximation
planted dense subgraph
4:1-4:16
Regular Paper
Moses
Charikar
Moses Charikar
Yonatan
Naamad
Yonatan Naamad
Anthony
Wirth
Anthony Wirth
10.4230/LIPIcs.APPROX-RANDOM.2016.4
Michael Alekhnovich, Sam Buss, Shlomo Moran, and Toniann Pitassi. Minimum propositional proof length is NP-hard to linearly approximate. The Journal of Symbolic Logic, 66(01):171-191, 2001.
Benny Applebaum, Boaz Barak, and Avi Wigderson. Public-key cryptography from different assumptions. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), pages 171-180. ACM, 2010.
Sanjeev Arora, Boaz Barak, Markus Brunnermeier, and Rong Ge. Computational complexity and information asymmetry in financial products. In Proceedings of the Innovations in (Theoretical) Computer Science Conference (ICS), pages 49-65, 2010.
Pranjal Awasthi, Moses Charikar, Kevin A. Lai, and Andrej Risteski. Label optimal regret bounds for online local learning. In Proceedings of the 28th Conference on Learning Theory (COLT), pages 150-166, 2015.
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.
Aditya Bhaskara, Moses Charikar, Eden Chlamtac, Uriel Feige, and Aravindan Vijayaraghavan. Detecting high log-densities: an O(n^1/4) approximation for densest k-subgraph. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), pages 201-210. ACM, 2010.
Ning Chen. On the approximability of influence in social networks. SIAM Journal on Discrete Mathematics, 23(3):1400-1415, 2009.
Ramkumar Chinchani, Duc Ha, Anusha Iyer, Hung Q Ngo, and Shambhu Upadhyaya. On the hardness of approximating the min-hack problem. Journal of Combinatorial Optimization, 9(3):295-311, 2005.
Eden Chlamtac, Michael Dinitz, and Robert Krauthgamer. Everywhere-sparse spanners via dense subgraphs. In Proceedings of the 53rd Annual Symposium on Foundations of Computer Science (FOCS), pages 758-767. IEEE, 2012.
Ferdinando Cicalese, Gennaro Cordasco, Luisa Gargano, Martin Milanič, Joseph Peters, and Ugo Vaccaro. Spread of influence in weighted networks under time and budget constraints. Theoretical Computer Science, 586:40-58, 2015.
Ferdinando Cicalese, Gennaro Cordasco, Luisa Gargano, Martin Milanič, and Ugo Vaccaro. Latency-bounded target set selection in social networks. Theoretical Computer Science, 535:1-15, 2014.
Amin Coja-Oghlan, Uriel Feige, Michael Krivelevich, and Daniel Reichman. Contagious sets in expanders. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1953-1987. SIAM, 2015.
Irit Dinur and Shmuel Safra. On the hardness of approximating label-cover. Information Processing Letters, 89(5):247-254, 2004.
Pedro Domingos and Matt Richardson. Mining the network value of customers. In Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 57-66. ACM, 2001.
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Oded Goldreich and Shafi Goldwasser. On the possibility of basing cryptography on the assumption that P≠NP., 1998.
Michael Goldwasser and Rajeev Motwani. Intractability of assembly sequencing: Unit disks in the plane. Springer, 1997.
David Kempe, Jon Kleinberg, and Éva Tardos. Maximizing the spread of influence through a social network. In Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 137-146. ACM, 2003.
Mihai Pătraşcu and Ryan Williams. On the possibility of faster sat algorithms. In Proceedings of the 21st Annual ACM-SIAM symposium on Discrete Algorithms (SODA), pages 1065-1075. Society for Industrial and Applied Mathematics, 2010.
Matthew Richardson and Pedro Domingos. Mining knowledge-sharing sites for viral marketing. In Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 61-70. ACM, 2002.
Christopher Umans. Hardness of approximating Σ₂^p minimization problems. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS), pages 465-474. IEEE, 1999.
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Approximation Algorithms for Parallel Machine Scheduling with Speed-up Resources
We consider the problem of scheduling with renewable speed-up resources. Given m identical machines, n jobs and c different discrete resources, the task is to schedule each job non-preemptively onto one of the machines so as to minimize the makespan. In our problem, a job has its original processing time, which could be reduced by utilizing one of the resources. As resources are different, the amount of the time reduced for each job is different depending on the resource it uses. Once a resource is being used by one job, it can not be used simultaneously by any other job until this job is finished, hence the scheduler should take into account the job-to-machine assignment together with the resource-to-job assignment.
We observe that, the classical unrelated machine scheduling problem is actually a special case of our problem when m=c, i.e., the number of resources equals the number of machines. Extending the techniques for the unrelated machine scheduling, we give a 2-approximation algorithm when both m and c are part of the input. We then consider two special cases for the problem, with m or c being a constant, and derive PTASes (Polynomial Time Approximation Schemes) respectively. We also establish the relationship between the two parameters m and c, through which we are able to transform the PTAS for the case when m is constant to the case when c is a constant. The relationship between the two parameters reveals the structure within the problem, and may be of independent interest.
approximation algorithms
scheduling
linear programming
5:1-5:12
Regular Paper
Lin
Chen
Lin Chen
Deshi
Ye
Deshi Ye
Guochuan
Zhang
Guochuan Zhang
10.4230/LIPIcs.APPROX-RANDOM.2016.5
N. Alon, Y. Azar, G.J. Woeginger, and T. Yadid. Approximation schemes for scheduling. In 8th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'97), pages 493-500, 1997. URL: http://dx.doi.org/10.1109/SFCS.1975.23.
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J. Chen and A. Miranda. A polynomial time approximation scheme for general multiprocessor job scheduling. SIAM journal on computing, 31(1):1-17, 2001. URL: http://dx.doi.org/10.1145/361604.361612.
http://dx.doi.org/10.1145/361604.361612
A. Grigoriev, M. Sviridenko, and M. Uetz. Machine scheduling with resource dependent processing times. Mathematical programming, 110(1):209-228, 2007. URL: http://dx.doi.org/10.1145/361604.361612.
http://dx.doi.org/10.1145/361604.361612
K. Jansen, M. Maack, and M. Rau. Approximation schemes for machine scheduling with resource (in-)dependent processing times. In 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1526-1542, 2016. URL: http://dx.doi.org/10.1109/SFCS.1975.23.
http://dx.doi.org/10.1109/SFCS.1975.23
K. Jansen and M. Mastrolilli. Scheduling unrelated parallelmachines: linear programming strikes back. Technical report, University of Kiel, 2010. Technical Report Bericht-Nr. 1004. URL: http://dx.doi.org/10.1109/SFCS.1975.23.
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K. Jansen and L. Porkolab. General multiprocessor task scheduling: Approximate solutions in linear time. In Workshop on Algorithms and Data Structures (WADS'99), pages 110-121, 1999. URL: http://dx.doi.org/10.1109/SFCS.1975.23.
http://dx.doi.org/10.1109/SFCS.1975.23
K. Jansen and L. Porkolab. Improved Approximation Schemes for Scheduling Unrelated Parallel Machines. Mathematics of Operations Research, 26(2):324-338, 2001. URL: http://dx.doi.org/10.1145/361604.361612.
http://dx.doi.org/10.1145/361604.361612
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http://dx.doi.org/10.1145/361604.361612
H. Kellerer and V.A. Strusevich. Scheduling parallel dedicated machines with the speeding-up resource. Naval Research Logistics, 55(5):377-389, 2008. URL: http://dx.doi.org/10.1145/361604.361612.
http://dx.doi.org/10.1145/361604.361612
J. K. Lenstra, D. B. Shmoys, and Eva Tardos. Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programing, 46:259-271, 1990. URL: http://dx.doi.org/10.1145/361604.361612.
http://dx.doi.org/10.1145/361604.361612
H. Xu, L. Chen, D. Ye, and G. Zhang. Scheduling on two identical machines with a speed-up resource. Information Processing Letters, 111(7):831-835, 2011. URL: http://dx.doi.org/10.1145/361604.361612.
http://dx.doi.org/10.1145/361604.361612
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The Densest k-Subhypergraph Problem
The Densest k-Subgraph (DkS) problem, and its corresponding minimization problem Smallest p-Edge Subgraph (SpES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out).
In this paper we generalize both DkS and SpES from graphs to hypergraphs. We consider the Densest k-Subhypergraph problem (given a hypergraph (V, E), find a subset W subseteq V of k vertices so as to maximize the number of hyperedges contained in W) and define the Minimum p-Union problem (given a hypergraph, choose p of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an O(n^{4(4-sqrt{3})/13 + epsilon}) <= O(n^{0.697831+epsilon})-approximation (for arbitrary constant epsilon > 0) for Densest k-Subhypergraph and an ~O(n^{2/5})-approximation for Minimum p-Union. We also give an O(sqrt{m})-approximation for Minimum p-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.
Hypergraphs
Approximation algorithms
6:1-6:19
Regular Paper
Eden
Chlamtac
Eden Chlamtac
Michael
Dinitz
Michael Dinitz
Christian
Konrad
Christian Konrad
Guy
Kortsarz
Guy Kortsarz
George
Rabanca
George Rabanca
10.4230/LIPIcs.APPROX-RANDOM.2016.6
Noga Alon, Sanjeev Arora, Rajsekar Manokaran, Dana Moshkovitz, and Omri Weinstein. Inapproximability of densest k-subgraph from average case hardness. Unpublished manuscript, 2011.
Benny Applebaum. Pseudorandom generators with long stretch and low locality from random local one-way functions. SIAM Journal on Computing, 42(5):2008-2037, 2013. URL: http://dx.doi.org/10.1137/120884857.
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Benny Applebaum, Boaz Barak, and Avi Wigderson. Public-key cryptography from different assumptions. In Proceedings of the Forty-second ACM Symposium on Theory of Computing, STOC'10, pages 171-180, 2010.
S. Arora, B. Barak, M. Brunnermeier, and R. Ge. Complicational complexity and information asymmetry in finnancial products. Submitted, 2016.
Sanjeev Arora and Rong Ge. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: 14th International Workshop, APPROX 2011, and 15th International Workshop, RANDOM 2011, Princeton, NJ, USA, August 17-19, 2011. Proceedings, chapter New Tools for Graph Coloring, pages 1-12. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22935-0_1.
http://dx.doi.org/10.1007/978-3-642-22935-0_1
Aditya Bhaskara. Finding Dense Structures in Graphs and Matrices. PhD thesis, Princeton University, 2012.
Aditya Bhaskara, Moses Charikar, Eden Chlamtac, Uriel Feige, and Aravindan Vijayaraghavan. Detecting high log-densities: an O(n^1/4) approximation for densest k-subgraph. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 201-210, 2010.
Moses Charikar. Greedy approximation algorithms for finding dense components in a graph. In Approximation Algorithms for Combinatorial Optimization, Third International Workshop, APPROX 2000, Saarbrücken, Germany, September 5-8, 2000, Proceedings, pages 84-95, 2000. URL: http://dx.doi.org/10.1007/3-540-44436-X_10.
http://dx.doi.org/10.1007/3-540-44436-X_10
Eden Chlamtac, Michael Dinitz, and Robert Krauthgamer. Everywhere-sparse spanners via dense subgraphs. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 758-767, 2012.
Julia Chuzhoy, Yury Makarychev, Aravindan Vijayaraghavan, and Yuan Zhou. Approximation algorithms and hardness of the k-route cut problem. ACM Trans. Algorithms, 12(1):2:1-2:40, December 2015. URL: http://dx.doi.org/10.1145/2644814.
http://dx.doi.org/10.1145/2644814
Michael Dinitz, Guy Kortsarz, and Zeev Nutov. Improved Approximation Algorithm for Steiner k-Forest with Nearly Uniform Weights. In Klaus Jansen, José D. P. Rolim, Nikhil R. Devanur, and Cristopher Moore, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014), volume 28 of Leibniz International Proceedings in Informatics (LIPIcs), pages 115-127, Dagstuhl, Germany, 2014. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://drops.dagstuhl.de/opus/volltexte/2014/4692, URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.115.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.115
Uriel Feige. Relations between average case complexity and approximation complexity. In Proceedings of the Thiry-fourth Annual ACM Symposium on Theory of Computing, STOC'02, pages 534-543, New York, NY, USA, 2002. ACM. URL: http://dx.doi.org/10.1145/509907.509985.
http://dx.doi.org/10.1145/509907.509985
Uriel Feige, Guy Kortsarz, and David Peleg. The dense k-subgraph problem. Algorithmica, 29(3):410-421, 2001.
Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 468-485, 2012.
Anupam Gupta, Mohammadtaghi Hajiaghayi, Viswanath Nagarajan, and R. Ravi. Dial a ride from k-forest. ACM Trans. Algorithms, 6(2):41:1-41:21, April 2010. URL: http://dx.doi.org/10.1145/1721837.1721857.
http://dx.doi.org/10.1145/1721837.1721857
Subhash Khot. Ruling out ptas for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM Journal on Computing, 36(4):1025-1071, 2006. URL: http://dx.doi.org/10.1137/S0097539705447037.
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Christian Konrad and Adi Rosén. Approximating semi-matchings in streaming and in two-party communication. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 637-649, 2013.
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Anand Louis and Yury Makarychev. Approximation Algorithms for Hypergraph Small Set Expansion and Small Set Vertex Expansion. In Klaus Jansen, José D. P. Rolim, Nikhil R. Devanur, and Cristopher Moore, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014), volume 28 of Leibniz International Proceedings in Informatics (LIPIcs), pages 339-355, Dagstuhl, Germany, 2014. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://drops.dagstuhl.de/opus/volltexte/2014/4707, URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.339.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.339
Anand Louis, Prasad Raghavendra, and Santosh Vempala. The complexity of approximating vertex expansion. 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 0:360-369, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.46.
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Yuru Makarychev. Personal communication.
Zeev Nutov. Approximating steiner networks with node-weights. SIAM Journal on Computing, 39(7):3001-3022, 2010. URL: http://dx.doi.org/10.1137/080729645.
http://dx.doi.org/10.1137/080729645
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Online Row Sampling
Finding a small spectral approximation for a tall n x d matrix A is a fundamental numerical primitive. For a number of reasons, one often seeks an approximation whose rows are sampled from those of A. Row sampling improves interpretability, saves space when A is sparse, and preserves row structure, which is especially important, for example, when A represents a graph.
However, correctly sampling rows from A can be costly when the matrix is large and cannot be stored and processed in memory. Hence, a number of recent publications focus on row sampling in the streaming setting, using little more space than what is required to store the outputted approximation [Kelner Levin 2013] [Kapralov et al. 2014].
Inspired by a growing body of work on online algorithms for machine learning and data analysis, we extend this work to a more restrictive online setting: we read rows of A one by one and immediately decide whether each row should be kept in the spectral approximation or discarded, without ever retracting these decisions. We present an extremely simple algorithm that approximates A up to multiplicative error epsilon and additive error delta using O(d log d log (epsilon ||A||_2^2/delta) / epsilon^2) online samples, with memory overhead proportional to the cost of storing the spectral approximation. We also present an algorithm that uses O(d^2) memory but only requires O(d log (epsilon ||A||_2^2/delta) / epsilon^2) samples, which we show is optimal.
Our methods are clean and intuitive, allow for lower memory usage than prior work, and expose new theoretical properties of leverage score based matrix approximation.
spectral sparsification
leverage score sampling
online sparsification
7:1-7:18
Regular Paper
Michael B.
Cohen
Michael B. Cohen
Cameron
Musco
Cameron Musco
Jakub
Pachocki
Jakub Pachocki
10.4230/LIPIcs.APPROX-RANDOM.2016.7
Ahmed Alaoui and Michael W Mahoney. Fast randomized kernel ridge regression with statistical guarantees. In Advances in Neural Information Processing Systems \intcalcSub20151987 (NIPS), pages 775-783, 2015.
Joshua Batson, Daniel A Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM Journal on Computing, 41(6):1704-1721, 2012.
Antoine Bordes and Léon Bottou. The huller: a simple and efficient online SVM. In Machine Learning: ECML 2005, pages 505-512. Springer, 2005.
Christos Boutsidis, Dan Garber, Zohar Karnin, and Edo Liberty. Online principal components analysis. In Proceedings of the \nth\intcalcSub20151989 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 887-901, 2015.
Christos Boutsidis and David P Woodruff. Optimal CUR matrix decompositions. In Proceedings of the \nth\intcalcSub20141968 Annual ACM Symposium on Theory of Computing (STOC), pages 353-362, 2014.
Kenneth L. Clarkson and David P. Woodruff. Low rank approximation and regression in input sparsity time. In Proceedings of the \nth\intcalcSub20131968 Annual ACM Symposium on Theory of Computing (STOC), pages 81-90, 2013.
Michael B Cohen, Sam Elder, Cameron Musco, Christopher Musco, and Madalina Persu. Dimensionality reduction for k-means clustering and low rank approximation. In Proceedings of the \nth\intcalcSub20151968 Annual ACM Symposium on Theory of Computing (STOC), pages 163-172, 2015.
Michael B Cohen, Yin Tat Lee, Cameron Musco, Christopher Musco, Richard Peng, and Aaron Sidford. Uniform sampling for matrix approximation. In Proceedings of the \nth\intcalcSub20152009 Conference on Innovations in Theoretical Computer Science (ITCS), pages 181-190, 2015.
Michael B Cohen, Cameron Musco, and Christopher Musco. Ridge leverage scores for low-rank approximation. http://arxiv.org/abs/1511.07263, 2015.
Koby Crammer, Ofer Dekel, Joseph Keshet, Shai Shalev-Shwartz, and Yoram Singer. Online passive-aggressive algorithms. The Journal of Machine Learning Research, 7:551-585, 2006.
Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. In Proceedings of the \nth\intcalcSub20141959 Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 561-570, 2014.
Jonathan A Kelner and Alex Levin. Spectral sparsification in the semi-streaming setting. Theory of Computing Systems, 53(2):243-262, 2013.
Ioannis Koutis, Gary L Miller, and Richard Peng. Approaching optimality for solving SDD linear systems. In Proceedings of the \nth\intcalcSub20101959 Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 235-244, 2010.
Yin Tat Lee and He Sun. Constructing linear-sized spectral sparsification in almost-linear time. In Proceedings of the \nth\intcalcSub20151959 Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 250-269, 2015.
Mu Li, Gary L Miller, and Richard Peng. Iterative row sampling. In Proceedings of the \nth\intcalcSub20131959 Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 127-136, 2013.
Edo Liberty, Ram Sriharsha, and Maxim Sviridenko. An algorithm for online k-means clustering. In Proceedings of the Eighteenth Workshop on Algorithm Engineering and Experiments (ALENEX), pages 81-89, 2016.
Michael W. Mahoney and Xiangrui Meng. Low-distortion subspace embeddings in input-sparsity time and applications to robust linear regression. In Proceedings of the \nth\intcalcSub20131968 Annual ACM Symposium on Theory of Computing (STOC), pages 91-100, 2013.
Jelani Nelson and Huy L. Nguyen. OSNAP: Faster numerical linear algebra algorithms via sparser subspace embeddings. In Proceedings of the \nth\intcalcSub20131959 Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 117-126, 2013.
Daniel A Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. SIAM Journal on Computing, 40(6):1913-1926, 2011.
Daniel A Spielman and Shang-Hua Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the \nth\intcalcSub20041968 Annual ACM Symposium on Theory of Computing (STOC), pages 81-90, 2004.
Daniel A Spielman and Shang-Hua Teng. Nearly linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. SIAM Journal on Matrix Analysis and Applications, 35(3):835-885, 2014.
Joel Tropp. Freedman’s inequality for matrix martingales. Electronic Communications in Probability, 16:262-270, 2011.
Creative Commons Attribution 3.0 Unported license
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Oblivious Rounding and the Integrality Gap
The following paradigm is often used for handling NP-hard combinatorial optimization problems. One first formulates the problem as an integer program, then one relaxes it to a linear program (LP, or more generally, a convex program), then one solves the LP relaxation in polynomial time, and finally one rounds the optimal LP solution, obtaining a feasible solution to the original problem. Many of the commonly used rounding schemes (such as randomized rounding, threshold rounding and others) are "oblivious" in the sense that the rounding is performed based on the LP solution alone, disregarding the objective function. The goal of our work is to better understand in which cases oblivious rounding suffices in order to obtain approximation ratios that match the integrality gap of the underlying LP. Our study is information theoretic - the rounding is restricted to be oblivious but not restricted to run in polynomial time. In this information theoretic setting we characterize the approximation ratio achievable by oblivious rounding. It turns out to equal the integrality gap of the underlying LP on a problem that is the closure of the original combinatorial optimization problem. We apply our findings to the study of the approximation ratios obtainable by oblivious rounding for the maximum welfare problem, showing that when valuation functions are submodular oblivious rounding can match the integrality gap of the configuration LP (though we do not know what this integrality gap is), but when valuation functions are gross substitutes oblivious rounding cannot match the integrality gap (which is 1).
Welfare-maximization
8:1-8:23
Regular Paper
Uriel
Feige
Uriel Feige
Michal
Feldman
Michal Feldman
Inbal
Talgam-Cohen
Inbal Talgam-Cohen
10.4230/LIPIcs.APPROX-RANDOM.2016.8
Ashwinkumar Badanidiyuru, Shahar Dobzinski, Hu Fu, Robert Kleinberg, Noam Nisan, and Tim Roughgarden. Sketching valuation functions. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1025-1035, 2012.
Sushil Bikhchandani and John W. Mamer. Competitive equilibrium in an exchange economy with indivisibilities. Journal of Economic Theory, 74(2):385-413, 1997.
Robert Carr and Santosh Vempala. Randomized metarounding. Random Structures and Algorithms, 20(3):343-352, 2002.
Deeparnab Chakrabarty and Gagan Goel. On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and GAP. SIAM J. Comput., 39(6):2189-2211, 2010.
Nan Du, Yingyu Liang, Maria-Florina Balcan, and Le Song. Learning time-varying coverage functions. In Proceedings of the 27th Neural Information Processing Systems Conference, pages 3374-3382, 2014.
Shaddin Dughmi, Tim Roughgarden, and Qiqi Yan. From convex optimization to randomized mechanisms: Toward optimal combinatorial auctions. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, pages 149-158, 2011.
Paul Dütting, Thomas Kesselheim, and Éva Tardos. Algorithms as mechanisms: The price of anarchy of relax-and-round. In Proceedings of the 16th ACM Conference on Economics and Computation, pages 187-201, 2015.
Uriel Feige. On maximizing welfare when utility functions are subadditive. SIAM J. Comput., 39(1):122-142, 2009.
Uriel Feige and Shlomo Jozeph. Oblivious algorithms for the maximum directed cut problem. Algorithmica, 71(2):409-428, 2015.
Uriel Feige and Jan Vondrák. The submodular welfare problem with demand queries. Theory of Computing, 6(1):247-290, 2010.
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Avinatan Hassidim and Yaron Singer. Submodular optimization under noise. Manuscript, 2016.
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Kazuo Murota. Discrete Convex Analysis. Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics, 2003.
Noam Nisan and Ilya Segal. The communication requirements of efficient allocations and supporting prices. Journal of Economic Theory, 129:192-224, 2006.
Renato Paes Leme. Gross substitutability: An algorithmic survey. Working paper, 2014.
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Prasad Raghavendra and David Steurer. How to round any CSP. In Proceedings of the 50th Symposium on Foundations of Computer Science, pages 586-594, 2009.
Tim Roughgarden, Inbal Talgam-Cohen, and Jan Vondràk. When are welfare guarantees robust? Working paper, 2016.
Aravind Srinivasan. Budgeted allocations in the full-information setting. In Proceedings of the 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pages 247-253, 2008.
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A Deterministic Fully Polynomial Time Approximation Scheme For Counting Integer Knapsack Solutions Made Easy
Given n elements with nonnegative integer weights w=(w_1,...,w_n), an integer capacity C and positive integer ranges u=(u_1,...,u_n), we consider the counting version of the classic integer knapsack problem: find the number of distinct multisets whose weights add up to at most C. We give a deterministic algorithm that estimates the number of solutions to within relative error epsilon in time polynomial in n, log U and 1/epsilon, where U=max_i u_i. More precisely, our algorithm runs in O((n^3 log^2 U)/epsilon) log (n log U)/epsilon) time. This is an improvement of n^2 and 1/epsilon (up to log terms) over the best known deterministic algorithm by Gopalan et al. [FOCS, (2011), pp. 817-826]. Our algorithm is relatively simple, and its analysis is rather elementary. Our results are achieved by means of a careful formulation of the problem as a dynamic program, using the notion of binding constraints.
Approximate counting
integer knapsack
dynamic programming
bounding constraints
$K$-approximating sets and functions
9:1-9:11
Regular Paper
Nir
Halman
Nir Halman
10.4230/LIPIcs.APPROX-RANDOM.2016.9
M. E. Dyer. Approximate counting by dynamic programming. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), June 9-11, 2003, San Diego, CA, USA, pages 693-699, 2003.
P. Gopalan, A. Klivans, and R. Meka. Polynomial-time approximation schemes for Knapsack and related counting problems using branching programs. CoRR, abs/1008.3187, 2010.
P. Gopalan, A. Klivans, R. Meka, D. Štefankovič, S. Vempala, and E. Vigoda. An FPTAS for #Knapsack and related counting problems. In IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pages 817-826, 2011.
N. Halman, D. Klabjan, C.-L. Li, J. Orlin, and D. Simchi-Levi. Fully polynomial time approximation schemes for stochastic dynamic programs. SIAM Journal on Discrete Mathematics, 28:1725-1796, 2014.
N. Halman, D. Klabjan, M. Mostagir, J. Orlin, and D. Simchi-Levi. A fully polynomial time approximation scheme for single-item stochastic inventory control with discrete demand. Mathematics of Operations Research, 34:674-685, 2009.
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D. Štefankovič, S. Vempala, and E. Vigoda. A deterministic polynomial-time approximation scheme for counting knapsack solutions. SIAM Journal on Computing, 41:356-366, 2012.
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A Competitive Flow Time Algorithm for Heterogeneous Clusters Under Polytope Constraints
Modern data centers consist of a large number of heterogeneous resources such as CPU, memory, network bandwidth, etc. The resources are pooled into clusters for various reasons such as scalability, resource consolidation, and privacy. Clusters are often heterogeneous so that they can better serve jobs with different characteristics submitted from clients. Each job benefits differently depending on how much resource is allocated to the job, which in turn translates to how quickly the job gets completed.
In this paper, we formulate this setting, which we term Multi-Cluster Polytope Scheduling (MCPS). In MCPS, a set of n jobs arrive over time to be executed on m clusters. Each cluster i is associated with a polytope P_i, which constrains how fast one can process jobs assigned to the cluster. For MCPS, we seek to optimize the popular objective of minimizing average weighted flow time of jobs in the online setting. We give a constant competitive algorithm with small constant resource augmentation for a large class of polytopes, which capture many interesting problems that arise in practice. Further, our algorithm is non-clairvoyant. Our algorithm and analysis combine and generalize techniques developed in the recent results for the classical unrelated machines scheduling and the polytope scheduling problem [10,12,11].
Polytope constraints
average flow time
multi-clusters
online scheduling
and competitive analysis
10:1-10:15
Regular Paper
Sungjin
Im
Sungjin Im
Janardhan
Kulkarni
Janardhan Kulkarni
Benjamin
Moseley
Benjamin Moseley
Kamesh
Munagala
Kamesh Munagala
10.4230/LIPIcs.APPROX-RANDOM.2016.10
Faraz Ahmad, Srimat T. Chakradhar, Anand Raghunathan, and T. N. Vijaykumar. Tarazu: optimizing mapreduce on heterogeneous clusters. In ASPLOS, pages 61-74. ACM, 2012. URL: http://dx.doi.org/10.1145/2150976.2150984.
http://dx.doi.org/10.1145/2150976.2150984
Amazon EC2-Spot-Instances. URL: http://aws.amazon.com/ec2/spot-instances/.
http://aws.amazon.com/ec2/spot-instances/
Jivitej S. Chadha, Naveen Garg, Amit Kumar, and V. N. Muralidhara. A competitive algorithm for minimizing weighted flow time on unrelated machines with speed augmentation. In STOC, pages 679-684, 2009.
R. Cole, V. Gkatzelis, and G. Goel. Mechanism design for fair division: allocating divisible items without payments. In ACM EC, pages 251-268, 2013.
Jeff Edmonds and Kirk Pruhs. Scalably scheduling processes with arbitrary speedup curves. In ACM-SIAM Symposium on Discrete Algorithms, pages 685-692, 2009.
Kyle Fox, Sungjin Im, and Benjamin Moseley. Energy efficient scheduling of parallelizable jobs. In SODA, pages 948-957, 2013.
N. Garg and A. Kumar. Better algorithms for minimizing average flow-time on related machines. In ICALP (1), 2006.
A. Ghodsi, M. Zaharia, B. Hindman, A. Konwinski, I. Stoica, and S. Shenker. Dominant resource fairness: Fair allocation of multiple resource types. In NSDI, 2011.
Robert Grandl, Ganesh Ananthanarayanan, Srikanth Kandula, Sriram Rao, and Aditya Akella. Multi-resource packing for cluster schedulers. In ACM SIGCOMM 2014 Conference, SIGCOMM'14, Chicago, IL, USA, August 17-22, 2014, pages 455-466, 2014. URL: http://dx.doi.org/10.1145/2619239.2626334.
http://dx.doi.org/10.1145/2619239.2626334
Sungjin Im, Janardhan Kulkarni, and Kamesh Munagala. Competitive algorithms from competitive equilibria: Non-clairvoyant scheduling under polyhedral constraints. In STOC, 2014.
Sungjin Im, Janardhan Kulkarni, and Kamesh Munagala. Competitive flow time algorithms for polyhedral scheduling. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 506-524, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.38.
http://dx.doi.org/10.1109/FOCS.2015.38
Sungjin Im, Janardhan Kulkarni, Kamesh Munagala, and Kirk Pruhs. Selfishmigrate: A scalable algorithm for non-clairvoyantly scheduling heterogeneous processors. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 531-540, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.63.
http://dx.doi.org/10.1109/FOCS.2014.63
Sungjin Im, Benjamin Moseley, and Kirk Pruhs. A tutorial on amortized local competitiveness in online scheduling. SIGACT News, 42(2):83-97, 2011. URL: http://dx.doi.org/10.1145/1998037.1998058.
http://dx.doi.org/10.1145/1998037.1998058
Bala Kalyanasundaram and Kirk Pruhs. Speed is as powerful as clairvoyance. JACM, 47(4):617-643, 2000.
Gunho Lee, Byung-Gon Chun, and Randy H Katz. Heterogeneity-aware resource allocation and scheduling in the cloud. In Proceedings of the 3rd USENIX Workshop on Hot Topics in Cloud Computing, HotCloud, volume 11, 2011.
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Lucian Popa, Gautam Kumar, Mosharaf Chowdhury, Arvind Krishnamurthy, Sylvia Ratnasamy, and Ion Stoica. Faircloud: sharing the network in cloud computing. In ACM SIGCOMM, pages 187-198, 2012.
Matei Zaharia, Andy Konwinski, Anthony D. Joseph, Randy Katz, and Ion Stoica. Improving mapreduce performance in heterogeneous environments. In OSDI, pages 29-42, Berkeley, CA, USA, 2008. USENIX Association. URL: http://dl.acm.org/citation.cfm?id=1855741.1855744.
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https://creativecommons.org/licenses/by/3.0/legalcode
Revisiting Connected Dominating Sets: An Optimal Local Algorithm?
In this paper we consider the classical Connected Dominating Set (CDS) problem. Twenty years ago, Guha and Khuller developed two algorithms for this problem - a centralized greedy approach with an approximation guarantee of H(D) +2, and a local greedy approach with an approximation guarantee of 2(H(D)+1) (where H() is the harmonic function, and D is the maximum degree in the graph). A local greedy algorithm uses significantly less information about the graph, and can be useful in a variety of contexts. However, a fundamental question remained - can we get a local greedy algorithm with the same performance guarantee as the global greedy algorithm without the penalty of the multiplicative factor of "2" in the approximation factor? In this paper, we answer that question in the affirmative.
graph algorithms
approximation algorithms
dominating sets
local information algorithms
11:1-11:12
Regular Paper
Samir
Khuller
Samir Khuller
Sheng
Yang
Sheng Yang
10.4230/LIPIcs.APPROX-RANDOM.2016.11
Konstantin Avrachenkov, Prithwish Basu, Giovanni Neglia, Bernardete Ribeiro, and Don Towsley. Pay few, influence most: Online myopic network covering. In Computer Communications Workshops (INFOCOM WKSHPS), 2014 IEEE Conference on, pages 813-818. IEEE, 2014.
Christian Borgs, Michael Brautbar, Jennifer Chayes, Sanjeev Khanna, and Brendan Lucier. The power of local information in social networks. In Internet and Network Economics, pages 406-419. Springer, 2012.
Ding-Zhu Du and Peng-Jun Wan. Connected dominating set: theory and applications, volume 77. Springer Science &Business Media, 2012.
Devdatt Dubhashi, Alessandro Mei, Alessandro Panconesi, Jaikumar Radhakrishnan, and Aravind Srinivasan. Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons. In Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, pages 717-724. Society for Industrial and Applied Mathematics, 2003.
Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM (JACM), 45(4):634-652, 1998.
Sudipto Guha and Samir Khuller. Approximation algorithms for connected dominating sets. Algorithmica, 20(4):374-387, 1998.
Sudipto Guha and Samir Khuller. Improved methods for approximating node weighted steiner trees and connected dominating sets. Information and computation, 150(1):57-74, 1999.
Samir Khuller, Manish Purohit, and Kanthi K Sarpatwar. Analyzing the optimal neighborhood: algorithms for budgeted and partial connected dominating set problems. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1702-1713. SIAM, 2014.
Yuzhen Liu and Weifa Liang. Approximate coverage in wireless sensor networks. In Local Computer Networks, 2005. 30th Anniversary. The IEEE Conference on, pages 68-75. IEEE, 2005.
Adish Singla, Eric Horvitz, Pushmeet Kohli, Ryen White, and Andreas Krause. Information gathering in networks via active exploration. In Proceedings of the 24th International Conference on Artificial Intelligence, pages 981-988. AAAI Press, 2015.
Creative Commons Attribution 3.0 Unported license
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Online Energy Storage Management: an Algorithmic Approach
Motivated by the importance of energy storage networks in smart grids, we provide an algorithmic study of the online energy storage management problem in a network setting, the first to the best of our knowledge. Given online power supplies, either entirely renewable supplies or those in combination with traditional supplies, we want to route power from the supplies to demands using storage units subject to a decay factor. Our goal is to maximize the total utility of satisfied demands less the total production cost of routed power. We model renewable supplies with the zero production cost function and traditional supplies with convex production cost functions. For two natural storage unit settings, private and public, we design poly-logarithmic competitive algorithms in the network flow model using the dual fitting and online primal dual methods for convex problems. Furthermore, we show strong hardness results for more general settings of the problem. Our techniques may be of independent interest in other routing and storage management problems.
Online Algorithms
Competitive Analysis
Routing
Storage
Approximation Algorithms
Power Control
12:1-12:23
Regular Paper
Anthony
Kim
Anthony Kim
Vahid
Liaghat
Vahid Liaghat
Junjie
Qin
Junjie Qin
Amin
Saberi
Amin Saberi
10.4230/LIPIcs.APPROX-RANDOM.2016.12
Rajeev Alur, Sampath Kannan, Kevin Tian, and Yifei Yuan. On the complexity of shortest path problems on discounted cost graphs. In Proceedings of the 7th International Conference on Language and Automata Theory and Applications, LATA'13, pages 44-55, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg. URL: http://dx.doi.org/10.1007/978-3-642-37064-9_6.
http://dx.doi.org/10.1007/978-3-642-37064-9_6
James Aspnes, Yossi Azar, Amos Fiat, Serge Plotkin, and Orli Waarts. On-line routing of virtual circuits with applications to load balancing and machine scheduling. J. ACM, 44(3):486-504, May 1997. URL: http://dx.doi.org/10.1145/258128.258201.
http://dx.doi.org/10.1145/258128.258201
B. Awerbuch, Y. Azar, and S. Plotkin. Throughput-competitive online routing. In Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, FOCS'93, pages 32-40, 1993.
Yossi Azar, Ilan Reuven Cohen, and Debmalya Panigrahi. Online covering with convex objectives and applications. arXiv:1412.3507, Dec 2014. URL: http://arxiv.org/abs/1412.3507.
http://arxiv.org/abs/1412.3507
Moshe Babaioff, Michael Dinitz, Anupam Gupta, Nicole Immorlica, and Kunal Talwar. Secretary problems: Weights and discounts. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'09, pages 1245-1254, Philadelphia, PA, USA, 2009. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=1496770.1496905.
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M.E. Baran and F.F. Wu. Network reconfiguration in distribution systems for loss reduction and load balancing. IEEE Transactions on Power Delivery, 4(2):1401-1407, apr 1989. URL: http://dx.doi.org/10.1109/61.25627.
http://dx.doi.org/10.1109/61.25627
Marcin Bienkowski, Jaroslaw Byrka, Marek Chrobak, Lukasz Jeż, Dorian Nogneng, and Jiří Sgall. Better approximation bounds for the joint replenishment problem. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'14, pages 42-54. SIAM, 2014. URL: http://dl.acm.org/citation.cfm?id=2634074.2634078.
http://dl.acm.org/citation.cfm?id=2634074.2634078
E. Bitar, R. Rajagopal, P. Khargonekar, and K. Poolla. The Role of Co-Located Storage for Wind Power Producers in Conventional Electricity Markets. In Proc. of American Control Conference (ACC), pages 3886-3891, 2011.
E. Bitar and Yunjian Xu. On incentive compatibility of deadline differentiated pricing for deferrable demand. In Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, pages 5620-5627, Dec 2013. URL: http://dx.doi.org/10.1109/CDC.2013.6760775.
http://dx.doi.org/10.1109/CDC.2013.6760775
N. Buchbinder, T. Kimbrelt, R. Levi, K. Makarychev, and M. Sviridenko. Online make-to-order joint replenishment model: Primal dual competitive algorithms. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'08, pages 952-961, Philadelphia, PA, USA, 2008. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=1347082.1347186.
http://dl.acm.org/citation.cfm?id=1347082.1347186
Niv Buchbinder, Shahar Chen, Anupam Gupta, Viswanath Nagarajan, and Joseph (Seffi) Naor. Online convex covering and packing problems. arXiv:1412.8347, Dec 2014. URL: http://arxiv.org/abs/1412.8347.
http://arxiv.org/abs/1412.8347
Niv Buchbinder and Joseph (Seffi) Naor. Improved bounds for online routing and packing via a primal-dual approach. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS'06, pages 293-304, Washington, DC, USA, 2006. IEEE Computer Society. URL: http://dx.doi.org/10.1109/FOCS.2006.39.
http://dx.doi.org/10.1109/FOCS.2006.39
California Independent System Operator. Annual Report on Market Issues and Performance, 2014. URL: http://www.caiso.com/Documents/2014AnnualReport_MarketIssues_Performance.pdf.
http://www.caiso.com/Documents/2014AnnualReport_MarketIssues_Performance.pdf
T-H. Hubert Chan, Zhiyi Huang, and Ning Kang. Online convex covering and packing problems. arXiv:1502.01802, Apr 2015. URL: http://arxiv.org/abs/1502.01802.
http://arxiv.org/abs/1502.01802
Chi-Kin Chau, Guanglin Zhang, and Minghua Chen. Cost Minimizing Online Algorithms for Energy Storage Management with Worst-case Guarantee. IEEE Transactions on Smart Grid, nov 2015. URL: http://arxiv.org/abs/1511.07559,
http://arxiv.org/abs/1511.07559
In-Koo Cho. Competitive equilibrium in a radial network. RAND Journal of Economics, pages 438-460, 2003.
Paul Denholm, Erik Ela, Brendan Kirby, and Michael Milligan. The role of energy storage with renewable electricity generation. Technical Report NREL/TP-6A2-47187, National Renewable Energy Laboratory, January 2010.
Nikhil R. Devanur and Kamal Jain. Online matching with concave returns. In Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC'12, pages 137-144, New York, NY, USA, 2012. ACM. URL: http://dx.doi.org/10.1145/2213977.2213992.
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European Commission. Energy Roadmap 2050, 2011. URL: https://ec.europa.eu/energy/en/topics/energy-strategy/2050-energy-strategy.
https://ec.europa.eu/energy/en/topics/energy-strategy/2050-energy-strategy
J Duncan Glover, Mulukutla Sarma, and Thomas Overbye. Power System Analysis &Design, Fifth Edition. Cengage Learning, 2012.
L. Huang, J. Walrand, and K. Ramchandran. Optimal Demand Response with Energy Storage Management. In Proc. of IEEE Third International Conference on Smart Grid Communications (SmartGridComm), pages 61-66, 2012. URL: http://dx.doi.org/10.1109/SmartGridComm.2012.6485960.
http://dx.doi.org/10.1109/SmartGridComm.2012.6485960
Zhiyi Huang and Anthony Kim. Welfare maximization with production costs: A primal dual approach. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'15, pages 59-72. SIAM, 2015. URL: http://dl.acm.org/citation.cfm?id=2722129.2722135.
http://dl.acm.org/citation.cfm?id=2722129.2722135
Mark Z. Jacobson, Mark A. Delucchi, Guillaume Bazouin, Zack A. F. Bauer, Christa C. Heavey, Emma Fisher, Sean B. Morris, Diniana J. Y. Piekutowski, Taylor A. Vencill, and Tim W. Yeskoo. 100% clean and renewable wind, water, and sunlight (WWS) all-sector energy roadmaps for the 50 United States. Energy Environ. Sci., 8(7):2093-2117, Jul 2015. URL: http://dx.doi.org/10.1039/C5EE01283J.
http://dx.doi.org/10.1039/C5EE01283J
Thomas Kesselheim, Robert Kleinberg, and Eva Tardos. Smooth online mechanisms: A game-theoretic problem in renewable energy markets. In Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC'15, pages 203-220, New York, NY, USA, 2015. ACM. URL: http://dx.doi.org/10.1145/2764468.2764487.
http://dx.doi.org/10.1145/2764468.2764487
Frank Kreikebaum, Debrup Das, Yi Yang, Frank Lambert, and Deepak Divan. Smart Wires — A distributed, low-cost solution for controlling power flows and monitoring transmission lines. In 2010 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT Europe), pages 1-8. IEEE, oct 2010. URL: http://dx.doi.org/10.1109/ISGTEUROPE.2010.5638853.
http://dx.doi.org/10.1109/ISGTEUROPE.2010.5638853
Retsef Levi, Robin O. Roundy, and David B. Shmoys. Primal-dual algorithms for deterministic inventory problems. Math. Oper. Res., 31(2):267-284, February 2006. URL: http://dx.doi.org/10.1287/moor.1050.0178.
http://dx.doi.org/10.1287/moor.1050.0178
David Lindley. The energy storage problem. Nature, 463(7), January 2010.
S. H. Low. Convex Relaxation of Optimal Power Flow, Part I: Formulations and Equivalence. ArXiv e-prints, May 2014. URL: http://arxiv.org/abs/1405.0766.
http://arxiv.org/abs/1405.0766
S. H. Low. Convex Relaxation of Optimal Power Flow, Part II: Exactness. ArXiv e-prints, May 2014. URL: http://arxiv.org/abs/1405.0766.
http://arxiv.org/abs/1405.0766
Lian Lu, Jinlong Tu, Chi-Kin Chau, Minghua Chen, and Xiaojun Lin. Online energy generation scheduling for microgrids with intermittent energy sources and co-generation. ACM SIGMETRICS Performance Evaluation Review, 41(1):53, jun 2013. URL: http://dl.acm.org/citation.cfm?id=2494232.2465551, URL: http://dx.doi.org/10.1145/2494232.2465551.
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Viswanath Nagarajan and Cong Shi. Approximation algorithms for inventory problems with submodular or routing costs. arXiv:1504.06560, April 2015. URL: http://arxiv.org/abs/1504.06560.
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National Renewable Energy Laboratory. The Value of Energy Storage for Grid Applications, 2013. URL: http://www.nrel.gov/docs/fy13osti/58465.pdf.
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Julia Pahl and Stefan Voß. Integrating deterioration and lifetime constraints in production and supply chain planning: A survey. European Journal of Operational Research, 238(3):654-674, 2014. URL: http://dx.doi.org/10.1016/j.ejor.2014.01.060.
http://dx.doi.org/10.1016/j.ejor.2014.01.060
J. Qin, R. Sevlian, D. Varodayan, and R. Rajagopal. Optimal Electric Energy Storage Operation. In Proc. of IEEE Power and Energy Society General Meeting, pages 1-6, 2012. URL: http://dx.doi.org/10.1109/PESGM.2012.6345242.
http://dx.doi.org/10.1109/PESGM.2012.6345242
J. Qin, H. I. Su, and R. Rajagopal. Storage in Risk Limiting Dispatch: Control and Approximation. In Proc. of American Control Conference (ACC), pages 4202-4208, 2013.
Junjie Qin, Yinlam Chow, Jiyan Yang, and Ram Rajagopal. Distributed Online Modified Greedy Algorithm for Networked Storage Operation under Uncertainty. Smart Grid, IEEE Transactions on, PP(99):1, jun 2014. URL: http://arxiv.org/abs/1406.4615, http://arxiv.org/abs/1406.4615, URL: http://dx.doi.org/10.1109/TSG.2015.2422780.
http://dx.doi.org/10.1109/TSG.2015.2422780
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Brian Stott, Jorge Jardim, and Ongun Alsaç. Dc power flow revisited. Power Systems, IEEE Transactions on, 24(3):1290-1300, 2009.
H. I. Su and A. El Gamal. Modeling and Analysis of the Role of Energy Storage for Renewable Integration: Power Balancing. IEEE Transactions on Power Systems, 28(4):4109-4117, 2013. URL: http://dx.doi.org/10.1109/TPWRS.2013.2266667.
http://dx.doi.org/10.1109/TPWRS.2013.2266667
Kevin D. Wayne. Generalized maximum flow algorithms. PhD thesis, Cornell University, 1999.
L. Xie, Y. Gu, A. Eskandari, and M. Ehsani. Fast MPC-Based Coordination of Wind Power and Battery Energy Storage Systems. Journal of Energy Engineering, 138(2):43-53, 2012. URL: http://dx.doi.org/10.1061/(ASCE)EY.1943-7897.0000071.
http://dx.doi.org/10.1061/(ASCE)EY.1943-7897.0000071
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
LP-Relaxations for Tree Augmentation
In the Tree Augmentation Problem (TAP) the goal is to augment a tree T by a minimum size edge set F from a given edge set E such that T+F is 2-edge-connected. The best approximation ratio known for TAP is 1.5. In the more general Weighted TAP problem, F should be of minimum weight. Weighted TAP admits several 2-approximation algorithms w.r.t. the standard cut-LP relaxation. The problem is equivalent to the problem of covering a laminar set family. Laminar set families play an important role in the design of approximation algorithms for connectivity network design problems. In fact, Weighted TAP is the simplest connectivity network design problem for which a ratio better than 2 is not known. Improving this "natural" ratio is a major open problem, which may have implications on many other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than 2, which is an old open problem even for TAP. In this paper we introduce two different LP-relaxations, and for each of them give a simple algorithm that computes a feasible solution for TAP of size at most 7/4 times the optimal LP value. This gives some hope to break the ratio 2 for the weighted case.
Tree Augmentation; LP-relaxation; Laminar family; Approximation algorithms
13:1-13:16
Regular Paper
Guy
Kortsarz
Guy Kortsarz
Zeev
Nutov
Zeev Nutov
10.4230/LIPIcs.APPROX-RANDOM.2016.13
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Bi-Criteria Approximation Algorithm for k-Means
We consider the classical k-means clustering problem in the setting of bi-criteria approximation, in which an algorithm is allowed to output beta*k > k clusters, and must produce a clustering with cost at most alpha times the to the cost of the optimal set of k clusters. We argue that this approach is natural in many settings, for which the exact number of clusters is a priori unknown, or unimportant up to a constant factor.
We give new bi-criteria approximation algorithms, based on linear programming and local search, respectively, which attain a guarantee alpha(beta) depending on the number beta*k of clusters that may be opened. Our guarantee alpha(beta) is always at most 9 + epsilon and improves rapidly with beta (for example: alpha(2) < 2.59, and alpha(3) < 1.4). Moreover, our algorithms have only polynomial dependence on the dimension of the input data, and so are applicable in high-dimensional settings.
k-means clustering
bicriteria approximation algorithms
linear programming
local search
14:1-14:20
Regular Paper
Konstantin
Makarychev
Konstantin Makarychev
Yury
Makarychev
Yury Makarychev
Maxim
Sviridenko
Maxim Sviridenko
Justin
Ward
Justin Ward
10.4230/LIPIcs.APPROX-RANDOM.2016.14
Alexander A. Ageev and Maxim Sviridenko. Pipage rounding: A new method of constructing algorithms with proven performance guarantee. J. of Combinatorial Optimization, 8(3):307-328, 2004.
Ankit Aggarwal, Amit Deshpande, and Ravi Kannan. Adaptive sampling for k-means clustering. In Proc. of the 12th International Workshop APPROX, pages 15-28, 2009.
D. Aloise, A. Deshpande, P. Hansen, and P. Popat. NP-hardness of Euclidean sum-of-squares clustering. In Machine Learning, volume 75, pages 245-249, 2009.
Aris Anagnostopoulos, Anirban Dasgupta, and Ravi Kumar. A constant-factor approximation algorithm for co-clustering. In Theory of Computing, volume 8(1), pages 597-622, 2012.
D. Arthur and S. Vassilvitskii. k-means++: the advantages of careful seeding. In Proc. of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1027-1035, 2007.
Vijay Arya, Naveen Garg, Rohit Khandekar, Adam Meyerson, Kamesh Munagala, and Vinayaka Pandit. Local search heuristics for k-median and facility location problems. SIAM J. on Computing, 33(3):544-562, 2004.
Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The hardness of approximation of Euclidean k-means. In Proc. of the 31st International Symposium on Computational Geometry, pages 754-767, 2015.
Jarosław Byrka, Thomas Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An improved approximation for k-median, and positive correlation in budgeted optimization. In Proc. of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 737-756, 2015. Updated version: URL: http://arxiv.org/abs/1406.2951.
http://arxiv.org/abs/1406.2951
Moses Charikar and Sudipto Guha. Improved combinatorial algorithms for facility location problems. SIAM J. on Computing, 34(4):803-824, 2005.
Fabian A. Chudak and David B. Shmoys. Improved approximation algorithms for the uncapacitated facility location problem. SIAM J. on Computing, 33(1):1-25, 2003.
Michael B. Cohen, Sam Elder, Cameron Musco, Christopher Musco, and Madalina Persu. Dimensionality reduction for k-means clustering and low rank approximation. In Proc. of the 47th Annual ACM on Symposium on Theory of Computing, pages 163-172, 2015.
Vincent Cohen-Addad, Philip N. Klein, and Claire Mathieu. Local search yields approximation schemes for k-means and k-median in euclidean and minor-free metrics. CoRR, abs/1603.09535, 2016. URL: http://arxiv.org/abs/1603.09535.
http://arxiv.org/abs/1603.09535
S. Dasgupta. The hardness of k-means clustering. In Technical Report CS2007-0890, University of California, San Diego., 2007.
Dan Feldman, Morteza Monemizadeh, and Christian Sohler. A PTAS for k-means clustering based on weak coresets. In Proc. of the 23rd Annual Symposium on Computational Geometry, pages 11-18, 2007.
Zachary Friggstad, Mohsen Rezapour, and Mohammad R. Salavatipour. Local search yields a PTAS for k-means in doubling metrics. CoRR, abs/1603.08976, 2016. URL: http://arxiv.org/abs/1603.08976.
http://arxiv.org/abs/1603.08976
Mary Inaba, Naoki Katoh, and Hiroshi Imai. Applications of weighted Voronoi diagrams and randomization to variance-based k-clustering: (extended abstract). In Proc. of the 10th Annual Symposium on Computational Geometry, pages 332-339, 1994.
Kamal Jain and Vijay V. Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. J. ACM, 48(2):274-296, March 2001.
Tapas Kanungo, David M. Mount, Nathan S. Netanyahu, Angela Y. Wu, and Christine D. Piatko. A local search approximation algorithm for k-means clustering. Computational Geometry, 28(2-3):89-112, 2004.
Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. In Proc. of the 45th Annual ACM Symposium on Theory of Computing, pages 901-910, 2013.
Jyh-Han Lin and Jeffrey Scott Vitter. Approximation algorithms for geometric median problems. Information Processing Lett., 44(5):245-249, December 1992.
Jyh-Han Lin and Jeffrey Scott Vitter. ε-approximations with minimum packing constraint violation. In Proc. of the 24th Annual ACM Symposium on Theory of Computing, pages 771-782, 1992.
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S. Lloyd. Least squares quantization in PCM. IEEE Trans. on Information Theory, 28(2):129-137, Mar 1982.
Meena Mahajan, Prajakta Nimbhorkar, and Kasturi Varadarajan. The planar k-means problem is NP-hard. In Proc. of the 3rd International Workshop on Algorithms and Computation, pages 274-285, 2009.
Jirı Matoušek. On approximate geometric k-clustering. Discrete &Computational Geometry, 24(1):61-84, 2000.
R. Ostrovsky, Y. Rabani, L. Schulman, and C. Swamy. The effectiveness of Lloyd-type methods for the k-means problem. In Proc. of the 47th Annual IEEE Symposium on Foundations of Computer Science, pages 165-174, 2006.
Maxim Sviridenko. An improved approximation algorithm for the metric uncapacitated facility location problem. In Integer Programming and Combinatorial Optimization: Proc. of the 9th IPCO Conference, pages 240-257, 2002.
Creative Commons Attribution 3.0 Unported license
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Near-Optimal UGC-hardness of Approximating Max k-CSP_R
In this paper, we prove an almost-optimal hardness for Max k-CSP_R based on Khot's Unique Games Conjecture (UGC). In Max k-CSP_R, we are given a set of predicates each of which depends on exactly k variables. Each variable can take any value from 1, 2, ..., R. The goal is to find an assignment to variables that maximizes the number of satisfied predicates.
Assuming the Unique Games Conjecture, we show that it is NP-hard to approximate Max k-CSP_R to within factor 2^{O(k log k)}(log R)^{k/2}/R^{k - 1} for any k, R.
To the best of our knowledge, this result improves on all the known hardness of approximation results when 3 <= k = o(log R/log log R). In this case, the previous best hardness result was NP-hardness of approximating within a factor O(k/R^{k-2}) by Chan. When k = 2, our result matches the best known UGC-hardness result of Khot, Kindler, Mossel and O'Donnell.
In addition, by extending an algorithm for Max 2-CSP_R by Kindler, Kolla and Trevisan, we provide an Omega(log R/R^{k - 1})-approximation algorithm for Max k-CSP_R. This algorithm implies that our inapproximability result is tight up to a factor of 2^{O(k \log k)}(\log R)^{k/2 - 1}. In comparison, when 3 <= k is a constant, the previously known gap was $O(R)$, which is significantly larger than our gap of O(polylog R).
Finally, we show that we can replace the Unique Games Conjecture assumption with Khot's d-to-1 Conjecture and still get asymptotically the same hardness of approximation.
inapproximability
unique games conjecture
constraint satisfaction problem
invariance principle
15:1-15:28
Regular Paper
Pasin
Manurangsi
Pasin Manurangsi
Preetum
Nakkiran
Preetum Nakkiran
Luca
Trevisan
Luca Trevisan
10.4230/LIPIcs.APPROX-RANDOM.2016.15
Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for unique games and related problems. J. ACM, 62(5):42, 2015. URL: http://dx.doi.org/10.1145/2775105.
http://dx.doi.org/10.1145/2775105
P. Austrin and E. Mossel. Approximation resistant predicates from pairwise independence. In Computational Complexity, 2008. CCC'08. 23rd Annual IEEE Conference on, pages 249-258, June 2008. URL: http://dx.doi.org/10.1109/CCC.2008.20.
http://dx.doi.org/10.1109/CCC.2008.20
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Anupam Gupta and Kunal Talwar. Approximating unique games. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, pages 99-106, 2006. URL: http://dl.acm.org/citation.cfm?id=1109557.1109569.
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http://dx.doi.org/10.4086/toc.2008.v004a005
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Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs
In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a closed walk of minimum cost in a directed graph visiting every vertex. We consider the approximability of ATSP on topologically restricted graphs. It has been shown by Oveis Gharan and Saberi [SODA, 2011] that there exists polynomial-time constant-factor approximations on planar graphs and more generally graphs of constant orientable genus. This result was extended to non-orientable genus by Erickson and Sidiropoulos [SoCG, 2014].
We show that for any class of nearly-embeddable graphs, ATSP admits a polynomial-time constant-factor approximation. More precisely, we show that for any fixed non-negative k, there exist positive alpha and beta, such that ATSP on n-vertex k-nearly-embeddable graphs admits an alpha-approximation in time O(n^beta). The class of k-nearly-embeddable graphs contains graphs with at most k apices, k vortices of width at most k, and an underlying surface of either orientable or non-orientable genus at most k. Prior to our work, even the case of graphs with a single apex was open. Our algorithm combines tools from rounding the Held-Karp LP via thin trees with dynamic programming.
We complement our upper bounds by showing that solving ATSP exactly on graphs of pathwidth k (and hence on k-nearly embeddable graphs) requires time n^{Omega(k)}, assuming the Exponential-Time Hypothesis (ETH). This is surprising in light of the fact that both TSP on undirected graphs and Minimum Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth.
asymmetric TSP
approximation algorithms
nearly-embeddable graphs
Held-Karp LP
exponential time hypothesis
16:1-16:54
Regular Paper
Dániel
Marx
Dániel Marx
Ario
Salmasi
Ario Salmasi
Anastasios
Sidiropoulos
Anastasios Sidiropoulos
10.4230/LIPIcs.APPROX-RANDOM.2016.16
Nima Anari and Shayan Oveis Gharan. Effective-resistance-reducing flows, spectrally thin trees, and asymmetric tsp. In 55th Annual IEEE Symposium on Foundations of Computer Science, FOCS, 2015.
Arash Asadpour, Michel X Goemans, Aleksander Madry, Shayan Oveis Gharan, and Amin Saberi. An O(log n/log log n)-approximation Algorithm for the Asymmetric Traveling Salesman Problem. In SODA, volume 10, pages 379-389. SIAM, 2010.
Markus Bläser. A new approximation algorithm for the asymmetric tsp with triangle inequality. ACM Transactions on Algorithms (TALG), 4(4):47, 2008.
Moses Charikar, Michel X Goemans, and Howard Karloff. On the integrality ratio for asymmetric tsp. In Foundations of Computer Science, 2004. Proceedings. 45th Annual IEEE Symposium on, pages 101-107. IEEE, 2004.
Jianer Chen, Xiuzhen Huang, Iyad A Kanj, and Ge Xia. Strong computational lower bounds via parameterized complexity. Journal of Computer and System Sciences, 72(8):1346-1367, 2006.
Holger Dell and Dániel Marx. Kernelization of packing problems. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 68-81. SIAM, 2012.
Reinhard Diestel. Graph theory graduate texts in mathematics; 173. Springer-Verlag Berlin and Heidelberg GmbH &, 2000.
Jeff Erickson and Anastasios Sidiropoulos. A near-optimal approximation algorithm for asymmetric tsp on embedded graphs. In Proceedings of the thirtieth annual symposium on Computational geometry, page 130. ACM, 2014.
Uriel Feige and Mohit Singh. Improved approximation ratios for traveling salesperson tours and paths in directed graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 104-118. Springer, 2007.
Fedor V Fomin, Petr A Golovach, Daniel Lokshtanov, and Saket Saurabh. Almost optimal lower bounds for problems parameterized by clique-width. SIAM Journal on Computing, 43(5):1541-1563, 2014.
Alan M Frieze and Giulia Galbiati. On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks, 12(1):23-39, 1982.
Alan M. Frieze, Giulia Galbiati, and Francesco Maffioli. On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks, 12(1):23-39, 1982. URL: http://dx.doi.org/10.1002/net.3230120103.
http://dx.doi.org/10.1002/net.3230120103
Shayan Oveis Gharan and Amin Saberi. The asymmetric traveling salesman problem on graphs with bounded genus. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 967-975. SIAM, 2011.
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Michael Held and Richard Karp. The traveling salesman problem and minimum spanning trees. Operations Research, 18:1138-1162, 1970.
Michael Held and Richard M Karp. The traveling-salesman problem and minimum spanning trees. Operations Research, 18(6):1138-1162, 1970.
Klaus Jansen, Stefan Kratsch, Dániel Marx, and Ildikó Schlotter. Bin packing with fixed number of bins revisited. Journal of Computer and System Sciences, 79(1):39-49, 2013.
Haim Kaplan, Moshe Lewenstein, Nira Shafrir, and Maxim Sviridenko. Approximation algorithms for asymmetric tsp by decomposing directed regular multigraphs. Journal of the ACM (JACM), 52(4):602-626, 2005.
Ken-ichi Kawarabayashi and Bojan Mohar. Some recent progress and applications in graph minor theory. Graphs and Combinatorics, 23(1):1-46, 2007.
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Planar Matching in Streams Revisited
We present data stream algorithms for estimating the size or weight of the maximum matching in low arboricity graphs. A large body of work has focused on improving the constant approximation factor for general graphs when the data stream algorithm is permitted O(n polylog n) space where n is the number of nodes. This space is necessary if the algorithm must return the matching. Recently, Esfandiari et al. (SODA 2015) showed that it was possible to estimate the maximum cardinality of a matching in a planar graph up to a factor of 24+epsilon using O(epsilon^{-2} n^{2/3} polylog n) space. We first present an algorithm (with a simple analysis) that improves this to a factor 5+epsilon using the same space. We also improve upon the previous results for other graphs with bounded arboricity. We then present a factor 12.5 approximation for matching in planar graphs that can be implemented using O(log n) space in the adjacency list data stream model where the stream is a concatenation of the adjacency lists of the graph. The main idea behind our results is finding "local" fractional matchings, i.e., fractional matchings where the value of any edge e is solely determined by the edges sharing an endpoint with e. Our work also improves upon the results for the dynamic data stream model where the stream consists of a sequence of edges being inserted and deleted from the graph. We also extend our results to weighted graphs, improving over the bounds given by Bury and Schwiegelshohn (ESA 2015), via a reduction to the unweighted problem that increases the approximation by at most a factor of two.
data streams
planar graphs
arboricity
matchings
17:1-17:12
Regular Paper
Andrew
McGregor
Andrew McGregor
Sofya
Vorotnikova
Sofya Vorotnikova
10.4230/LIPIcs.APPROX-RANDOM.2016.17
Kook Jin Ahn and Sudipto Guha. Linear programming in the semi-streaming model with application to the maximum matching problem. Inf. Comput., 222:59-79, 2013. URL: http://dx.doi.org/10.1016/j.ic.2012.10.006.
http://dx.doi.org/10.1016/j.ic.2012.10.006
Ziv Bar-Yossef, Ravi Kumar, and D. Sivakumar. Reductions in streaming algorithms, with an application to counting triangles in graphs. In Proc. of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, January 6-8, 2002, San Francisco, CA, USA., pages 623-632, 2002. URL: http://dl.acm.org/citation.cfm?id=545381.545464.
http://dl.acm.org/citation.cfm?id=545381.545464
Luciana S. Buriol, Gereon Frahling, Stefano Leonardi, Alberto Marchetti-Spaccamela, and Christian Sohler. Counting triangles in data streams. In Proceedings of the Twenty-Fifth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, June 26-28, 2006, Chicago, Illinois, USA, pages 253-262, 2006. URL: http://dx.doi.org/10.1145/1142351.1142388.
http://dx.doi.org/10.1145/1142351.1142388
Marc Bury and Chris Schwiegelshohn. Sublinear estimation of weighted matchings in dynamic data streams. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 263-274, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_23.
http://dx.doi.org/10.1007/978-3-662-48350-3_23
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 Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1326-1344, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch92.
http://dx.doi.org/10.1137/1.9781611974331.ch92
Kenneth L. Clarkson and David P. Woodruff. Numerical linear algebra in the streaming model. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 205-214, 2009. URL: http://dx.doi.org/10.1145/1536414.1536445.
http://dx.doi.org/10.1145/1536414.1536445
Michael Crouch and Daniel S. Stubbs. Improved streaming algorithms for weighted matching, via unweighted matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, September 4-6, 2014, Barcelona, Spain, pages 96-104, 2014. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.96.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.96
Michael S. Crouch, Andrew McGregor, and Daniel Stubbs. Dynamic graphs in the sliding-window model. In Algorithms - ESA 2013 - 21st Annual European Symposium, Sophia Antipolis, France, September 2-4, 2013. Proceedings, pages 337-348, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40450-4_29.
http://dx.doi.org/10.1007/978-3-642-40450-4_29
Andrzej Czygrinow, Michal Hanckowiak, and Edyta Szymanska. Fast distributed approximation algorithm for the maximum matching problem in bounded arboricity graphs. In Algorithms and Computation, 20th International Symposium, ISAAC 2009, Honolulu, Hawaii, USA, December 16-18, 2009. Proceedings, pages 668-678, 2009. URL: http://dx.doi.org/10.1007/978-3-642-10631-6_68.
http://dx.doi.org/10.1007/978-3-642-10631-6_68
Jack Edmonds. Maximum matching and a polyhedron with 0,1-vertices. Journal of Research of the National Bureau of Standards, 69:125-130, 1965.
Leah Epstein, Asaf Levin, Julián Mestre, and Danny Segev. Improved approximation guarantees for weighted matching in the semi-streaming model. SIAM J. Discrete Math., 25(3):1251-1265, 2011. URL: http://dx.doi.org/10.1137/100801901.
http://dx.doi.org/10.1137/100801901
Hossein Esfandiari, Mohammad Taghi Hajiaghayi, Vahid Liaghat, Morteza Monemizadeh, and Krzysztof Onak. Streaming algorithms for estimating the matching size in planar graphs and beyond. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1217-1233, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.81.
http://dx.doi.org/10.1137/1.9781611973730.81
Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Theor. Comput. Sci., 348(2):207-216, 2005. URL: http://dx.doi.org/10.1016/j.tcs.2005.09.013.
http://dx.doi.org/10.1016/j.tcs.2005.09.013
Mina Ghashami, Edo Liberty, Jeff M. Phillips, and David P. Woodruff. Frequent directions: Simple and deterministic matrix sketching. CoRR, abs/1501.01711, 2015. URL: http://arxiv.org/abs/1501.01711.
http://arxiv.org/abs/1501.01711
Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 468-485, 2012. URL: http://portal.acm.org/citation.cfm?id=2095157&CFID=63838676&CFTOKEN=79617016.
http://portal.acm.org/citation.cfm?id=2095157&CFID=63838676&CFTOKEN=79617016
Elena Grigorescu, Morteza Monemizadeh, and Samson Zhou. Estimating weighted matchings in o(n) space. CoRR, abs/1604.07467, 2016. URL: http://arxiv.org/abs/1604.07467.
http://arxiv.org/abs/1604.07467
Venkatesan Guruswami and Krzysztof Onak. Superlinear lower bounds for multipass graph processing. In Proc. of the 28th Conference on Computational Complexity, CCC 2013, Palo Alto, California, USA, 5-7 June, 2013, pages 287-298, 2013. URL: http://dx.doi.org/10.1109/CCC.2013.37.
http://dx.doi.org/10.1109/CCC.2013.37
Michael Kapralov. Better bounds for matchings in the streaming model. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 1679-1697, 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.121.
http://dx.doi.org/10.1137/1.9781611973105.121
Michael Kapralov, Sanjeev Khanna, and Madhu Sudan. Approximating matching size from random streams. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 734-751, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.55.
http://dx.doi.org/10.1137/1.9781611973402.55
Christian Konrad. Maximum matching in turnstile streams. In Algorithms - ESA 2015 - 23rd Annual European Symposium, Patras, Greece, September 14-16, 2015, Proceedings, pages 840-852, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48350-3_70.
http://dx.doi.org/10.1007/978-3-662-48350-3_70
Christian Konrad, Frédéric Magniez, and Claire Mathieu. Maximum matching in semi-streaming with few passes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Cambridge, MA, USA, August 15-17, 2012. Proceedings, pages 231-242, 2012. URL: http://dx.doi.org/10.1007/978-3-642-32512-0_20.
http://dx.doi.org/10.1007/978-3-642-32512-0_20
Christian Konrad and Adi Rosén. Approximating semi-matchings in streaming and in two-party communication. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 637-649, 2013. URL: http://dx.doi.org/10.1007/978-3-642-39206-1_54.
http://dx.doi.org/10.1007/978-3-642-39206-1_54
Madhusudan Manjunath, Kurt Mehlhorn, Konstantinos Panagiotou, and He Sun. Approximate counting of cycles in streams. In Algorithms - ESA 2011 - 19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011. Proceedings, pages 677-688, 2011. URL: http://dx.doi.org/10.1007/978-3-642-23719-5_57.
http://dx.doi.org/10.1007/978-3-642-23719-5_57
Andrew McGregor. Finding graph matchings in data streams. APPROX-RANDOM, pages 170-181, 2005.
Andrew McGregor. Graph stream algorithms: a survey. SIGMOD Record, 43(1):9-20, 2014. URL: http://dx.doi.org/10.1145/2627692.2627694.
http://dx.doi.org/10.1145/2627692.2627694
Mariano Zelke. Weighted matching in the semi-streaming model. Algorithmica, 62(1-2):1-20, 2012. URL: http://dx.doi.org/10.1007/s00453-010-9438-5.
http://dx.doi.org/10.1007/s00453-010-9438-5
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A Robust and Optimal Online Algorithm for Minimum Metric Bipartite Matching
We study the Online Minimum Metric Bipartite Matching Problem. In this problem, we are given point sets S and R which correspond to the server and request locations; here |S|=|R|=n. All these locations are points from some metric space and the cost of matching a server to a request is given by the distance between their locations in this space. In this problem, the request points arrive one at a time. When a request arrives, we must immediately and irrevocably match it to a "free" server. The matching obtained after all the requests are processed is the online matching M. The cost of M is the sum of the cost of its edges. The performance of any online algorithm is the worst-case ratio of the cost of its online solution M to the minimum-cost matching.
We present a deterministic online algorithm for this problem. Our algorithm is the first to simultaneously achieve optimal performances in the well-known adversarial and the random arrival models. For the adversarial model, we obtain a competitive ratio of 2n-1 + o(1); it is known that no deterministic algorithm can do better than 2n-1. In the random arrival model, our algorithm obtains a competitive ratio of 2H_n - 1 + o(1); where H_n is the n-th Harmonic number. We also prove that any online algorithm will have a competitive ratio of at least 2H_n - 1-o(1) in this model.
We use a new variation of the offline primal-dual method for computing minimum cost matching to compute the online matching. Our primal-dual method is based on a relaxed linear-program. Under metric costs, this specific relaxation helps us relate the cost of the online matching with the offline matching leading to its robust properties.
Online Algorithms
Metric Bipartite Matching
18:1-18:16
Regular Paper
Sharath
Raghvendra
Sharath Raghvendra
10.4230/LIPIcs.APPROX-RANDOM.2016.18
Pankaj K. Agarwal and R. Sharathkumar. Approximation algorithms for bipartite matching with metric and geometric costs. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31-June 03, 2014, pages 555-564, 2014.
Pankaj K. Agarwal and Kasturi R. Varadarajan. A near-linear constant-factor approximation for euclidean bipartite matching? In Proceedings of the 20th ACM Symposium on Computational Geometry, Brooklyn, New York, USA, June 8-11, 2004, pages 247-252, 2004.
Antonios Antoniadis, Neal Barcelo, Michael Nugent, Kirk Pruhs, and Michele Scquizzato. A o(n)-competitive deterministic algorithm for online matching on a line. In Approximation and Online Algorithms - 12th International Workshop, WAOA 2014, Wrocław, Poland, September 11-12, 2014, pages 11-22, 2014.
N. Bansal, N. Buchbinder, A Madry, and J. Naor. A polylogarithmic-competitive algorithm for the k-server problem. In Proceedings of the IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pages 267-276, Oct 2011.
Nikhil Bansal, Niv Buchbinder, Anupam Gupta, and Joseph Naor. An o(log² k)-competitive algorithm for metric bipartite matching. In Algorithms - ESA 2007, 15th Annual European Symposium, Eilat, Israel, October 8-10, 2007, Proceedings, pages 522-533, 2007.
A. Gupta and K. Lewi. The online metric matching problem for doubling metrics. In Automata, Languages, and Programming, volume 7391 of LNCS, pages 424-435. Springer, 2012.
Piotr Indyk. A near linear time constant factor approximation for euclidean bichromatic matching (cost). In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, New Orleans, Louisiana, USA, January 7-9, 2007, pages 39-42, 2007.
Bala Kalyanasundaram and Kirk Pruhs. Online weighted matching. J. Algorithms, 14(3):478-488, 1993.
Chinmay Karande, Aranyak Mehta, and Pushkar Tripathi. Online bipartite matching with unknown distributions. In Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC'11, pages 587-596, New York, NY, USA, 2011. ACM.
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.
Elias Koutsoupias and Akash Nanavati. The online matching problem on a line. In Roberto Solis-Oba and Klaus Jansen, editors, Approximation and Online Algorithms: First International Workshop, WAOA 2003, Budapest, Hungary, September 16-18, 2003. Revised Papers, pages 179-191. Springer Berlin Heidelberg, Berlin, Heidelberg, 2004.
Elias Koutsoupias and Christos H. Papadimitriou. On the k-server conjecture. J. ACM, 42(5):971-983, September 1995.
M. Mahdian and Q. Yan. Online bipartite matching with random arrivals: An approach based on strongly factor-revealing lps. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, STOC'11, pages 597-606, 2011.
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A. Meyerson, A. Nanavati, and L. Poplawski. Randomized online algorithms for minimum metric bipartite matching. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithm, pages 954-959, 2006.
R. Sharathkumar and Pankaj K. Agarwal. Algorithms for the transportation problem in geometric settings. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 306-317, 2012.
R. Sharathkumar and Pankaj K. Agarwal. A near-linear time ε-approximation algorithm for geometric bipartite matching. In Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC'12, pages 385-394. ACM, 2012. URL: http://dx.doi.org/10.1145/2213977.2214014.
http://dx.doi.org/10.1145/2213977.2214014
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Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One
We show the first dimension-preserving search-to-decision reductions for approximate SVP and CVP. In particular, for any gamma <= 1 + O(log n/n), we obtain an efficient dimension-preserving reduction from gamma^{O(n/log n)}-SVP to gamma-GapSVP and an efficient dimension-preserving reduction from gamma^{O(n)}-CVP to gamma-GapCVP. These results generalize the known equivalences of the search and decision versions of these problems in the exact case when gamma = 1. For SVP, we actually obtain something slightly stronger than a search-to-decision reduction - we reduce gamma^{O(n/log n)}-SVP to gamma-unique SVP, a potentially easier problem than gamma-GapSVP.
Lattices
SVP
CVP
19:1-19:18
Regular Paper
Noah
Stephens-Davidowitz
Noah Stephens-Davidowitz
10.4230/LIPIcs.APPROX-RANDOM.2016.19
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.
Divesh Aggarwal and Chandan Dubey. Improved hardness results for unique Shortest Vector Problem, 2015. URL: http://eccc.hpi-web.de/report/2013/076/.
http://eccc.hpi-web.de/report/2013/076/
Dorit Aharonov and Oded Regev. Lattice problems in NP intersect coNP. Journal of the ACM, 52(5):749-765, 2005. Preliminary version in FOCS'04.
Miklós Ajtai. Generating hard instances of lattice problems. In STOC, pages 99-108. ACM, 1996.
Miklós Ajtai. The Shortest Vector Problem in L2 is NP-hard for randomized reductions. In STOC, 1998. URL: http://dx.doi.org/10.1145/276698.276705.
http://dx.doi.org/10.1145/276698.276705
Miklós Ajtai, Ravi Kumar, and D. Sivakumar. A sieve algorithm for the shortest lattice vector problem. In STOC, pages 601-610, 2001.
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
Shi Bai, Weiqiang Wen, and Damien Stehlé. Improved reduction from the Bounded Distance Decoding problem to the unique Shortest Vector Problem in lattices. In ICALP, 2016.
W. Banaszczyk. New bounds in some transference theorems in the geometry of numbers. Mathematische Annalen, 296(4):625-635, 1993. URL: http://dx.doi.org/10.1007/BF01445125.
http://dx.doi.org/10.1007/BF01445125
U. Betke, M. Henk, and J.M. Wills. Successive-minima-type inequalities. Discrete &Computational Geometry, 9(1):165-175, 1993. URL: http://dx.doi.org/10.1007/BF02189316.
http://dx.doi.org/10.1007/BF02189316
Zvika Brakerski and Vinod Vaikuntanathan. Efficient fully homomorphic encryption from (standard) LWE. In FOCS, pages 97-106. IEEE, 2011.
Zvika Brakerski and Vinod Vaikuntanathan. Lattice-based FHE as secure as PKE. In ITCS, pages 1-12, 2014.
Jin-Yi Cai and Ajay Nerurkar. Approximating the SVP to within a factor (1+1/dim^ε) is NP-hard under randomized reductions. Journal of Computer and System Sciences, 59(2):221-239, 1999. URL: http://dx.doi.org/10.1006/jcss.1999.1649.
http://dx.doi.org/10.1006/jcss.1999.1649
Daniel Dadush and Gabor Kun. Lattice sparsification and the approximate Closest Vector Problem. In SODA, 2013.
Daniel Dadush, Chris Peikert, and Santosh Vempala. Enumerative lattice algorithms in any norm via M-ellipsoid coverings. In FOCS, pages 580-589. IEEE, 2011.
Daniel Dadush, Oded Regev, and Noah Stephens-Davidowitz. On the Closest Vector Problem with a distance guarantee. In CCC, pages 98-109, 2014. URL: http://dx.doi.org/10.1109/CCC.2014.18.
http://dx.doi.org/10.1109/CCC.2014.18
Craig Gentry. Fully homomorphic encryption using ideal lattices. In STOC, pages 169-178. ACM, New York, 2009.
Craig Gentry, Chris Peikert, and Vinod Vaikuntanathan. Trapdoors for hard lattices and new cryptographic constructions. In STOC, pages 197-206, 2008.
Oded Goldreich, Daniele Micciancio, Shmuel Safra, and Jean-Paul 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
Ishay Haviv and Oded Regev. Tensor-based hardness of the Shortest Vector Problem to within almost polynomial factors. Theory of Computing, 8(23):513-531, 2012. Preliminary version in STOC'07.
Antoine Joux and Jacques Stern. Lattice reduction: A toolbox for the cryptanalyst. Journal of Cryptology, 11(3):161-185, 1998.
G. A. Kabatjanskiĭ and V. I. Levenšteĭn. Bounds for packings on the sphere and in space. Problemy Peredači Informacii, 14(1):3-25, 1978.
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
Subhash Khot. Hardness of approximating the Shortest Vector Problem in lattices. Journal of the ACM, 52(5):789-808, September 2005. Preliminary version in FOCS'04.
S. Ravi Kumar and D. Sivakumar. On the unique shortest lattice vector problem. Theoretical Computer Science, 255(1‚Äì2):641-648, 2001. URL: http://dx.doi.org/10.1016/S0304-3975(00)00387-X.
http://dx.doi.org/10.1016/S0304-3975(00)00387-X
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.
Yi-Kai Liu, Vadim Lyubashevsky, and Daniele Micciancio. On Bounded Distance Decoding for general lattices. In RANDOM, 2006.
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http://dx.doi.org/10.1007/978-3-642-03356-8_34
Vadim Lyubashevsky, Chris Peikert, and Oded Regev. On ideal lattices and Learning with Errors over rings. In EUROCRYPT, 2010.
Daniele Micciancio. The Shortest Vector Problem is NP-hard to approximate to within some constant. SIAM Journal on Computing, 30(6):2008-2035, March 2001. Preliminary version in FOCS 1998.
Daniele Micciancio. Efficient reductions among lattice problems. In SODA, pages 84-93. ACM, New York, 2008.
Daniele Micciancio and Shafi Goldwasser. Complexity of Lattice Problems: a cryptographic perspective, volume 671 of The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Boston, Massachusetts, March 2002.
Daniele Micciancio and Oded Regev. Worst-case to average-case reductions based on Gaussian measures. SIAM Journal on Computing, 37(1):267-302, 2007.
Phong Q Nguyen and Jacques Stern. The two faces of lattices in cryptology. In Cryptography and lattices, pages 146-180. Springer, 2001.
Andrew M Odlyzko. The rise and fall of knapsack cryptosystems. Cryptology and computational number theory, 42:75-88, 1990.
Chris Peikert. Public-key cryptosystems from the worst-case Shortest Vector Problem. In STOC, pages 333-342. ACM, 2009.
Chris Peikert and Alon Rosen. Lattices that admit logarithmic worst-case to average-case connection factors. In STOC, 2007.
Xavier Pujol and Damien Stehlé. Solving the shortest lattice vector problem in time 2^2.465 n. IACR Cryptology ePrint Archive, 2009:605, 2009.
Oded Regev. On lattices, Learning with Errors, random linear codes, and cryptography. Journal of the ACM, 56(6):Art. 34, 40, 2009. URL: http://dx.doi.org/10.1145/1568318.1568324.
http://dx.doi.org/10.1145/1568318.1568324
Noah Stephens-Davidowitz. Dimension-preserving reductions between lattice problems, 2015. URL: http://noahsd.com/latticeproblems.pdf.
http://noahsd.com/latticeproblems.pdf
Noah Stephens-Davidowitz. Discrete Gaussian sampling reduces to CVP and SVP. In SODA, 2016.
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Proving Weak Approximability Without Algorithms
A boolean predicate is said to be strongly approximation resistant if, given a near-satisfiable instance of its maximum constraint satisfaction problem, it is hard to find an assignment such that the fraction of constraints satisfied deviates significantly from the expected fraction of constraints satisfied by a random assignment. A predicate which is not strongly approximation resistant is known as weakly approximable.
We give a new method for proving the weak approximability of predicates, using a simple SDP relaxation, without designing and analyzing new rounding algorithms for each predicate. Instead, we use the recent characterization of strong approximation resistance by Khot et al. [STOC 2014], and show how to prove that for a given predicate, certain necessary conditions for strong resistance derived from their characterization, are violated. By their result, this implies the existence of a good rounding algorithm, proving weak approximability.
We show how this method can be used to obtain simple proofs of (weak approximability analogues of) various known results on approximability, as well as new results on weak approximability of symmetric predicates.
approximability
constraint satisfaction problems
approximation resistance
linear programming
semidefinite programming
20:1-20:15
Regular Paper
Ridwan
Syed
Ridwan Syed
Madhur
Tulsiani
Madhur Tulsiani
10.4230/LIPIcs.APPROX-RANDOM.2016.20
Per Austrin and Elchanan Mossel. Approximation resistant predicates from pairwise independence. In Proceedings of the 23rd IEEE Conference on Computational Complexity, pages 249-258, Los Alamitos, CA, USA, 2008. IEEE Computer Society. URL: http://front.math.ucdavis.edu/0802.2300, URL: http://dx.doi.org/10.1109/CCC.2008.20.
http://dx.doi.org/10.1109/CCC.2008.20
Siu On Chan. Approximation resistance from pairwise independent subgroups. In Proceedings of the 45th ACM Symposium on Theory of Computing, pages 447-456, 2013. URL: http://dx.doi.org/10.1145/2488608.2488665.
http://dx.doi.org/10.1145/2488608.2488665
Mahdi Cheraghchi, Johan Håstad, Marcus Isaksson, and Ola Svensson. Approximating linear threshold predicates. ACM Transactions on Computation Theory (TOCT), 4(1):2, 2012.
Lars Engebretsen, Jonas Holmerin, and Alexander Russell. Inapproximability Results for Equations over Finite Groups. Theor. Comput. Sci., 312(1):17-45, 2004. URL: http://dx.doi.org/10.1016/S0304-3975(03)00401-8.
http://dx.doi.org/10.1016/S0304-3975(03)00401-8
M.X. Goemans and D.P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42(6):1115-1145, 1995. Preliminary version in Proc. of STOC'94.
Venkatesan Guruswami and Euiwoong Lee. Towards a characterization of approximation resistance for symmetric CSPs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2015, August 24-26, 2015, Princeton, NJ, USA, pages 305-322, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.305.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.305
Venkatesan Guruswami, Daniel Lewin, Madhu Sudan, and Luca Trevisan. A tight characterization of NP with 3 query PCPs. In Proceedings of the 39th IEEE Symposium on Foundations of Computer Science, pages 8-17, 1998. URL: http://dx.doi.org/10.1109/SFCS.1998.743424.
http://dx.doi.org/10.1109/SFCS.1998.743424
Gustav Hast. Beating a Random Assignment. PhD thesis, Royal Institute of Technology, Sweden, 2005.
Johan Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, 2001.
Johan Håstad. Every 2-CSP Allows Nontrivial Approximation. Computational Complexity, 17(4):549-566, 2008.
Subhash Khot. Hardness Results for Coloring 3-Colorable 3-Uniform Hypergraphs. In Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, pages 23-32, 2002. URL: http://dx.doi.org/10.1109/SFCS.2002.1181879.
http://dx.doi.org/10.1109/SFCS.2002.1181879
Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the 34th ACM Symposium on Theory of Computing, pages 767-775, 2002.
Subhash Khot, Madhur Tulsiani, and Pratik Worah. A characterization of strong approximation resistance. In Proceedings of the 46th ACM Symposium on Theory of Computing, pages 634-643. ACM, 2014.
Avner Magen, Siavosh Benabbas, and Per Austrin. On quadratic threshold CSPs. Discrete Mathematics &Theoretical Computer Science, 14, 2012.
Prasad Raghavendra. Optimal algorithms and inapproximability results for every CSP? In Proceedings of the 40th ACM Symposium on Theory of Computing, pages 245-254, 2008.
Alex Samorodnitsky and Luca Trevisan. A PCP characterization of NP with optimal amortized query complexity. In Proceedings of the 32nd ACM Symposium on Theory of Computing, pages 191-199, 2000.
Alex Samorodnitsky and Luca Trevisan. Gowers uniformity, influence of variables, and PCPs. In Proceedings of the 38th ACM Symposium on Theory of Computing, pages 11-20, 2006.
Johan Håstad. On the Efficient Approximability of Constraint Satisfaction Problems. In Surveys in Combinatorics, volume 346, pages 201-222. Cambridge University Press, 2007.
Uri Zwick. Approximation Algorithms for Constraint Satisfaction Problems Involving at Most Three Variables per Constraint. In Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms, pages 201-210, 1998.
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Every Property of Outerplanar Graphs is Testable
A D-disc around a vertex v of a graph G=(V,E) is the subgraph induced by all vertices of distance at most D from v. We show that the structure of an outerplanar graph on n vertices is determined, up to modification (insertion or deletion) of at most epsilon n edges, by a set of D-discs around the vertices, for D=D(epsilon) that is independent of the size of the graph. Such a result was already known for planar graphs (and any hyperfinite graph class), in the limited case of bounded degree graphs (that is, their maximum degree is bounded by some fixed constant, independent of |V|). We prove this result with no assumption on the degree of the graph.
A pure combinatorial consequence of this result is that two outerplanar graphs that share the same local views are close to be isomorphic.
We also obtain the following property testing results in the sparse graph model:
* graph isomorphism is testable for outerplanar graphs by poly(log n) queries.
* every graph property is testable for outerplanar graphs by poly(log n) queries.
We note that we can replace outerplanar graphs by a slightly more general family of k-edge-outerplanar graphs. The only previous general testing results, as above, where known for forests (Kusumoto and Yoshida), and for some power-law graphs that are extremely close to be bounded degree hyperfinite (by Ito).
Property testing
Isomorphism
Outerplanar graphs
21:1-21:19
Regular Paper
Jasine
Babu
Jasine Babu
Areej
Khoury
Areej Khoury
Ilan
Newman
Ilan Newman
10.4230/LIPIcs.APPROX-RANDOM.2016.21
Noga Alon, Eldar Fischer, Ilan Newman, and Asaf Shapira. A combinatorial characterization of the testable graph properties: It’s all about regularity. SIAM J. Comput., 39(1):143-167, 2009. URL: http://dx.doi.org/10.1137/060667177.
http://dx.doi.org/10.1137/060667177
Itai Benjamini, Oded Schramm, and Asaf Shapira. Every minor-closed property of sparse graphs is testable. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008, pages 393-402, 2008. URL: http://dx.doi.org/10.1145/1374376.1374433.
http://dx.doi.org/10.1145/1374376.1374433
Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, 1998. URL: http://dx.doi.org/10.1145/285055.285060.
http://dx.doi.org/10.1145/285055.285060
Avinatan Hassidim, Jonathan A. Kelner, Huy N. Nguyen, and Krzysztof Onak. Local graph partitions for approximation and testing. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, pages 22-31, 2009. URL: http://dx.doi.org/10.1109/FOCS.2009.77.
http://dx.doi.org/10.1109/FOCS.2009.77
Hiro Ito. Every property is testable on a natural class of scale-free multigraphs. CoRR, abs/1504.00766, 2015. URL: http://arxiv.org/abs/1504.00766.
http://arxiv.org/abs/1504.00766
Mitsuru Kusumoto and Yuichi Yoshida. Testing forest-isomorphism in the adjacency list model. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings, Part I, pages 763-774, 2014. URL: http://dx.doi.org/10.1007/978-3-662-43948-7_63.
http://dx.doi.org/10.1007/978-3-662-43948-7_63
Reut Levi and Dana Ron. A quasi-polynomial time partition oracle for graphs with an excluded minor. ACM Trans. Algorithms, 11(3):24:1-24:13, 2015. URL: http://dx.doi.org/10.1145/2629508.
http://dx.doi.org/10.1145/2629508
Richard J. Lipton and Robert Endre Tarjan. Applications of a planar separator theorem. SIAM Journal on Computing, 9(3):615-627, 1980.
Ilan Newman and Christian Sohler. Every property of hyperfinite graphs is testable. SIAM J. Comput., 42(3):1095-1112, 2013. URL: http://dx.doi.org/10.1137/120890946.
http://dx.doi.org/10.1137/120890946
Krzysztof Onak. New sublinear methods in the struggle against classical problems. PhD Thesis, Massachusetts Institute of Technology, 2010.
Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252-271, 1996. URL: http://dx.doi.org/10.1137/S0097539793255151.
http://dx.doi.org/10.1137/S0097539793255151
Christian Sohler. Private Communication, 2015.
Creative Commons Attribution 3.0 Unported license
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The Condensation Phase Transition in the Regular k-SAT Model
Much of the recent work on phase transitions in discrete structures has been inspired by ingenious but non-rigorous approaches from physics. The physics predictions typically come in the form of distributional fixed point problems that mimic Belief Propagation, a message passing algorithm. In this paper we show how the Belief Propagation calculation can be turned into a rigorous proof of such a prediction, namely the existence and location of a condensation phase transition in the regular k-SAT model.
random k-SAT
phase transitions
Belief Propagation
condensation
22:1-22:18
Regular Paper
Victor
Bapst
Victor Bapst
Amin
Coja-Oghlan
Amin Coja-Oghlan
10.4230/LIPIcs.APPROX-RANDOM.2016.22
D. Achlioptas and A. Coja-Oghlan. Algorithmic barriers from phase transitions. In Proc. 49th FOCS, pages 793-802, 2008.
D. Achlioptas and C. Moore. Random k-sat: two moments suffice to cross a sharp threshold. SIAM Journal on Computing, 36:740-762, 2006.
D. Achlioptas, A. Naor, and Y. Peres. Rigorous location of phase transitions in hard optimization problems. Nature, 435:759-764, 2005.
D. Achlioptas and Y. Peres. The threshold for random k-sat is 2^k ln 2 - o(k). Journal of the AMS, 17:947-973, 2004.
J. Banks and C. Moore. Information-theoretic thresholds for community detection in sparse networks. Information-theoretic thresholds for community detection in sparse networks, arXiv:1601.02658, 2016.
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V. Bapst, A. Coja-Oghlan, S. Hetterich, F. Raßmann, and D. Vilenchik. The condensation phase transition in random graph coloring. Communications in Mathematical Physics, 341:543-606, 2016.
V. Bapst, A. Coja-Oghlan, and F. Raßmann. A positive temperature phase transition in random hypergraph 2-coloring. Annals of Applied Probability, 26:1362?1406, 2016.
B. Barak. Structure vs. combinatorics in computational complexity, 2013.
M. Bayati, D. Gamarnik, and P. Tetali. Combinatorial approach to the interpolation method and scaling limits in sparse random graphs. Annals of Probability, 41:4080-4115, 2013.
A. Coja-Oghlan and K. Panagiotou. Going after the k-sat threshold. In Proc. 45th STOC, pages 705-714, 2013.
A. Coja-Oghlan and K. Panagiotou. The asymptotic k-sat threshold. Advances in Mathematics, 288:985-1068, 2016.
A. Coja-Oghlan and L. Zdeborová. The condensation transition in random hypergraph 2-coloring. In Proc. 23rd SODA, pages 241-250, 2012.
P. Contucci, S. Dommers, C. Giardina, and S. Starr. Antiferromagnetic potts model on the Erdős-Rényi random graph. Communications in Mathematical Physics, 323:517-554, 2013.
A. Dembo, A. Montanari, A. Sly, and N. Sun. The replica symmetric solution for potts models on d-regular graphs. Comm. Math. Phys., 327:551-575, 2014.
J. Ding, A. Sly, and N. Sun. Maximum independent sets on random regular graphs. Maximum independent sets on random regular graphs, arXiv:1310.4787, 2013.
J. Ding, A. Sly, and N. Sun. Satisfiability threshold for random regular nae-sat. In Proc. 46th STOC, pages 814-822, 2014.
J. Ding, A. Sly, and N. Sun. Proof of the satisfiability conjecture for large k. In Proc. 47th STOC, pages 59-68, 2015.
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D. Gamarnik and M. Sudan. Performance of the survey propagation-guided decimation algorithm for the random nae-k-sat problem. Performance of the Survey Propagation-guided decimation algorithm for the random NAE-K-SAT problem, arXiv:1402.0052, 2014.
F. Krzakala, A. Montanari, F. Ricci-Tersenghi, G. Semerjian, and L. Zdeborova. Gibbs states and and the set of solutions of random constraint satisfaction problems. Proc. National Academy of Sciences, 104:10318-10323, 2007.
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On Higher-Order Fourier Analysis over Non-Prime Fields
The celebrated Weil bound for character sums says that for any low-degree polynomial P and any additive character chi, either chi(P) is a constant function or it is distributed close to uniform. The goal of higher-order Fourier analysis is to understand the connection between the algebraic and analytic properties of polynomials (and functions, generally) at a more detailed level. For instance, what is the tradeoff between the equidistribution of chi(P) and its "structure"?
Previously, most of the work in this area was over fields of prime order. We extend the tools of higher-order Fourier analysis to analyze functions over general finite fields. Let K be a field extension of a prime finite field F_p. Our technical results are:
1. If P: K^n -> K is a polynomial of degree <= d, and E[chi(P(x))] > |K|^{-s} for some s > 0 and non-trivial additive character chi, then P is a function of O_{d, s}(1) many non-classical polynomials of weight degree < d. The definition of non-classical polynomials over non-prime fields is one of the contributions of this work.
2. Suppose K and F are of bounded order, and let H be an affine subspace of K^n. Then, if P: K^n -> K is a polynomial of degree d that is sufficiently regular, then (P(x): x in H) is distributed almost as uniformly as possible subject to constraints imposed by the degree of P. Such a theorem was previously known for H an affine subspace over a prime field.
The tools of higher-order Fourier analysis have found use in different areas of computer science, including list decoding, algorithmic decomposition and testing. Using our new results, we revisit some of these areas.
(i) For any fixed finite field K, we show that the list decoding radius of the generalized Reed Muller code over K equals the minimum distance of the code.
(ii) For any fixed finite field K, we give a polynomial time algorithm to decide whether a given polynomial P: K^n -> K can be decomposed as a particular composition of lesser degree polynomials.
(iii) For any fixed finite field K, we prove that all locally characterized affine-invariant properties of functions f: K^n -> K are testable with one-sided error.
finite fields
higher order fourier analysis
coding theory
property testing
23:1-23:29
Regular Paper
Arnab
Bhattacharyya
Arnab Bhattacharyya
Abhishek
Bhowmick
Abhishek Bhowmick
Chetan
Gupta
Chetan Gupta
10.4230/LIPIcs.APPROX-RANDOM.2016.23
Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron. Testing Reed-Muller codes. IEEE Trans. Inform. Theory, 51(11):4032-4039, 2005. URL: http://dx.doi.org/10.1109/TIT.2005.856958.
http://dx.doi.org/10.1109/TIT.2005.856958
S. Arora and M. Sudan. Improved low-degree testing and its applications. Combinatorica, 23(3):365-426, 2003.
László Babai, Lance Fortnow, Leonid A. Levin, and Mario Szegedy. Checking computations in polylogarithmic time. In Proc. 23rd Annual ACM Symposium on the Theory of Computing, pages 21-32, New York, 1991. ACM Press.
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Arnab Bhattacharyya. Polynomial decompositions in polynomial time. In Proc. 22nd Annual European Symposium on Algorithms, pages 125-136, 2014.
Arnab Bhattacharyya and Abhishek Bhowmick. Using higher-order fourier analysis over general fields. CoRR, abs/1505.00619, 2015. URL: http://arxiv.org/abs/1505.00619.
http://arxiv.org/abs/1505.00619
Arnab Bhattacharyya, Victor Chen, Madhu Sudan, and Ning Xie. Testing linear-invariant non-linear properties. Theory Comput., 7(1):75-99, 2011.
Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, and Shachar Lovett. Every locally characterized affine-invariant property is testable. In Proc. 45th Annual ACM Symposium on the Theory of Computing, pages 429-436, 2013.
Arnab Bhattacharyya, Eldar Fischer, and Shachar Lovett. Testing low complexity affine-invariant properties. In Proc. 24th ACM-SIAM Symposium on Discrete Algorithms, pages 1337-1355, 2013.
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Arnab Bhattacharyya, Pooya Hatami, and Madhur Tulsiani. Algorithmic regularity for polynomials and applications. In Proc. 26th ACM-SIAM Symposium on Discrete Algorithms, pages 1870-1889, 2015.
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http://arxiv.org/abs/1506.02047
Abhishek Bhowmick and Shachar Lovett. List decoding Reed-Muller codes over small fields. In Proc. 47th Annual ACM Symposium on the Theory of Computing, pages 277-285, New York, NY, USA, 2015. ACM.
Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. J. Comp. Sys. Sci., 47:549-595, 1993. Earlier version in STOC'90.
A. Bogdanov and E. Viola. Pseudorandom bits for polynomials. In Proc. 48^th IEEE Symp. on Foundations of Computer Science (FOCS'07), 2007.
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P. Elias. List decoding for noisy channels. Technical Report 335, Research Laboratory of Electronics, MIT, 1957.
Uriel Feige, Shafi Goldwasser, László Lovász, Shmuel Safra, and Mario Szegedy. Interactive proofs and the hardness of approximating cliques. J. ACM, 43(2):268-292, 1996.
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O. Goldreich, R. Rubinfeld, and M. Sudan. Learning polynomials with queries: The highly noisy case. SIAM J. Discrete Math., 13(4):535-570, 2000.
Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45:653-750, 1998.
Oded Goldreich and Tali Kaufman. Proximity oblivious testing and the role of invariances. In Studies in Complexity and Cryptography, pages 173-190. Springer, 2011.
Oded Goldreich and Dana Ron. On proximity oblivious testing. SIAM J. Comput., 40(2):534-566, 2011.
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Parikshit Gopalan, Ryan O'Donnell, Rocco A. Servedio, Amir Shpilka, and Karl Wimmer. Testing Fourier dimensionality and sparsity. In Proc. 36th Annual International Conference on Automata, Languages, and Programming, pages 500-512, 2009.
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Ben Green and Terence Tao. The distribution of polynomials over finite fields, with applications to the Gowers norms. Contrib. Discrete Math., 4(2), 2009.
Elena Grigorescu, Tali Kaufman, and Madhu Sudan. Succinct representation of codes with applications to testing. SIAM Journal on Discrete Mathematics, 26(4):1618-1634, 2012.
Alan Guo, Swastik Kopparty, and Madhu Sudan. New affine-invariant codes from lifting. In Proceedings of the 4th conference on Innovations in Theoretical Computer Science, pages 529-540. ACM, 2013.
V. Guruswami. List Decoding of Error-Correcting Codes, volume 3282 of Lecture Notes in Computer Science. Springer, 2004.
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Hamed Hatami and Shachar Lovett. Estimating the distance from testable affine-invariant properties. In Proc. 54th Annual IEEE Symposium on Foundations of Computer Science, pages 237-242. IEEE, 2013.
Tali Kaufman and Simon Litsyn. Almost orthogonal linear codes are locally testable. In Proc. 46th Annual IEEE Symposium on Foundations of Computer Science, pages 317-326. IEEE, 2005.
Tali Kaufman and Shachar Lovett. Worst case to average case reductions for polynomials. In Proc. 49th Annual IEEE Symposium on Foundations of Computer Science, pages 166-175, 2008.
Tali Kaufman and Dana Ron. Testing polynomials over general fields. SIAM J. on Comput., 36(3):779-802, 2006.
Tali Kaufman and Madhu Sudan. Algebraic property testing: the role of invariance. In Proc. 40th Annual ACM Symposium on the Theory of Computing, pages 403-412, 2008.
Shachar Lovett, Roy Meshulam, and Alex Samorodnitsky. Inverse conjecture for the Gowers norm is false. In Proc. 40th Annual ACM Symposium on the Theory of Computing, pages 547-556, New York, NY, USA, 2008. ACM.
R. Pellikaan and X. Wu. List decoding of q-ary Reed-Muller codes. IEEE Transactions on Information Theory, 50(4):679-682, 2004.
Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. on Comput., 25:252-271, 1996.
M. Sudan. Decoding of Reed-Solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997. URL: https://citeseer.ist.psu.edu/sudan97decoding.html.
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Terence Tao. Higher Order Fourier Analysis, volume 142 of Graduate Studies in Mathematics. American Mathematical Society, 2012.
Terence Tao and Tamar Ziegler. The inverse conjecture for the Gowers norm over finite fields via the correspondence principle. Analysis &PDE, 3(1):1-20, 2010.
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Andre Weil. Sur les courbes algébriques et les varietes qui s'en deduisent. Actualites Sci. et Ind., 1041, 1948.
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Yuichi Yoshida. A characterization of locally testable affine-invariant properties via decomposition theorems. In Proc. 46th Annual ACM Symposium on the Theory of Computing, pages 154-163, 2014.
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Bounded Independence vs. Moduli
Let k = k(n) be the largest integer such that there exists a k-wise uniform distribution over {0,1}^n that is supported on the set S_m := {x in {0,1}^n: sum_i x_i equiv 0 mod m}, where m is any integer. We show that Omega(n/m^2 log m) <= k <= 2n/m + 2. For k = O(n/m) we also show that any k-wise uniform distribution puts probability mass at most 1/m + 1/100 over S_m. For any fixed odd m there is k \ge (1 - Omega(1))n such that any k-wise uniform distribution lands in S_m with probability exponentially close to |S_m|/2^n; and this result is false for any even m.
Bounded independence
Modulus
24:1-24:9
Regular Paper
Ravi
Boppana
Ravi Boppana
Johan
Håstad
Johan Håstad
Chin Ho
Lee
Chin Ho Lee
Emanuele
Viola
Emanuele Viola
10.4230/LIPIcs.APPROX-RANDOM.2016.24
Louay M. J. Bazzi. Polylogarithmic independence can fool DNF formulas. SIAM J. Comput., 38(6):2220-2272, 2009.
Mark Braverman. Polylogarithmic independence fools AC^0 circuits. J. of the ACM, 57(5), 2010.
Neal Carothers. A short course on approximation theory. Available at http://personal.bgsu.edu/∼carother/Approx.html.
Suresh Chari, Pankaj Rohatgi, and Aravind Srinivasan. Improved algorithms via approximations of probability distributions. J. Comput. System Sci., 61(1):81-107, 2000. URL: http://dx.doi.org/10.1006/jcss.1999.1695.
http://dx.doi.org/10.1006/jcss.1999.1695
E. Cheney. Introduction to approximation theory. McGraw-Hill, New York, New York, 1966.
Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, and Emanuele Viola. Bounded independence fools halfspaces. SIAM J. on Computing, 39(8):3441-3462, 2010.
Ilias Diakonikolas, Daniel Kane, and Jelani Nelson. Bounded independence fools degree-2 threshold functions. In 51th IEEE Symp. on Foundations of Computer Science (FOCS). IEEE, 2010.
Guy Even, Oded Goldreich, Michael Luby, Noam Nisan, and Boban Velickovic. Efficient approximation of product distributions. Random Struct. Algorithms, 13(1):1-16, 1998.
Parikshit Gopalan, Ryan O'Donnell, Yi Wu, and David Zuckerman. Fooling functions of halfspaces under product distributions. In 25th IEEE Conf. on Computational Complexity (CCC), pages 223-234. IEEE, 2010.
Chin Ho Lee and Emanuele Viola. Some limitations of the sum of small-bias distributions. Available at http://www.ccs.neu.edu/home/viola/, 2015.
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Raghu Meka and David Zuckerman. Small-bias spaces for group products. In 13th Workshop on Randomization and Computation (RANDOM), volume 5687 of Lecture Notes in Computer Science, pages 658-672. Springer, 2009.
Alexander A. Razborov. A simple proof of Bazzi’s theorem. ACM Transactions on Computation Theory (TOCT), 1(1), 2009.
Avishay Tal. Tight bounds on The Fourier Spectrum of AC⁰. Electronic Colloquium on Computational Complexity, Technical Report TR14-174, 2014.
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Emanuele Viola and Avi Wigderson. Norms, XOR lemmas, and lower bounds for polynomials and protocols. Theory of Computing, 4:137-168, 2008.
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Approximating Subadditive Hadamard Functions on Implicit Matrices
An important challenge in the streaming model is to maintain small-space approximations of entrywise functions performed on a matrix that is generated by the outer product of two vectors given as a stream. In other works, streams typically define matrices in a standard way via a sequence of updates, as in the
work of Woodruff [22] and others. We describe the matrix formed by the outer product, and other matrices that do not fall into this category, as implicit matrices. As such, we consider the general problem of computing over such implicit matrices with Hadamard functions, which are functions applied entrywise on a matrix. In this paper, we apply this generalization to provide new techniques for identifying independence between two data streams. The previous state of the art algorithm of Braverman and Ostrovsky [9] gave a (1 +- epsilon)-approximation for the L_1 distance between the joint and product of the marginal distributions, using space O(log^{1024}(nm) epsilon^{-1024}), where m is the length of the stream and n denotes the size of the universe from which stream elements are drawn. Our general techniques include the L_1 distance as a special case, and we give an improved space bound of O(log^{12}(n) log^{2}({nm}/epsilon) epsilon^{-7}).
Streaming Algorithms
Measuring Independence
Hadamard Functions
Implicit Matrices
25:1-25:19
Regular Paper
Vladimir
Braverman
Vladimir Braverman
Alan
Roytman
Alan Roytman
Gregory
Vorsanger
Gregory Vorsanger
10.4230/LIPIcs.APPROX-RANDOM.2016.25
Noga Alon, Alexandr Andoni, Tali Kaufman, Kevin Matulef, Ronitt Rubinfeld, and Ning Xie. Testing k-wise and almost k-wise independence. In Proceedings of the 39th annual ACM Symposium on Theory of Computing, 2007.
Noga Alon, Oded Goldreich, and Yishay Mansour. Almost k-wise independence versus k-wise independence. Information Processing Letters, 88(3):107-110, 2003.
Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. In Proceedings of the 28th annual ACM Symposium on Theory of Computing, 1996.
Tuğkan Batu, Eldar Fischer, Lance Fortnow, Ravi Kumar, Ronitt Rubinfeld, and Patrick White. Testing random variables for independence and identity. In Proceedings of the 42nd annual IEEE Symposium on Foundations of Computer Science, 2001.
Tuğkan Batu, Lance Fortnow, Ronitt Rubinfeld, Warren D. Smith, and Patrick White. Testing that distributions are close. In Proceedings of the 41st annual IEEE Symposium on Foundations of Computer Science, 2000.
Tuğkan Batu, Lance Fortnow, Ronitt Rubinfeld, Warren D. Smith, and Patrick White. Testing closeness of discrete distributions. Journal of the ACM, 60(1):4, 2013.
Tuğkan Batu, Ravi Kumar, and Ronitt Rubinfeld. Sublinear algorithms for testing monotone and unimodal distributions. In Proceedings of the 36th annual ACM Symposium on Theory of Computing, 2004.
Vladimir Braverman, Kai-Min Chung, Zhenming Liu, Michael Mitzenmacher, and Rafail Ostrovsky. AMS Without 4-Wise Independence on Product Domains. In Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science, 2010.
Vladimir Braverman and Rafail Ostrovsky. Measuring independence of datasets. In Proceedings of the 42nd annual ACM Symposium on Theory of Computing, 2010.
Vladimir Braverman and Rafail Ostrovsky. Zero-one frequency laws. In Proceedings of the 42nd annual ACM Symposium on Theory of Computing, 2010.
Vladimir Braverman and Rafail Ostrovsky. Generalizing the layering method of Indyk and Woodruff: Recursive sketches for frequency-based vectors on streams. In International Workshop on Approximation Algorithms for Combinatorial Optimization Problems. Springer, 2013.
Dominique Guillot, Apoorva Khare, and Bala Rajaratnam. Complete characterization of hadamard powers preserving loewner positivity, monotonicity, and convexity. Journal of Mathematical Analysis and Applications, 425(1):489-507, 2015.
Roger A. Horn and Charles R. Johnson. Topics in matrix analysis. Cambridge University Press, Cambridge, 1991.
Piotr Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation. Journal of the ACM, 53(3):307-323, 2006.
Piotr Indyk and Andrew McGregor. Declaring independence via the sketching of sketches. In Proceedings of the 19th annual ACM-SIAM Symposium on Discrete Algorithms, 2008.
Ralph Kimball and Joe Caserta. The Data Warehouse ETL Toolkit: Practical Techniques for Extracting, Cleaning, Conforming, and Delivering Data. John Wiley &Sons, 2004.
Erich L. Lehmann and Joseph P. Romano. Testing statistical hypotheses. Springer Science &Business Media, 2006.
Andrew McGregor and Hoa T. Vu. Evaluating bayesian networks via data streams. In Proceedings of the 21st International Computing and Combinatorics Conference, 2015.
Viswanath Poosala and Yannis E. Ioannidis. Selectivity estimation without the attribute value independence assumption. In Proceedings of the 23rd International Conference on Very Large Data Bases, 1997.
Ronitt Rubinfeld and Rocco A. Servedio. Testing monotone high-dimensional distributions. In Proceedings of the 37th annual ACM Symposium on Theory of Computing, 2005.
Amit Sahai and Salil Vadhan. A complete problem for statistical zero knowledge. Journal of the ACM, 50(2):196-249, 2003.
David P. Woodruff. Sketching as a tool for numerical linear algebra. CoRR, abs/1411.4357, 2014. URL: http://arxiv.org/abs/1411.4357.
http://arxiv.org/abs/1411.4357
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Local Convergence and Stability of Tight Bridge-Addable Graph Classes
A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if G is bridge-addable and G_n is a uniform n-vertex graph from G, then G_n is connected with probability at least (1+o(1))e^{-1/2}. The constant e^{-1/2} is best possible since it is reached for the class of forests.
In this paper we prove a form of uniqueness in this statement: if G is a bridge-addable class and the random graph G_n is connected with probability close to e^{-1/2}, then G_n is asymptotically close to a uniform forest in some "local" sense. For example, if the probability converges to e^{-1/2}, then G_n converges for the Benjamini-Schramm topology, to the uniform infinite random forest F_infinity. This result is reminiscent of so-called "stability results" in extremal graph theory, with the difference that here the "stable" extremum is not a graph but a graph class.
bridge-addable classes
random graphs
stability
local convergence
random forests
26:1-26:11
Regular Paper
Guillaume
Chapuy
Guillaume Chapuy
Guillem
Perarnau
Guillem Perarnau
10.4230/LIPIcs.APPROX-RANDOM.2016.26
Louigi Addario-Berry, Colin McDiarmid, and Bruce Reed. Connectivity for bridge-addable monotone graph classes. Combin. Probab. Comput., 21(6):803-815, 2012.
Paul Balister, Béla Bollobás, and Stefanie Gerke. Connectivity of addable graph classes. J. Combin. Theory Ser. B, 98(3):577-584, 2008.
Itai Benjamini and Oded Schramm. Recurrence of distributional limits of finite planar graphs. Electron. J. Probab., 6:no. 23, 13 pp. (electronic), 2001.
Guillaume Chapuy and Guillem Perarnau. Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture. Extended abstract in the proceedings of SODA 2016. Long version submitted for publication, see arXiv:1504.06344., 2015.
Guillaume Chapuy and Guillem Perarnau. Local convergence and stability of tight bridge-addable graph classes. In preparation., 2016.
Paul Erdős. On some new inequalities concerning extremal properties of graphs. In Theory of Graphs (Proc. Colloq., Tihany, 1966), pages 77-81, 1966.
Paul Erdős. Some recent results on extremal problems in graph theory. Results, Theory of Graphs (Internat. Sympos., Rome, 1966), Gordon and Breach, New York, pages 117-123, 1967.
Paul Erdős and M Simonovits. A limit theorem in graph theory. In Studia Sci. Math. Hung. Citeseer, 1966.
Mihyun Kang and Konstantinos Panagiotou. On the connectivity of random graphs from addable classes. J. Combin. Theory Ser. B, 103(2):306-312, 2013.
László Lovász. Large networks and graph limits, volume 60 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2012.
Colin McDiarmid, Angelika Steger, and Dominic J. A. Welsh. Random graphs from planar and other addable classes. In Topics in discrete mathematics, volume 26 of Algorithms Combin., pages 231-246. Springer, Berlin, 2006.
Alfréd Rényi. Some remarks on the theory of trees. Magyar Tud. Akad. Mat. Kutató Int. Közl., 4:73-85, 1959.
Miklós Simonovits. A method for solving extremal problems in graph theory, stability problems. In Theory of Graphs (Proc. Colloq., Tihany, 1966), pages 279-319, 1968.
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Belief Propagation on Replica Symmetric Random Factor Graph Models
According to physics predictions, the free energy of random factor graph models that satisfy a certain "static replica symmetry" condition can be calculated via the Belief Propagation message passing scheme [Krzakala et al. PNAS, 2007]. Here we prove this conjecture for a wide class of random factor graph models. Specifically, we show that the messages constructed just as in the case of acyclic factor graphs asymptotically satisfy the Belief Propagation equations and that the free energy density is given by the Bethe free energy formula.
Gibbs distributions
Belief Propagation
Bethe Free Energy
Random k-SAT
27:1-27:15
Regular Paper
Amin
Coja-Oghlan
Amin Coja-Oghlan
Will
Perkins
Will Perkins
10.4230/LIPIcs.APPROX-RANDOM.2016.27
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Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem
An important theorem of Banaszczyk (Random Structures & Algorithms 1998) states that for any sequence of vectors of l_2 norm at most 1/5 and any convex body K of Gaussian measure 1/2 in R^n, there exists a signed combination of these vectors which lands inside K. A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. to find an efficient algorithm which constructs the signed combination.
We make progress towards this goal along several fronts. As our first contribution, we show an equivalence between Banaszczyk's theorem and the existence of O(1)-subgaussian distributions over signed combinations. For the case of symmetric convex bodies, our equivalence implies the existence of a universal signing algorithm (i.e. independent of the body), which simply samples from the subgaussian sign distribution and checks to see if the associated combination lands inside the body. For asymmetric convex bodies, we provide a novel recentering procedure, which allows us to reduce to the case where the body is symmetric.
As our second main contribution, we show that the above framework can be efficiently implemented when the vectors have length O(1/sqrt{log n}), recovering Banaszczyk's results under this stronger assumption. More precisely, we use random walk techniques to produce the required O(1)-subgaussian signing distributions when the vectors have length O(1/sqrt{log n}), and use a stochastic gradient ascent method to implement the recentering procedure for asymmetric bodies.
Discrepancy
Vector Balancing
Convex Geometry
28:1-28:12
Regular Paper
Daniel
Dadush
Daniel Dadush
Shashwat
Garg
Shashwat Garg
Shachar
Lovett
Shachar Lovett
Aleksandar
Nikolov
Aleksandar Nikolov
10.4230/LIPIcs.APPROX-RANDOM.2016.28
David Applegate and Ravi Kannan. Sampling and integration of near log-concave functions. In Proceedings of the 23rd annual ACM symposium on Theory of Computing, pages 156-163, 1991.
Wojciech Banaszczyk. Balancing vectors and Gaussian measures of n-dimensional convex bodies. Random Structures Algorithms, 12(4):351-360, 1998.
Wojciech Banaszczyk. On series of signed vectors and their rearrangements. Random Struct. Algorithms, 40(3):301-316, 2012. URL: http://dx.doi.org/10.1002/rsa.20373.
http://dx.doi.org/10.1002/rsa.20373
Nikhil Bansal. Constructive algorithms for discrepancy minimization. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science FOCS 2010, pages 3-10. IEEE Computer Soc., Los Alamitos, CA, 2010.
Nikhil Bansal, Daniel Dadush, and Shashwat Garg. An algorithm for Komlós conjecture matching Banaszczyk’s bound. To appears in FOCS, 2016.
Franck Barthe, Olivier Guédon, Shahar Mendelson, and Assaf Naor. A probabilistic approach to the geometry of the lⁿ_p-ball. Ann. Probab., 33(2):480-513, 2005.
József Beck. Roth’s estimate of the discrepancy of integer sequences is nearly sharp. Combinatorica, 1(4):319-325, 1981.
József Beck and Tibor Fiala. "Integer-making" theorems. Discrete Appl. Math., 3(1):1-8, 1981.
Boris Bukh. An improvement of the Beck-Fiala theorem. CoRR, abs/1306.6081, 2013.
Bernard Chazelle. The discrepancy method. Cambridge University Press, Cambridge, 2000. Randomness and complexity.
William Chen, Anand Srivastav, Giancarlo Travaglini, et al. A Panorama of Discrepancy Theory, volume 2107. Springer, 2014.
Ben Cousins and Santosh Vempala. A cubic algorithm for computing Gaussian volume. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1215-1228. ACM, New York, 2014.
Ronen Eldan and Mohit Singh. Efficient algorithms for discrepancy minimization in convex sets. CoRR, abs/1409.2913, 2014.
Esther Ezra and Shachar Lovett. On the Beck-Fiala conjecture for random set systems. Electronic Colloquium on Computational Complexity (ECCC), 22:190, 2015.
Nicholas J. A. Harvey, Roy Schwartz, and Mohit Singh. Discrepancy without partial colorings. In Approximation, randomization, and combinatorial optimization, volume 28 of LIPIcs. Leibniz Int. Proc. Inform., pages 258-273. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2014.
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Shachar Lovett and Raghu Meka. Constructive discrepancy minimization by walking on the edges. SIAM J. Comput., 44(5):1573-1582, 2015. Preliminary version in FOCS 2012.
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Aleksandar Nikolov. The Komlós conjecture holds for vector colorings. arXiv preprint arXiv:1301.4039, 2013.
Aleksandar Nikolov and Kunal Talwar. Approximating hereditary discrepancy via small width ellipsoids. In Symposium on Discrete Algorithms, SODA, pages 324-336, 2015.
Thomas Rothvoss. Constructive discrepancy minimization for convex sets. In 55th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2014, pages 140-145. IEEE Computer Soc., Los Alamitos, CA, 2014.
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Aravind Srinivasan. Improving the discrepancy bound for sparse matrices: better approximations for sparse lattice approximation problems. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (New Orleans, LA, 1997), pages 692-701. ACM, New York, 1997.
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http://dx.doi.org/10.1007/978-3-642-54075-2
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On the Beck-Fiala Conjecture for Random Set Systems
Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,Sigma), where each element x in X lies in t randomly selected sets of Sigma, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |\Sigma| >= |X| the hereditary discrepancy of (X,Sigma) is with high probability O(sqrt{t log t}); and when |X| >> |\Sigma|^t the hereditary discrepancy of (X,Sigma) is with high probability O(1). The first bound combines the Lovasz Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.
Discrepancy theory
Beck-Fiala conjecture
Random set systems
29:1-29:10
Regular Paper
Esther
Ezra
Esther Ezra
Shachar
Lovett
Shachar Lovett
10.4230/LIPIcs.APPROX-RANDOM.2016.29
Noga Alon and Joel H. Spencer. The probabilistic method. Wiley-Interscience series in discrete mathematics and optimization. Wiley, New York, Chichester, Weinheim, 2000.
Wojciech Banaszczyk. Balancing vectors and Gaussian measures of n-dimensional convex bodies. Random Structures &Algorithms, 12(4):351-360, 1998.
Nikhil Bansal, Daniel Dadush, and Shashwat Garg. An algorithm for Komlós conjecture matching Banaszczyk’s bound. arXiv preprint arXiv:1605.02882, 2016.
József Beck and Tibor Fiala. Integer-making theorems. Discrete Applied Mathematics, 3(1):1-8, 1981.
Debe Bednarchak and Martin Helm. A note on the Beck-Fiala theorem. Combinatorica, 17(1):147-149, 1997.
Boris Bukh. An improvement of the Beck-Fiala theorem. arXiv preprint arXiv:1306.6081, 2013.
Bernard Chazelle. The discrepancy method: randomness and complexity. Cambridge University Press, Cambridge, New York, 2000.
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Martin Helm. On the Beck-Fiala theorem. Discrete mathematics, 207(1):73-87, 1999.
Jiri Matoušek. Geometric discrepancy: An illustrated guide, volume 18. Springer Science &Business Media, 2009.
Robin A Moser. A constructive proof of the Lovász local lemma. In Proceedings of the forty-first annual ACM symposium on Theory of computing, pages 343-350. ACM, 2009.
Robin A Moser and Gábor Tardos. A constructive proof of the general Lovász local lemma. Journal of the ACM (JACM), 57(2):11, 2010.
Joel Spencer. Six standard deviations suffice. Transactions of the American Mathematical Society, 289(2):679-706, 1985.
Richard M. Wilson. A diagonal form for the incidence matrices of t-subsets vs. k-subsets. European Journal of Combinatorics, 11(6):609-615, 1990.
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The Niceness of Unique Sink Orientations
Random Edge is the most natural randomized pivot rule for the simplex algorithm. Considerable progress has been made recently towards fully understanding its behavior. Back in 2001, Welzl introduced the concepts of reachmaps and niceness of Unique Sink Orientations (USO), in an effort to better understand the behavior of Random Edge. In this paper, we initiate the systematic study of these concepts. We settle the questions that were asked by Welzl about the niceness of (acyclic) USO. Niceness implies natural upper bounds for Random Edge and we provide evidence that these are tight or almost tight in many interesting cases. Moreover, we show that Random Edge is polynomial on at least n^{Omega(2^n)} many (possibly cyclic) USO. As a bonus, we describe a derandomization of Random Edge which achieves the same asymptotic upper bounds with respect to niceness.
random edge
unique sink orientation
random walk
reachmap
niceness
30:1-30:14
Regular Paper
Bernd
Gärtner
Bernd Gärtner
Antonis
Thomas
Antonis Thomas
10.4230/LIPIcs.APPROX-RANDOM.2016.30
Ilan Adler, Christos H. Papadimitriou, and Aviad Rubinstein. On simplex pivoting rules and complexity theory. In Jon Lee and Jens Vygen, editors, Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Bonn, Germany, June 23-25, 2014. Proceedings, volume 8494 of Lecture Notes in Computer Science, pages 13-24. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-319-07557-0_2.
http://dx.doi.org/10.1007/978-3-319-07557-0_2
Yoshikazu Aoshima, David Avis, Theresa Deering, Yoshitake Matsumoto, and Sonoko Moriyama. On the existence of Hamiltonian paths for history based pivot rules on acyclic unique sink orientations of hypercubes. Discrete Applied Mathematics, 160(15):2104-2115, 2012. URL: http://dx.doi.org/10.1016/j.dam.2012.05.023.
http://dx.doi.org/10.1016/j.dam.2012.05.023
József Balogh and Robin Pemantle. The Klee-Minty random edge chain moves with linear speed. Random Structures &Algorithms, 30(4):464-483, 2007. URL: http://dx.doi.org/10.1002/rsa.20127.
http://dx.doi.org/10.1002/rsa.20127
Yann Disser and Martin Skutella. The simplex algorithm is NP-mighty. 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 858-872. SIAM, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.59.
http://dx.doi.org/10.1137/1.9781611973730.59
John Fearnley and Rahul Savani. The complexity of the simplex method. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 201-208. ACM, 2015. URL: http://dx.doi.org/10.1145/2746539.2746558.
http://dx.doi.org/10.1145/2746539.2746558
John Fearnley and Rahul Savani. The complexity of all-switches strategy improvement. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 130-139. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch10.
http://dx.doi.org/10.1137/1.9781611974331.ch10
Jan Foniok, Bernd Gärtner, Lorenz Klaus, and Markus Sprecher. Counting unique-sink orientations. Discrete Applied Mathematics, 163, Part 2:155-164, 2014. URL: http://dx.doi.org/10.1016/j.dam.2013.07.017.
http://dx.doi.org/10.1016/j.dam.2013.07.017
Oliver Friedmann. A subexponential lower bound for Zadeh’s pivoting rule for solving linear programs and games. In Oktay Günlük and Gerhard J. Woeginger, editors, Integer Programming and Combinatoral Optimization - 15th International Conference, IPCO 2011, New York, NY, USA, June 15-17, 2011. Proceedings, volume 6655 of Lecture Notes in Computer Science, pages 192-206. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-20807-2_16.
http://dx.doi.org/10.1007/978-3-642-20807-2_16
Oliver Friedmann, Thomas Dueholm Hansen, and Uri Zwick. Subexponential lower bounds for randomized pivoting rules for the simplex algorithm. In Lance Fortnow and Salil P. Vadhan, editors, Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 283-292. ACM, 2011. URL: http://dx.doi.org/10.1145/1993636.1993675.
http://dx.doi.org/10.1145/1993636.1993675
Bernd Gärtner. The Random-Facet simplex algorithm on combinatorial cubes. Random Structures &Algorithms, 20(3), 2002. URL: http://dx.doi.org/10.1002/rsa.10034.
http://dx.doi.org/10.1002/rsa.10034
Bernd Gärtner, Walter D. Morris Jr., and Leo Rüst. Unique sink orientations of grids. Algorithmica, 51(2):200-235, 2008. URL: http://dx.doi.org/10.1007/s00453-007-9090-x.
http://dx.doi.org/10.1007/s00453-007-9090-x
Bernd Gärtner and Ingo Schurr. Linear programming and unique sink orientations. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, pages 749-757. ACM Press, 2006. URL: http://dl.acm.org/citation.cfm?id=1109557.1109639.
http://dl.acm.org/citation.cfm?id=1109557.1109639
Bernd Gärtner and Antonis Thomas. The complexity of recognizing unique sink orientations. 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 341-353. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.341.
http://dx.doi.org/10.4230/LIPIcs.STACS.2015.341
Bernd Gärtner and Antonis Thomas. The Niceness of Unique Sink Orientations. CoRR, June 2016. URL: http://arxiv.org/abs/1606.07709.
http://arxiv.org/abs/1606.07709
Peter L. Hammer, Bruno Simeone, Thomas M. Liebling, and Dominique de Werra. From linear separability to unimodality: A hierarchy of pseudo-boolean functions. SIAM J. Discrete Math., 1(2):174-184, 1988. URL: http://dx.doi.org/10.1137/0401019.
http://dx.doi.org/10.1137/0401019
Thomas Dueholm Hansen, Mike Paterson, and Uri Zwick. Improved upper bounds for random-edge and random-jump on abstract cubes. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 874-881. SIAM, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.65.
http://dx.doi.org/10.1137/1.9781611973402.65
Thomas Dueholm Hansen and Uri Zwick. Random-edge is slower than random-facet on abstract cubes. To appear in ICALP 2016., 2016. URL: http://cs.au.dk/~tdh/papers/Random-Edge-AUSO.pdf.
http://cs.au.dk/~tdh/papers/Random-Edge-AUSO.pdf
Gil Kalai. A subexponential randomized simplex algorithm (extended abstract). In S. Rao Kosaraju, Mike Fellows, Avi Wigderson, and John A. Ellis, editors, Proceedings of the 24th Annual ACM Symposium on Theory of Computing, May 4-6, 1992, Victoria, British Columbia, Canada, pages 475-482. ACM, 1992. URL: http://dx.doi.org/10.1145/129712.129759.
http://dx.doi.org/10.1145/129712.129759
Lorenz Klaus and Hiroyuki Miyata. Enumeration of PLCP-orientations of the 4-cube. European Journal of Combinatorics, 50:138-151, 2015. Combinatorial Geometries: Matroids, Oriented Matroids and Applications. Special Issue in Memory of Michel Las Vergnas. URL: http://dx.doi.org/10.1016/j.ejc.2015.03.010.
http://dx.doi.org/10.1016/j.ejc.2015.03.010
Victor Klee and George J. Minty. How good is the simplex algorithm? Inequalities III, pages 159-175, 1972.
Jiří Matoušek. Lower bounds for a subexponential optimization algorithm. Random Structures &Algorithms, 5(4):591-607, 1994. URL: http://dx.doi.org/10.1002/rsa.3240050408.
http://dx.doi.org/10.1002/rsa.3240050408
Jiří Matoušek. The number of unique-sink orientations of the hypercube*. Combinatorica, 26(1):91-99, 2006. URL: http://dx.doi.org/10.1007/s00493-006-0007-0.
http://dx.doi.org/10.1007/s00493-006-0007-0
Jiří Matoušek, Micha Sharir, and Emo Welzl. A subexponential bound for linear programming. Algorithmica, 16(4/5):498-516, 1996. URL: http://dx.doi.org/10.1007/BF01940877.
http://dx.doi.org/10.1007/BF01940877
Jiří Matoušek and Tibor Szabó. RANDOM EDGE can be exponential on abstract cubes. Advances in Mathematics, 204(1):262-277, 2006. URL: http://dx.doi.org/10.1109/FOCS.2004.56.
http://dx.doi.org/10.1109/FOCS.2004.56
Walter D. Morris Jr. Randomized pivot algorithms for P-matrix linear complementarity problems. Mathematical Programming, 92(2):285-296, 2002. URL: http://dx.doi.org/10.1007/s101070100268.
http://dx.doi.org/10.1007/s101070100268
Ingo Schurr and Tibor Szabó. Finding the sink takes some time: An almost quadratic lower bound for finding the sink of unique sink oriented cubes. Discrete & Computational Geometry, 31(4):627-642, 2004. URL: http://dx.doi.org/10.1007/s00454-003-0813-8.
http://dx.doi.org/10.1007/s00454-003-0813-8
Ingo Schurr and Tibor Szabó. Jumping doesn't help in abstract cubes. In Michael Jünger and Volker Kaibel, editors, Integer Programming and Combinatorial Optimization, 11th International IPCO Conference, 2005, volume 3509 of LNCS, pages 225-235. Springer, 2005. URL: http://dx.doi.org/10.1007/11496915_17.
http://dx.doi.org/10.1007/11496915_17
Alan Stickney and Layne Watson. Digraph models of Bard-type algorithms for the linear complementarity problem. Math. Oper. Res., 3(4):322-333, 1978. URL: http://dx.doi.org/10.1287/moor.3.4.322.
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http://dx.doi.org/10.1109/SFCS.2001.959931
Emo Welzl. i-Niceness. http://www.ti.inf.ethz.ch/ew/workshops/01-lc/problems/node7.html, 2001.
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Kathy Williamson-Hoke. Completely unimodal numberings of a simple polytope. Discrete Applied Mathematics, 20(1):69-81, 1988. URL: http://dx.doi.org/10.1016/0166-218X(88)90042-X.
http://dx.doi.org/10.1016/0166-218X(88)90042-X
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Uniqueness, Spatial Mixing, and Approximation for Ferromagnetic 2-Spin Systems
For anti-ferromagnetic 2-spin systems, a beautiful connection has been established, namely that the following three notions align perfectly: the uniqueness in infinite regular trees, the decay of correlations (also known as spatial mixing), and the approximability of the partition function. The uniqueness condition implies spatial mixing, and an FPTAS for the partition function exists based on spatial mixing. On the other hand, non-uniqueness implies some long range correlation, based on which NP-hardness reductions are built. These connections for ferromagnetic 2-spin systems are much less clear, despite their similarities to anti-ferromagnetic systems. The celebrated Jerrum-Sinclair Markov chain [JS93] works even if spatial mixing or uniqueness fails.
We provide some partial answers. We use (β,γ) to denote the (+,+) and (−,−) edge interactions and λ the external field, where βγ>1. If all fields satisfy λ<λ_c (assuming β≤γ), where λ_c=(γ/β)^{(Δ_c+1)/2} and Δ_c=(\sqrt{βγ}+1)/(\sqrt{βγ}−1), then a weaker version of spatial mixing holds in all trees. Moreover, if β≤1, then λ<λ_c is sufficient to guarantee strong spatial mixing and FPTAS. This improves the previous best algorithm, which is an FPRAS based on Markov chains and works for λ<γ/β [LLZ14a]. The bound λ_c is almost optimal. When β≤1, uniqueness holds in all infinite regular trees, if and only if λ≤λ^int_c, where λ^int_c=(γ/β)(⌈Δc⌉+1)/2. If we allow fields λ>λ^int′_c, where λ^int′_c=(γ/β)(⌊Δc⌋+2)/2, then approximating the partition function is #BIS-hard.
Approximate counting; Ising model; Spin systems; Correlation decay
31:1-31:26
Regular Paper
Heng
Guo
Heng Guo
Pinyan
Lu
Pinyan Lu
10.4230/LIPIcs.APPROX-RANDOM.2016.31
Jin-Yi Cai, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, Mark Jerrum, Daniel Štefankovič, and Eric Vigoda. #BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region. In RANDOM, pages 582-595, 2014.
Jin-Yi Cai and Michael Kowalczyk. Spin systems on k-regular graphs with complex edge functions. Theor. Comput. Sci., 461:2-16, 2012.
Martin E. Dyer, Leslie Ann Goldberg, Catherine S. Greenhill, and Mark Jerrum. The relative complexity of approximate counting problems. Algorithmica, 38(3):471-500, 2003.
Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. Inapproximability of the partition function for the antiferromagnetic Ising and Hard-Core models. CoRR,, 2012. URL: http://arxiv.org/abs/1203.2226.
http://arxiv.org/abs/1203.2226
Hans-Otto Georgii. Gibbs Measures and Phase Transitions, volume 9 of De Gruyter Studies in Mathematics. de Gruyter, Berlin, second edition, 2011.
Leslie Ann Goldberg and Mark Jerrum. The complexity of ferromagnetic Ising with local fields. Combinatorics, Probability & Computing, 16(1):43-61, 2007.
Leslie Ann Goldberg, Mark Jerrum, and Mike Paterson. The computational complexity of two-state spin systems. Random Struct. Algorithms, 23(2):133-154, 2003.
Mark Jerrum and Alistair Sinclair. Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput., 22(5):1087-1116, 1993.
Mark Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci., 43:169-188, 1986.
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Liang Li, Pinyan Lu, and Yitong Yin. Approximate counting via correlation decay in spin systems. In SODA, pages 922-940, 2012.
Liang Li, Pinyan Lu, and Yitong Yin. Correlation decay up to uniqueness in spin systems. In SODA, pages 67-84, 2013.
Jingcheng Liu, Pinyan Lu, and Chihao Zhang. The complexity of ferromagnetic two-spin systems with external fields. In RANDOM, pages 843-856, 2014.
Russell Lyons. The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys., 125(2):337-353, 1989.
Elchanan Mossel and Allan Sly. Exact thresholds for Ising-Gibbs samplers on general graphs. Annals of Probability, 41(1):294-328, 2013.
Alistair Sinclair, Piyush Srivastava, Daniel Štefankovič, and Yitong Yin. Spatial mixing and the connective constant: Optimal bounds. In SODA, pages 1549-1563, 2015.
Alistair Sinclair, Piyush Srivastava, and Marc Thurley. Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. In SODA, pages 941-953, 2012.
Allan Sly and Nike Sun. The computational hardness of counting in two-spin models on d-regular graphs. The Annals of Probability, 42(6):2383-2416, 2014.
Dror Weitz. Counting independent sets up to the tree threshold. In STOC, pages 140-149, 2006.
Jinshan Zhang, Heng Liang, and Fengshan Bai. Approximating partition functions of the two-state spin system. Inf. Process. Lett., 111(14):702-710, 2011.
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On Polynomial Approximations to AC^0
We make progress on some questions related to polynomial approximations of AC^0. It is known, from the works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that any AC^0 circuit of size s and depth d has an epsilon-error probabilistic polynomial over the reals of degree (log (s/epsilon))^{O(d)}. We improve this upper bound to (log s)^{O(d)}* log(1/epsilon), which is much better for small values of epsilon.
We give an application of this result by using it to resolve a question posed by Tal (ECCC 2014): we show that (log s)^{O(d)}* log(1/epsilon)-wise independence fools AC^0, improving on Tal's strengthening of Braverman's theorem (J. ACM 2010) that (log (s/epsilon))^{O(d)}-wise independence fools AC^0. Up to the constant implicit in the O(d), our result is tight. As far as we know, this is the first PRG construction for AC^0 that achieves optimal dependence on the error epsilon.
We also prove lower bounds on the best polynomial approximations to AC^0. We show that any polynomial approximating the OR function on n bits to a small constant error must have degree at least ~Omega(sqrt{log n}). This result improves exponentially on a recent lower bound demonstrated by Meka, Nguyen, and Vu (arXiv 2015).
Constant-depth Boolean circuits
Polynomials over reals
pseudo-random generators
k-wise independence
32:1-32:14
Regular Paper
Prahladh
Harsha
Prahladh Harsha
Srikanth
Srinivasan
Srikanth Srinivasan
10.4230/LIPIcs.APPROX-RANDOM.2016.32
Miklós Ajtai and Michael Ben-Or. A theorem on probabilistic constant depth computations. In Proc. 16th ACM Symp. on Theory of Computing (STOC), pages 471-474, 1984. URL: http://dx.doi.org/10.1145/800057.808715.
http://dx.doi.org/10.1145/800057.808715
Richard Beigel, Nick Reingold, and Daniel A. Spielman. The perceptron strikes back. In Proc. 6th IEEE Conf. on Structure in Complexity Theory, pages 286-291, 1991. URL: http://dx.doi.org/10.1109/SCT.1991.160270.
http://dx.doi.org/10.1109/SCT.1991.160270
Mark Braverman. Polylogarithmic independence fools AC⁰ circuits. J. ACM, 57(5), 2010. (Preliminary version in 24th IEEE Conference on Computational Complexity, 2009). URL: http://dx.doi.org/10.1145/1754399.1754401.
http://dx.doi.org/10.1145/1754399.1754401
Kevin P. Costello, Terence Tao, and Van Vu. Random symmetric matrices are almost surely nonsingular. Duke Math. J., 135(2):395-413, 2006. http://arxiv.org/abs/math/0505156, URL: http://dx.doi.org/10.1215/S0012-7094-06-13527-5.
http://dx.doi.org/10.1215/S0012-7094-06-13527-5
Paul Erdős. On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc., 51(12):898-902, 1945. URL: http://dx.doi.org/10.1090/S0002-9904-1945-08454-7.
http://dx.doi.org/10.1090/S0002-9904-1945-08454-7
Johan Håstad. Almost optimal lower bounds for small depth circuits. In Silvio Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 143-170. JAI Press, Greenwich, Connecticut, 1989. (Preliminary version in 18th STOC 1986). URL: http://www.csc.kth.se/~johanh/largesmalldepth.pdf.
http://www.csc.kth.se/~johanh/largesmalldepth.pdf
Johan Håstad. On the correlation of parity and small-depth circuits. SIAM J. Comput., 43(5):1699-1708, 2014. URL: http://dx.doi.org/10.1137/120897432.
http://dx.doi.org/10.1137/120897432
Russell Impagliazzo, William Matthews, and Ramamohan Paturi. A satisfiability algorithm for AC⁰. In Proc. 23rd Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 961-972, 2012. URL: http://arxiv.org/abs/1107.3127.
http://arxiv.org/abs/1107.3127
Swastik Kopparty and Srikanth Srinivasan. Certifying polynomials for AC⁰(parity) circuits, with applications. In Deepak D'Souza, Telikepalli Kavitha, and Jaikumar Radhakrishnan, editors, Proc. 32nd IARCS Annual Conf. on Foundations of Software Tech. and Theoretical Comp. Science (FSTTCS), volume 18 of LIPIcs, pages 36-47. Schloss Dagstuhl, 2012. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2012.36.
http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2012.36
Nathan Linial, Yishay Mansour, and Noam Nisan. Constant depth circuits, Fourier transform, and learnability. J. ACM, 40(3):607-620, 1993. (Preliminary version in 30th FOCS, 1989). URL: http://dx.doi.org/10.1145/174130.174138.
http://dx.doi.org/10.1145/174130.174138
John Edensor Littlewood and A. Cyril Offord. On the number of real roots of a random algebraic equation. J. London Math. Soc., s1-13(4):288-295, 1938. URL: http://dx.doi.org/10.1112/jlms/s1-13.4.288.
http://dx.doi.org/10.1112/jlms/s1-13.4.288
Michael Luby and Boban Velickovic. On deterministic approximation of DNF. Algorithmica, 16(4/5):415-433, 1996. (Preliminary version in 23rd STOC, 1991). URL: http://dx.doi.org/10.1007/BF01940873.
http://dx.doi.org/10.1007/BF01940873
Raghu Meka, Oanh Nguyen, and Van Vu. Anti-concentration for polynomials of independent random variables. arXiv 1507.00829, 2015. URL: https://arxiv.org/abs/1507.00829.
https://arxiv.org/abs/1507.00829
Noam Nisan and Avi Wigderson. Hardness vs. Randomness. J. Comput. Syst. Sci., 49(2):149-167, October 1994. (Preliminary version in 29th FOCS, 1988). URL: http://dx.doi.org/10.1016/S0022-0000(05)80043-1.
http://dx.doi.org/10.1016/S0022-0000(05)80043-1
Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. URL: http://analysisofbooleanfunctions.org/, URL: http://dx.doi.org/10.1017/CBO9781139814782.
http://dx.doi.org/10.1017/CBO9781139814782
Igor Carboni Oliveira and Rahul Santhanam. Majority is incompressible by AC⁰[p] circuits. In Proc. 30th Computational Complexity Conf., pages 124-157, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2015.124.
http://dx.doi.org/10.4230/LIPIcs.CCC.2015.124
Alexander A. Razborov. \foreignlanguagerussianНжние оценки размера схем ограниченной глубины в полном базисе, содержащем функцию логического сложения (Russian) [Lower bounds on the size of bounded depth circuits over a complete basis with logical addition]. Mathematicheskie Zametki, 41(4):598-607, 1987. (English translation in Mathematical Notes of the Academy of Sciences of the USSR, 41(4):333-338, 1987). URL: http://mi.mathnet.ru/eng/mz4883, URL: http://dx.doi.org/10.1007/BF01137685.
http://dx.doi.org/10.1007/BF01137685
Alexander A. Razborov and Emanuele Viola. Real advantage. ACM T. Comput. Theory, 5(4):17, 2013. URL: http://dx.doi.org/10.1145/2540089.
http://dx.doi.org/10.1145/2540089
Roman Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In Proc. 19th ACM Symp. on Theory of Computing (STOC), pages 77-82, 1987. URL: http://dx.doi.org/10.1145/28395.28404.
http://dx.doi.org/10.1145/28395.28404
Avishay Tal. Tight bounds on the fourier spectrum of AC⁰. Technical Report TR14-174, Elect. Colloq. on Comput. Complexity (ECCC), 2014.
Jun Tarui. Probablistic polynomials, AC⁰ functions, and the polynomial-time hierarchy. Theoret. Comput. Sci., 113(1):167-183, 1993. (Preliminary Version in 8th STACS, 1991). URL: http://dx.doi.org/10.1016/0304-3975(93)90214-E.
http://dx.doi.org/10.1016/0304-3975(93)90214-E
Seinosuke Toda and Mitsunori Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM J. Comput., 21(2):316-328, 1992. (Preliminary version in 6th Structure in Complexity Theory Conference, 1991). URL: http://dx.doi.org/10.1137/0221023.
http://dx.doi.org/10.1137/0221023
Luca Trevisan and Tongke Xue. A derandomized switching lemma and an improved derandomization of AC⁰. In Proc. 28th IEEE Conf. on Computational Complexity, pages 242-247, 2013. URL: http://dx.doi.org/10.1109/CCC.2013.32.
http://dx.doi.org/10.1109/CCC.2013.32
Ryan Williams. New algorithms and lower bounds for circuits with linear threshold gates. In Proc. 46th ACM Symp. on Theory of Computing (STOC), pages 194-202, 2014. http://arxiv.org/abs/1401.2444, URL: http://dx.doi.org/10.1145/2591796.2591858.
http://dx.doi.org/10.1145/2591796.2591858
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On the Structure of Quintic Polynomials
We study the structure of bounded degree polynomials over finite fields. Haramaty and Shpilka [STOC 2010] showed that biased degree three or four polynomials admit a strong structural property. We confirm that this is the case for degree five polynomials also. Let F=F_q be a prime field. Suppose f:F^n to F is a degree five polynomial with bias(f)=delta. We prove the following two structural properties for such f.
1. We have f= sum_{i=1}^{c} G_i H_i + Q, where G_i and H_is are nonconstant polynomials satisfying deg(G_i)+deg(H_i)<= 5 and Q is a degree <5 polynomial. Moreover, c does not depend on n.
2. There exists an Omega_{delta,q}(n) dimensional affine subspace V subseteq F^n such that f|_V is a constant.
Cohen and Tal [Random 2015] proved that biased polynomials of degree at most four are constant on a subspace of dimension Omega(n). Item 2.]extends this to degree five polynomials. A corollary to Item 2. is that any degree five affine disperser for dimension k is also an affine extractor for dimension O(k). We note that Item 2. cannot hold for degrees six or higher.
We obtain our results for degree five polynomials as a special case of structure theorems that we prove for biased degree d polynomials when d<|\F|+4. While the d<|F|+4 assumption seems very restrictive, we note that prior to our work such structure theorems were only known for d<|\F| by Green and Tao [Contrib. Discrete Math. 2009] and Bhowmick and Lovett [arXiv:1506.02047]. Using algorithmic regularity lemmas for polynomials developed by Bhattacharyya, et al. [SODA 2015], we show that whenever such a strong structure exists, it can be found algorithmically in time polynomial in n.
Higher-order Fourier analysis
Structure Theorem
Polynomials
Regularity lemmas
33:1-33:18
Regular Paper
Pooya
Hatami
Pooya Hatami
10.4230/LIPIcs.APPROX-RANDOM.2016.33
Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, and Shachar Lovett. Every locally characterized affine-invariant property is testable. In Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, STOC'13, pages 429-436, New York, NY, USA, 2013. ACM. URL: http://dx.doi.org/10.1145/2488608.2488662.
http://dx.doi.org/10.1145/2488608.2488662
Arnab Bhattacharyya, Pooya Hatami, and Madhur Tulsiani. Algorithmic regularity for polynomials and applications. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1870-1889, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.125.
http://dx.doi.org/10.1137/1.9781611973730.125
Abhishek Bhowmick and Shachar Lovett. Bias vs structure of polynomials in large fields, and applications in effective algebraic geometry and coding theory. CoRR, abs/1506.02047, 2015. URL: http://arxiv.org/abs/1506.02047.
http://arxiv.org/abs/1506.02047
Abhishek Bhowmick and Shachar Lovett. The list decoding radius of reed-muller codes over small fields. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC'15, pages 277-285, New York, NY, USA, 2015. ACM. URL: http://dx.doi.org/10.1145/2746539.2746543.
http://dx.doi.org/10.1145/2746539.2746543
Gil Cohen and Avishay Tal. Two structural results for low degree polynomials and applications. RANDOM, 2015.
Leonard Eugene Dickson. Linear groups: With an exposition of the Galois field theory. with an introduction by W. Magnus. Dover Publications, Inc., New York, 1958.
Ben Green and Terence Tao. The distribution of polynomials over finite fields, with applications to the Gowers norms. Contrib. Discrete Math., 4(2):1-36, 2009.
Elad Haramaty and Amir Shpilka. On the structure of cubic and quartic polynomials. In STOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing, pages 331-340. ACM, New York, 2010.
Hamed Hatami, Pooya Hatami, and James Hirst. Limits of Boolean functions on 𝔽ⁿ_p. Electron. J. Combin., 21(4):Paper 4.2, 15, 2014.
Hamed Hatami, Pooya Hatami, and Shachar Lovett. General systems of linear forms: equidistribution and true complexity. Advances in Mathematics, 292:446-477, 2016.
Tali Kaufman and Shachar Lovett. Worst case to average case reductions for polynomials. Foundations of Computer Science, IEEE Annual Symposium on, 0:166-175, 2008. URL: http://dx.doi.org/10.1109/FOCS.2008.17.
http://dx.doi.org/10.1109/FOCS.2008.17
Tali Kaufman, Shachar Lovett, and Ely Porat. Weight distribution and list-decoding size of reed-muller codes. Information Theory, IEEE Transactions on, 58(5):2689-2696, 2012.
Rudolf Lidl and Harald Niederreiter. Introduction to finite fields and their applications. Cambridge university press, 1994.
Shachar Lovett, Roy Meshulam, and Alex Samorodnitsky. Inverse conjecture for the Gowers norm is false. Theory Comput., 7:131-145, 2011. URL: http://dx.doi.org/10.4086/toc.2011.v007a009.
http://dx.doi.org/10.4086/toc.2011.v007a009
Tom Sanders. Additive structures in sumsets. Math. Proc. Cambridge Philos. Soc., 144(2):289-316, 2008. URL: http://dx.doi.org/10.1017/S030500410700093X.
http://dx.doi.org/10.1017/S030500410700093X
Terence Tao and Tamar Ziegler. The inverse conjecture for the Gowers norm over finite fields in low characteristic. Ann. Comb., 16(1):121-188, 2012. URL: http://dx.doi.org/10.1007/s00026-011-0124-3.
http://dx.doi.org/10.1007/s00026-011-0124-3
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Lower Bounds on Same-Set Inner Product in Correlated Spaces
Let P be a probability distribution over a finite alphabet Omega^L with all L marginals equal. Let X^(1), ..., X^(L), where X^(j) = (X_1^(j), ..., X_n^(j)) be random vectors such that for every coordinate i in [n] the tuples (X_i^(1), ..., X_i^(L)) are i.i.d. according to P.
The question we address is: does there exist a function c_P independent of n such that for every f: Omega^n -> [0, 1] with E[f(X^(1))] = m > 0 we have E[f(X^(1)) * ... * f(X^(n))] > c_P(m) > 0?
We settle the question for L=2 and when L>2 and P has bounded correlation smaller than 1.
same set hitting
product spaces
correlation
lower bounds
34:1-34:11
Regular Paper
Jan
Hazla
Jan Hazla
Thomas
Holenstein
Thomas Holenstein
Elchanan
Mossel
Elchanan Mossel
10.4230/LIPIcs.APPROX-RANDOM.2016.34
Creative Commons Attribution 3.0 Unported license
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Estimating Parameters Associated with Monotone Properties
There has been substantial interest in estimating the value of a graph parameter, i.e., of a real function defined on the set of finite graphs, by sampling a randomly chosen substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity q_z=q_z(epsilon) of an estimable parameter z is the size of the random sample required to ensure that the value of z(G) may be estimated within error epsilon with probability at least 2/3. In this paper, we study the sample complexity of estimating two graph parameters associated with a monotone graph property, improving previously known results. To obtain our results, we prove that the vertex set of any graph that satisfies a monotone property P may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of P}. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.
parameter estimation
parameter testing
edit distance to monotone graph properties
entropy of subgraph classes
speed of subgraph classes
35:1-35:13
Regular Paper
Carlos
Hoppen
Carlos Hoppen
Yoshiharu
Kohayakawa
Yoshiharu Kohayakawa
Richard
Lang
Richard Lang
Hanno
Lefmann
Hanno Lefmann
Henrique
Stagni
Henrique Stagni
10.4230/LIPIcs.APPROX-RANDOM.2016.35
Noga Alon, Richard A. Duke, Hanno Lefmann, Vojtěch Rödl, and Raphael Yuster. The algorithmic aspects of the regularity lemma. J. Algorithms, 16(1):80-109, 1994. URL: http://dx.doi.org/10.1006/jagm.1994.1005.
http://dx.doi.org/10.1006/jagm.1994.1005
Noga Alon, Eldar Fischer, Michael Krivelevich, and Mario Szegedy. Efficient testing of large graphs. Combinatorica, 20(4):451-476, 2000. URL: http://dx.doi.org/10.1007/s004930070001.
http://dx.doi.org/10.1007/s004930070001
Noga Alon, Eldar Fischer, Ilan Newman, and Asaf Shapira. A combinatorial characterization of the testable graph properties: it’s all about regularity. SIAM J. Comput., 39(1):143-167, 2009. URL: http://dx.doi.org/10.1137/060667177.
http://dx.doi.org/10.1137/060667177
Noga Alon and Asaf Shapira. A characterization of the (natural) graph properties testable with one-sided error. SIAM J. Comput., 37(6):1703-1727, 2008. URL: http://dx.doi.org/10.1137/06064888X.
http://dx.doi.org/10.1137/06064888X
Noga Alon and Asaf Shapira. Every monotone graph property is testable. SIAM J. Comput., 38(2):505-522, 2008. URL: http://dx.doi.org/10.1137/050633445.
http://dx.doi.org/10.1137/050633445
Noga Alon, Asaf Shapira, and Benny Sudakov. Additive approximation for edge-deletion problems. Ann. of Math. (2), 170(1):371-411, 2009. URL: http://dx.doi.org/10.4007/annals.2009.170.371.
http://dx.doi.org/10.4007/annals.2009.170.371
Jean-Pierre Barthélémy and Bernard Monjardet. The median procedure in cluster analysis and social choice theory. Math. Social Sci., 1(3):235-267, 1980/81. URL: http://dx.doi.org/10.1016/0165-4896(81)90041-X.
http://dx.doi.org/10.1016/0165-4896(81)90041-X
Béla Bollobás. Hereditary properties of graphs: asymptotic enumeration, global structure, and colouring. Doc. Math., pages 333-342 (electronic), 1998. Extra Vol. III.
Christian Borgs, Jennifer T. Chayes, László Lovász, Vera T. Sós, and Katalin Vesztergombi. Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math., 219(6):1801-1851, 2008. URL: http://dx.doi.org/10.1016/j.aim.2008.07.008.
http://dx.doi.org/10.1016/j.aim.2008.07.008
Irène Charon and Olivier Hudry. An updated survey on the linear ordering problem for weighted or unweighted tournaments. Ann. Oper. Res., 175:107-158, 2010. URL: http://dx.doi.org/10.1007/s10479-009-0648-7.
http://dx.doi.org/10.1007/s10479-009-0648-7
David Conlon and Jacob Fox. Bounds for graph regularity and removal lemmas. Geom. Funct. Anal., 22(5):1191-1256, 2012. URL: http://dx.doi.org/10.1007/s00039-012-0171-x.
http://dx.doi.org/10.1007/s00039-012-0171-x
Paul Erdős, Péter Frankl, and Vojtěch Rödl. The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs and Combinatorics, 2(1):113-121, 1986. URL: http://dx.doi.org/10.1007/BF01788085.
http://dx.doi.org/10.1007/BF01788085
Paul Erdős, Daniel J. Kleitman, and Bruce L. Rothschild. Asymptotic enumeration of K_n-free graphs. In Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo II, pages 19-27. Atti dei Convegni Lincei, No. 17. Accad. Naz. Lincei, Rome, 1976.
Eldar Fischer and Ilan Newman. Testing versus estimation of graph properties. SIAM J. Comput., 37(2):482-501 (electronic), 2007. URL: http://dx.doi.org/10.1137/060652324.
http://dx.doi.org/10.1137/060652324
Jacob Fox. A new proof of the graph removal lemma. Ann. of Math. (2), 174(1):561-579, 2011. URL: http://dx.doi.org/10.4007/annals.2011.174.1.17.
http://dx.doi.org/10.4007/annals.2011.174.1.17
Alan Frieze and Ravi Kannan. Quick approximation to matrices and applications. Combinatorica, 19(2):175-220, 1999. URL: http://dx.doi.org/10.1007/s004930050052.
http://dx.doi.org/10.1007/s004930050052
Z. Füredi. Extremal hypergraphs and combinatorial geometry. In S. D. Chatterji, editor, Proceedings of the International Congress of Mathematicians: August 3-11, 1994 Zürich, Switzerland, pages 1343-1352. Birkhäuser Basel, 1995. URL: http://dx.doi.org/10.1007/978-3-0348-9078-6_65.
http://dx.doi.org/10.1007/978-3-0348-9078-6_65
Oded Goldreich, editor. Property Testing - Current Research and Surveys [outgrow of a workshop at the Institute for Computer Science ITCS) at Tsinghua University, January 2010], volume 6390 of Lecture Notes in Computer Science. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-16367-8.
http://dx.doi.org/10.1007/978-3-642-16367-8
Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653-750, 1998. URL: http://dx.doi.org/10.1145/285055.285060.
http://dx.doi.org/10.1145/285055.285060
Oded Goldreich and Luca Trevisan. Three theorems regarding testing graph properties. Random Structures Algorithms, 23(1):23-57, 2003. URL: http://dx.doi.org/10.1002/rsa.10078.
http://dx.doi.org/10.1002/rsa.10078
William T. Gowers. Lower bounds of tower type for Szemerédi’s uniformity lemma. Geom. Funct. Anal., 7(2):322-337, 1997. URL: http://dx.doi.org/10.1007/PL00001621.
http://dx.doi.org/10.1007/PL00001621
László Lovász and Balázs Szegedy. Szemerédi’s lemma for the analyst. Geom. Funct. Anal., 17(1):252-270, 2007. URL: http://dx.doi.org/10.1007/s00039-007-0599-6.
http://dx.doi.org/10.1007/s00039-007-0599-6
Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. J. Comput. System Sci., 72(6):1012-1042, 2006. URL: http://dx.doi.org/10.1016/j.jcss.2006.03.002.
http://dx.doi.org/10.1016/j.jcss.2006.03.002
Endre Szemerédi. Regular partitions of graphs. In Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), volume 260 of Colloq. Internat. CNRS, pages 399-401. CNRS, Paris, 1978.
Creative Commons Attribution 3.0 Unported license
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Stable Matching with Evolving Preferences
We consider the problem of stable matching with dynamic preference lists. At each time-step, the preference list of some player may change by swapping random adjacent members. The goal of a central agency (algorithm) is to maintain an approximately stable matching, in terms of number of blocking pairs, at all time-steps. The changes in the preference lists are not reported to the algorithm, but must instead be probed explicitly. We design an algorithm that in expectation and with high probability maintains a matching that has at most O((log n)^2 blocking pairs.
Stable Matching
Dynamic Data
36:1-36:13
Regular Paper
Varun
Kanade
Varun Kanade
Nikos
Leonardos
Nikos Leonardos
Frédéric
Magniez
Frédéric Magniez
10.4230/LIPIcs.APPROX-RANDOM.2016.36
A. Anagnostopoulos, R. Kumar, M. Mahdian, and E. Upfal. Sorting and selection on dynamic data. Theoretical Computer Science, 412(24):2564-2576, 2011. Selected Papers from 36th International Colloquium on Automata, Languages and Programming. URL: http://dx.doi.org/10.1016/j.tcs.2010.10.003.
http://dx.doi.org/10.1016/j.tcs.2010.10.003
A. Anagnostopoulos, R. Kumar, M. Mahdian, E. Upfal, and F. Vandin. Algorithms on evolving graphs. In Proc. of 3rd Innovations in Theoretical Computer Science, 2012.
D. Eppstein, Z. Galil, and G. F. Italiano. Dynamic graph algorithms. In M. Atallah, editor, Algorithms and Theory of Computation Handbook, chapter 8. CRC Press, 1999.
D. Gale and L. S. Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, 69(1):9-15, 1962.
M. Gupta and R. Peng. Fully dynamic (1+ e)-approximate matchings. In Proc. of 54th IEEE Foundations of Computer Science, pages 548-557, Oct 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.65.
http://dx.doi.org/10.1109/FOCS.2013.65
D. Knuth. Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms. CRM proceedings &lecture notes. American Mathematical Society, 1997.
David Asher Levin, Yuval Peres, and Elizabeth Lee Wilmer. Markov Chains and Mixing Times. American Mathematical Society, 2009.
C. McDiarmid. Concentration. In M. Habib, C. McDiarmid, J. Ramirez-Alfonsin, and B. Reed, editors, Probabilistic Methods for Algorithmic Discrete Mathematics, volume 16 of Algorithms and Combinatorics, pages 195-248. Springer Berlin Heidelberg, 1998. URL: http://dx.doi.org/10.1007/978-3-662-12788-9_6.
http://dx.doi.org/10.1007/978-3-662-12788-9_6
R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge International Series on Parallel Computation. Cambridge University Press, 1995.
O. Neiman and S. Solomon. Simple deterministic algorithms for fully dynamic maximal matching. In Proc. of 45th ACM Symposium on Theory of Computing, pages 745-754, 2013. URL: http://dx.doi.org/10.1145/2488608.2488703.
http://dx.doi.org/10.1145/2488608.2488703
K. Onak and R. Rubinfeld. Maintaining a large matching and a small vertex cover. In Proc. of 42nd ACM Symposium on Theory of Computing, pages 457-464, 2010. URL: http://dx.doi.org/10.1145/1806689.1806753.
http://dx.doi.org/10.1145/1806689.1806753
L. B. Wilson. An analysis of the stable marriage assignment algorithm. BIT Numerical Mathematics, 12(4):569-575, 1972.
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An ~O(n) Queries Adaptive Tester for Unateness
We present an adaptive tester for the unateness property of Boolean functions. Given a function f:{0,1}^n -> {0,1} the tester makes O(n log(n)/epsilon) adaptive queries to the function. The tester always accepts a unate function, and rejects with probability at least 0.9 if a function is epsilon-far from being unate.
property testing
boolean functions
unateness
37:1-37:7
Regular Paper
Subhash
Khot
Subhash Khot
Igor
Shinkar
Igor Shinkar
10.4230/LIPIcs.APPROX-RANDOM.2016.37
Aleksandrs Belovs and Eric Blais. A polynomial lower bound for testing monotonicity. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, pages 1021-1032, New York, NY, USA, 2016. ACM. URL: http://dx.doi.org/10.1145/2897518.2897567.
http://dx.doi.org/10.1145/2897518.2897567
Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, and David P. Woodruff. Transitive-closure spanners of the hypercube and the hypergrid. Electronic Colloquium on Computational Complexity (ECCC), 16:46, 2009. URL: http://eccc.hpi-web.de/report/2009/046.
http://eccc.hpi-web.de/report/2009/046
Eric Blais. Testing juntas nearly optimally. In Proceedings of the Forty-first Annual ACM Symposium on Theory of Computing, STOC'09, pages 151-158, New York, NY, USA, 2009. ACM. URL: http://dx.doi.org/10.1145/1536414.1536437.
http://dx.doi.org/10.1145/1536414.1536437
Jop Briët, Sourav Chakraborty, David García-Soriano, and Arie Matsliah. Monotonicity testing and shortest-path routing on the cube. Combinatorica, 32(1):35-53, 2012.
Deeparnab Chakrabarty and C. Seshadhri. A o(n) monotonicity tester for boolean functions over the hypercube. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 411-418, 2013. URL: http://dx.doi.org/10.1145/2488608.2488660.
http://dx.doi.org/10.1145/2488608.2488660
Xi Chen, Anindya De, Rocco A. Servedio, and Li-Yang Tan. Boolean function monotonicity testing requires (almost) n^1/2 non-adaptive queries. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 519-528, 2015. URL: http://dx.doi.org/10.1145/2746539.2746570.
http://dx.doi.org/10.1145/2746539.2746570
Xi Chen, Rocco A. Servedio, and Li-Yang Tan. New algorithms and lower bounds for monotonicity testing. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 286-295, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.38.
http://dx.doi.org/10.1109/FOCS.2014.38
Yevgeniy Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, and Alex Samorodnitsky. Improved testing algorithms for monotonicity. In Randomization, Approximation, and Combinatorial Algorithms and Techniques, Third International Workshop on Randomization and Approximation Techniques in Computer Science, and Second International Workshop on Approximation Algorithms for Combinatorial Optimization Problems RANDOM-APPROX'99, Proceedings. Berkeley, CA, USA, August 8-11, 1999, pages 97-108, 1999.
Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. Monotonicity testing over general poset domains. In Proceedings of the Thiry-fourth Annual ACM Symposium on Theory of Computing, STOC'02, pages 474-483, New York, NY, USA, 2002. ACM.
Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samorodnitsky. Testing monotonicity. Combinatorica, 20(3):301-337, 2000. URL: http://dx.doi.org/10.1007/s004930070011.
http://dx.doi.org/10.1007/s004930070011
Subhash Khot, Dor Minzer, and Muli Safra. On monotonicity testing and boolean isoperimetric type theorems. In Proceedings of the 56th Annual Symposium on Foundations of Computer Science (FOCS 2015), 2015.
Guy Kindler and Shmuel Safra. Noise-resistant boolean-functions are juntas, 2003. Manuscript.
Eric Lehman and Dana Ron. On disjoint chains of subsets. J. Comb. Theory, Ser. A, 94(2):399-404, 2001. URL: http://dx.doi.org/10.1006/jcta.2000.3148.
http://dx.doi.org/10.1006/jcta.2000.3148
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A Local Algorithm for Constructing Spanners in Minor-Free Graphs
Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider this problem in the setting of local algorithms: one wants to quickly determine whether a given edge e is in a specific spanning tree, without computing the whole spanning tree, but rather by inspecting the local neighborhood of e. The challenge is to maintain consistency. That is, to answer queries about different edges according to the same spanning tree. Since it is known that this problem cannot be solved without essentially viewing all the graph, we consider the relaxed version of finding a spanning subgraph with (1+c)n edges instead of n-1 edges (where n is the number of vertices and c is a given approximation/sparsity parameter).
It is known that this relaxed problem requires inspecting order of n^{1/2} edges in general graphs (for any constant c), which motivates the study of natural restricted families of graphs. One such family is the family of graphs with an excluded minor (which in particular includes planar graphs). For this family there is an algorithm that achieves constant success probability, and inspects (d/c)^{poly(h)log(1/c)} edges (for each edge it is queried on), where d is the maximum degree in the graph and h is the size of the excluded minor. The distances between pairs of vertices in the spanning subgraph G' are at most a factor of poly(d, 1/c, h) larger than in G.
In this work, we show that for an input graph that is H-minor free for any H of size h, this task can be performed by inspecting only poly(d, 1/c, h) edges in G.
The distances between pairs of vertices in the spanning subgraph G' are at most a factor of h log(d)/c (up to poly-logarithmic factors) larger than in G. Furthermore, the error probability of the new algorithm is significantly improved to order of 1/n. This algorithm can also be easily adapted to yield an efficient algorithm for the distributed (message passing) setting.
spanners
sparse subgraphs
local algorithms
excluded-minor
38:1-38:15
Regular Paper
Reut
Levi
Reut Levi
Dana
Ron
Dana Ron
Ronitt
Rubinfeld
Ronitt Rubinfeld
10.4230/LIPIcs.APPROX-RANDOM.2016.38
N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7(4):567-583, 1986. URL: http://dx.doi.org/10.1016/0196-6774(86)90019-2.
http://dx.doi.org/10.1016/0196-6774(86)90019-2
N. Alon, R. Rubinfeld, S. Vardi, and N. Xie. Space-efficient local computation algorithms. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1132-1139, 2012.
Noga Alon, Paul D. Seymour, and Robin Thomas. A separator theorem for graphs with an excluded minor and its applications. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, May 13-17, 1990, Baltimore, Maryland, USA, pages 293-299, 1990. URL: http://dx.doi.org/10.1145/100216.100254.
http://dx.doi.org/10.1145/100216.100254
G. Even, M. Medina, and D. Ron. Deterministic stateless centralized local algorithms for bounded degree graphs. In Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, pages 394-405, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44777-2_33.
http://dx.doi.org/10.1007/978-3-662-44777-2_33
Ken-ichi Kawarabayashi, Philip N. Klein, and Christian Sommer. Linear-space approximate distance oracles for planar, bounded-genus and minor-free graphs. In Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Zurich, Switzerland, July 4-8, 2011, Proceedings, Part I, pages 135-146, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22006-7_12.
http://dx.doi.org/10.1007/978-3-642-22006-7_12
Ken-ichi Kawarabayashi and Bruce A. Reed. A separator theorem in minor-closed classes. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 153-162, 2010. URL: http://dx.doi.org/10.1109/FOCS.2010.22.
http://dx.doi.org/10.1109/FOCS.2010.22
R. Levi and D. Ron. A quasi-polynomial time partition oracle for graphs with an excluded minor. ACM Trans. Algorithms, 11(3):24:1-24:13, 2015.
R. Levi, D. Ron, and R. Rubinfeld. Local algorithms for sparse spanning graphs. In Proceedings of the Eighteenth International Workshop on Randomization and Computation (RANDOM), pages 826-842, 2014.
Reut Levi, Guy Moshkovitz, Dana Ron, Ronitt Rubinfeld, and Asaf Shapira. Constructing near spanning trees with few local inspections. Random Structures &Algorithms, pages n/a-n/a, 2016. URL: http://dx.doi.org/10.1002/rsa.20652.
http://dx.doi.org/10.1002/rsa.20652
W. Mader. Homomorphiesätze für graphen. Mathematische Annalen, 178:154-168, 1968.
Y. Mansour, A. Rubinstein, S. Vardi, and N. Xie. Converting online algorithms to local computation algorithms. In Automata, Languages and Programming: Thirty-Ninth International Colloquium (ICALP), pages 653-664, 2012.
Y. Mansour and S. Vardi. A local computation approximation scheme to maximum matching. In Proceedings of the Sixteenth International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 260-273, 2013.
D. Peleg and A. A. Schäffer. Graph spanners. Journal of Graph Theory, 13:99-116, 1989.
D. Peleg and J. D. Ullman. An optimal synchronizer for the hypercube. SIAM Journal on Computing, 18:229-243, 1989.
L. S. Ram and E. Vicari. Distributed small connected spanning subgraph: Breaking the diameter bound. Technical report, Zürich, 2011.
R. Rubinfeld, G. Tamir, S. Vardi, and N. Xie. Fast local computation algorithms. In Proceedings of The Second Symposium on Innovations in Computer Science (ICS), pages 223-238, 2011.
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Tight Bounds for Sketching the Operator Norm, Schatten Norms, and Subspace Embeddings
We consider the following oblivious sketching problem: given epsilon in (0,1/3) and n >= d/epsilon^2, design a distribution D over R^{k * nd} and a function f: R^k * R^{nd} -> R}, so that for any n * d matrix A, Pr_{S sim D} [(1-epsilon) |A|_{op} <= f(S(A),S) <= (1+epsilon)|A|_{op}] >= 2/3, where |A|_{op} = sup_{x:|x|_2 = 1} |Ax|_2 is the operator norm of A and S(A) denotes S * A, interpreting A as a vector in R^{nd}. We show a tight lower bound of k = Omega(d^2/epsilon^2) for this problem. Previously, Nelson and Nguyen (ICALP, 2014) considered the problem of finding a distribution D over R^{k * n} such that for any n * d matrix A, Pr_{S sim D}[forall x, (1-epsilon)|Ax|_2 <= |SAx|_2 <= (1+epsilon)|Ax|_2] >= 2/3, which is called an oblivious subspace embedding (OSE). Our result considerably strengthens theirs, as it (1) applies only to estimating the operator norm, which can be estimated given any OSE, and (2) applies to distributions over general linear operators S which treat A as a vector and compute S(A), rather than the restricted class of linear operators corresponding to matrix multiplication. Our technique also implies the first tight bounds for approximating the Schatten p-norm for even integers p via general linear sketches, improving the previous lower bound from k = Omega(n^{2-6/p}) [Regev, 2014] to k = Omega(n^{2-4/p}). Importantly, for sketching the operator norm up to a factor of alpha, where alpha - 1 = \Omega(1), we obtain a tight k = Omega(n^2/alpha^4) bound, matching the upper bound of Andoni and Nguyen (SODA, 2013), and improving the previous k = Omega(n^2/\alpha^6) lower bound. Finally, we also obtain the first lower bounds for approximating Ky Fan norms.
data streams
sketching
matrix norms
subspace embeddings
39:1-39:11
Regular Paper
Yi
Li
Yi Li
David P.
Woodruff
David P. Woodruff
10.4230/LIPIcs.APPROX-RANDOM.2016.39
Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. J. Comput. Syst. Sci., 58(1):137-147, 1999.
Alexandr Andoni and Huy L. Nguyen. Eigenvalues of a matrix in the streaming model. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1729-1737, 2013.
Alexandr Andoni, Huy L. Nguyên, Yury Polyanskiy, and Yihong Wu. Tight lower bound for linear sketches of moments. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 25-32, 2013.
Marc Bury and Chris Schwiegelshohn. Sublinear estimation of weighted matchings in dynamic data streams. In the Proceedings of ESA, 2015.
Emmanuel J. Candès and Benjamin Recht. Exact matrix completion via convex optimization. Commun. ACM, 55(6):111-119, 2012.
Kenneth L. Clarkson and David P. Woodruff. Numerical linear algebra in the streaming model. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 205-214, 2009.
Kenneth L. Clarkson and David P. Woodruff. Low rank approximation and regression in input sparsity time. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 81-90, 2013.
Amit Deshpande, Madhur Tulsiani, and Nisheeth K. Vishnoi. Algorithms and hardness for subspace approximation. In SODA, pages 482-496, 2011.
Moritz Hardt, Katrina Ligett, and Frank McSherry. A simple and practical algorithm for differentially private data release. In Advances in Neural Information Processing Systems 25, pages 2348-2356. 2012.
Yuri Ingster and I. A. Suslina. Nonparametric Goodness-of-Fit Testing Under Gaussian Models. Springer, 1st edition, 2002.
Robert Krauthgamer and Ori Sasson. Property testing of data dimensionality. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA., pages 18-27, 2003.
Rafał Latała. Estimates of moments and tails of Gaussian chaoses. Ann. Probab., 34(6):2315-2331, 2006.
Chao Li and Gerome Miklau. Measuring the achievable error of query sets under differential privacy. CoRR, abs/1202.3399, 2012.
Yi Li, Huy L. Nguyen, and David P. Woodruff. On sketching matrix norms and the top singular vector. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1562-1581, 2014.
Yi Li, Huy L. Nguyen, and David P. Woodruff. Turnstile streaming algorithms might as well be linear sketches. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 174-183, 2014.
Yi Li, Zhengyu Wang, and David P. Woodruff. Improved testing of low rank matrices. In The 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD'14, New York, NY, USA - August 24-27, 2014, pages 691-700, 2014.
Yi Li and David P. Woodruff. A tight lower bound for high frequency moment estimation with small error. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013. Proceedings, pages 623-638, 2013.
Yi Li and David P. Woodruff. On approximating functions of the singular values in a stream. In STOC, 2016.
Xiangrui Meng and Michael W. Mahoney. Low-distortion subspace embeddings in input-sparsity time and applications to robust linear regression. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 91-100, 2013.
Cameron Musco and Christopher Musco. Stronger approximate singular value decomposition via the block lanczos and power methods. CoRR, abs/1504.05477, 2015.
Jelani Nelson and Huy L. Nguyen. OSNAP: faster numerical linear algebra algorithms via sparser subspace embeddings. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 117-126, 2013.
Jelani Nelson and Huy L. Nguyên. Lower bounds for oblivious subspace embeddings. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 883-894, 2014.
Eric Price and David P. Woodruff. Applications of the shannon-hartley theorem to data streams and sparse recovery. In Proceedings of the 2012 IEEE International Symposium on Information Theory, ISIT 2012, Cambridge, MA, USA, July 1-6, 2012, pages 2446-2450, 2012.
Oded Regev. Personal communication, 2014.
Terence Tao. Topics in Random Matrix Theory. Graduate studies in mathematics. American Mathematical Society, 2012.
Alexandre B. Tsybakov. Introduction to Nonparametric Estimation. Springer, 1st edition, 2008.
Roman Vershynin. Introduction to the non-asymptotic analysis of random matrices. In Yonina C. Eldar and Gitta Kutyniok, editors, Compressed Sensing, pages 210-268. Cambridge University Press, 2012. Cambridge Books Online. URL: http://dx.doi.org/10.1017/CBO9780511794308.006.
http://dx.doi.org/10.1017/CBO9780511794308.006
Karl Wimmer, Yi Wu, and Peng Zhang. Optimal query complexity for estimating the trace of a matrix. CoRR, abs/1405.7112, 2014.
David P. Woodruff. Sketching as a tool for numerical linear algebra. Foundations and Trends in Theoretical Computer Science, 10(1-2):1-157, 2014.
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Bounds on the Norms of Uniform Low Degree Graph Matrices
The Sum Of Squares hierarchy is one of the most powerful tools we know of for solving combinatorial optimization problems. However, its performance is only partially understood. Improving our understanding of the sum of squares hierarchy is a major open problem in computational complexity theory.
A key component of analyzing the sum of squares hierarchy is understanding the behavior of certain matrices whose entries are random but not independent. For these matrices, there is a random input graph and each entry of the matrix is a low degree function of the edges of this input graph. Moreoever, these matrices are generally invariant (as a function of the input graph) when we permute the vertices of the input graph. In this paper, we bound the norms of all such matrices up to a polylogarithmic factor.
sum of squares hierarchy
matrix norm bounds
40:1-40:26
Regular Paper
Dhruv
Medarametla
Dhruv Medarametla
Aaron
Potechin
Aaron Potechin
10.4230/LIPIcs.APPROX-RANDOM.2016.40
Noga Alon, Michael Krivelevich, and Benny Sudakov. Finding a large hidden clique in a random graph. Random Structures and Algorithms, 13(3-4):457-466, 1998.
Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for unique games and related problems. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 563-572. IEEE, 2010.
Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM (JACM), 56(2):5, 2009.
Boaz Barak, Sam Hopkins, Jonathan Kelner, Ankur Moitra, Pravesh Kothari, and Aaron Potechin. A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem. ArXiv e-prints, April 2016. URL: http://arxiv.org/abs/1503.06447.
http://arxiv.org/abs/1503.06447
Boaz Barak, Prasad Raghavendra, and David Steurer. Rounding semidefinite programming hierarchies via global correlation. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 472-481. IEEE, 2011.
Boaz Barak and David Steurer. Sum-of-squares proofs and the quest toward optimal algorithms. In Proceedings of International Congress of Mathematicians (ICM), 2014.
Yash Deshpande and Andrea Montanari. Improved sum-of-squares lower bounds for hidden clique and hidden submatrix problems. CoRR, abs/1502.06590, 2015. URL: http://arxiv.org/abs/1502.06590.
http://arxiv.org/abs/1502.06590
Uriel Feige and Robert Krauthgamer. The probable value of the lovász-schrijver relaxations for maximum independent set. SIAM J. Comput., 32(2):345-370, 2003. URL: http://dx.doi.org/10.1137/S009753970240118X.
http://dx.doi.org/10.1137/S009753970240118X
Vitaly Feldman, Elena Grigorescu, Lev Reyzin, Santosh Vempala, and Ying Xiao. Statistical algorithms and a lower bound for detecting planted cliques. In Proceedings of the forty-fourth annual ACM symposium on Theory of Computing. ACM, 2013.
Vyacheslav L. Girko. Circular law. Theory of Probability and its Applications, 29:694-706, 1984.
Michel X Goemans and David P Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42(6):1115-1145, 1995.
Dima Grigoriev. Complexity of positivstellensatz proofs for the knapsack. Computational Complexity, 10(2):139-154, 2001.
Dima Grigoriev. Linear lower bound on degrees of positivstellensatz calculus proofs for the parity. Theoretical Computer Science, 259(1):613-622, 2001.
Venkatesan Guruswami and Ali Kemal Sinop. Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with psd objectives. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 482-491. IEEE, 2011.
Samuel B. Hopkins, Pravesh K. Kothari, and Aaron Potechin. Sos and planted clique: Tight analysis of MPW moments at all degrees and an optimal lower bound at degree four. CoRR, abs/1507.05230, 2015. URL: http://arxiv.org/abs/1507.05230.
http://arxiv.org/abs/1507.05230
Mark Jerrum. Large cliques elude the metropolis process. Random Structures &Algorithms, 3(4):347-359, 1992.
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Jean B Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11(3):796-817, 2001.
R. Meka, A. Potechin, and A. Wigderson. Sum-of-squares lower bounds for planted clique. ArXiv e-prints, March 2015. URL: http://arxiv.org/abs/1503.06447.
http://arxiv.org/abs/1503.06447
Karl Menger. Zur allgemeinen kurventheorie. Fundamenta Mathematicae, 10(1):96-115, 1927.
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Prasad Raghavendra and Tselil Schramm. Tight lower bounds for planted clique in the degree-4 SOS program. CoRR, abs/1507.05136, 2015. URL: http://arxiv.org/abs/1507.05136.
http://arxiv.org/abs/1507.05136
Grant Schoenebeck. Linear level lasserre lower bounds for certain k-csps. In Foundations of Computer Science, 2008. FOCS'08. IEEE 49th Annual IEEE Symposium on, pages 593-602. IEEE, 2008.
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Lower Bounds for CSP Refutation by SDP Hierarchies
For a k-ary predicate P, a random instance of CSP(P) with n variables and m constraints is unsatisfiable with high probability when m >= O(n). The natural algorithmic task in this regime is refutation: finding a proof that a given random instance is unsatisfiable. Recent work of Allen et al. suggests that the difficulty of refuting CSP(P) using an SDP is determined by a parameter cmplx(P), the smallest t for which there does not exist a t-wise uniform distribution over satisfying assignments to P. In particular they show that random instances of CSP(P) with m >> n^{cmplx(P)/2} can be refuted efficiently using an SDP.
In this work, we give evidence that n^{cmplx(P)/2} constraints are also necessary for refutation using SDPs. Specifically, we show that if P supports a (t-1)-wise uniform distribution over satisfying assignments, then the Sherali-Adams_+ and Lovasz-Schrijver_+ SDP hierarchies cannot refute a random instance of CSP(P) in polynomial time for any m <= n^{t/2-epsilon}.
constraint satisfaction problems
LP and SDP relaxations
average-case complexity
41:1-41:30
Regular Paper
Ryuhei
Mori
Ryuhei Mori
David
Witmer
David Witmer
10.4230/LIPIcs.APPROX-RANDOM.2016.41
Sarah R. Allen, Ryan O'Donnell, and David Witmer. How to refute a random CSP. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science, pages 689-708, 2015.
Benny Applebaum, Boaz Barak, and Avi Wigderson. Public-key cryptography from different assumptions. In Proceedings of the 42nd Annual ACM Symposium on Theory of Computing, pages 171-180, 2010.
Boaz Barak. Lecture 1 - Introduction. Notes from course "Sum of Squares upper bounds, lower bounds, and open questions".
Boaz Barak, Fernando G. S. L. Brandão, Aram W. Harrow, Jonathan Kelner, David Steurer, and Yuan Zhou. Hypercontractivity, Sum-of-Squares Proofs, and their Applications. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pages 307-326, 2012.
Boaz Barak, Siu On Chan, and Pravesh Kothari. Sum of squares lower bounds from pairwise independence. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, pages 97-106, 2015.
Boaz Barak, Guy Kindler, and David Steurer. On the Optimality of Semidefinite Relaxations for Average-Case and Generalized Constraint Satisfaction. In Proceedings of the 4th Innovations in Theoretical Computer Science conference, 2013.
Boaz Barak and Ankur Moitra. Tensor Prediction, Rademacher Complexity and Random 3-XOR. CoRR, abs/1501.06521, 2015. URL: http://arxiv.org/abs/1501.06521.
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Eli Ben-Sasson and Yonatan Bilu. A gap in average proof complexity. Electronic Colloquium on Computational Complexity (ECCC), 9(3), 2002.
Siavosh Benabbas, Konstantinos Georgiou, Avner Magen, and Madhur Tulsiani. SDP gaps from pairwise independence. Theory of Computing, 8:269-289, 2012.
Joshua Buresh-Oppenheim, Nicola Galesi, Shlomo Hoory, Avner Magen, and Toniann Pitassi. Rank bounds and integrality gaps for cutting planes procedures. Theory of Computing, 2:65-90, 2006. URL: http://dx.doi.org/10.4086/toc.2006.v002a004.
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Siu On Chan, James R. Lee, Prasad Raghavendra, and David Steurer. Approximate constraint satisfaction requires large LP relaxations. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, pages 350-359, 2013.
Amin Coja-Oghlan, Andreas Goerdt, and André Lanka. Strong Refutation Heuristics for Random k-SAT. In Klaus Jansen, Sanjeev Khanna, José D.P. Rolim, and Dana Ron, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 3122 of Lecture Notes in Computer Science, pages 310-321. Springer Berlin Heidelberg, 2004. URL: http://dx.doi.org/10.1007/978-3-540-27821-4_28.
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Sanjeeb Dash. On the Matrix Cuts of Lovász and Schrijver and their use in Integer Programming. PhD thesis, Rice University, 2001.
Jian Ding, Allan Sly, and Nike Sun. Proof of the satisfiability conjecture for large k. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, pages 59-68, 2015.
Uriel Feige. Relations Between Average Case Complexity and Approximation Complexity. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 534-543, 2002.
Uriel Feige and Eran Ofek. Easily refutable subformulas of large random 3CNF formulas. In Proceedings of the 31st International Colloquium on Automata, Languages and Programming, volume 3142 of Lecture Notes in Comput. Sci., pages 519-530. Springer, Berlin, 2004.
Vitaly Feldman, Will Perkins, and Santosh Vempala. On the Complexity of Random Satisfiability Problems with Planted Solutions. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, pages 77-86, 2015.
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James R. Lee, Prasad Raghavendra, and David Steurer. Lower bounds on the size of semidefinite programming relaxations. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, pages 567-576, 2015.
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Grant Schoenebeck. Linear Level Lasserre Lower Bounds for Certain k-CSPs. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, pages 593-602, 2008.
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A No-Go Theorem for Derandomized Parallel Repetition: Beyond Feige-Kilian
In this work we show a barrier towards proving a randomness-efficient parallel repetition, a promising avenue for achieving many tight inapproximability results. Feige and Kilian (STOC'95) proved an impossibility result for randomness-efficient parallel repetition for two prover games with small degree, i.e., when each prover has only few possibilities for the question of the other prover. In recent years, there have been indications that randomness-efficient parallel repetition (also called derandomized parallel repetition) might be possible for games with large degree, circumventing the impossibility result of Feige and Kilian. In particular, Dinur and Meir (CCC'11) construct games with large degree whose repetition can be derandomized using a theorem of Impagliazzo, Kabanets and Wigderson (SICOMP'12). However, obtaining derandomized parallel repetition theorems that would yield optimal inapproximability results has remained elusive.
This paper presents an explanation for the current impasse in progress, by proving a limitation on derandomized parallel repetition. We formalize two properties which we call "fortification-friendliness" and "yields robust embeddings". We show that any proof of derandomized parallel repetition achieving almost-linear blow-up cannot both (a) be fortification-friendly and (b) yield robust embeddings. Unlike Feige and Kilian, we do not require the small degree assumption.
Given that virtually all existing proofs of parallel repetition, including the derandomized parallel repetition result of Dinur and Meir, share these two properties, our no-go theorem highlights a major barrier to achieving almost-linear derandomized parallel repetition.
Derandomization
parallel repetition
Feige-Killian
fortification
42:1-42:29
Regular Paper
Dana
Moshkovitz
Dana Moshkovitz
Govind
Ramnarayan
Govind Ramnarayan
Henry
Yuen
Henry Yuen
10.4230/LIPIcs.APPROX-RANDOM.2016.42
L. Babai, L. Fortnow, and C. Lund. Nondeterministic exponential time has two-prover interactive protocols. In Foundations of Computer Science, 1990. Proceedings., 31st Annual Symposium on, pages 16-25. IEEE, 1990.
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R. Shaltiel. Derandomized parallel repetition theorems for free games. Computational Complexity, 22(3):565-594, 2013.
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Fast Synchronization of Random Automata
A synchronizing word for an automaton is a word that brings that automaton into one and the same state, regardless of the starting position. Cerny conjectured in 1964 that if a $n$-state deterministic automaton has a synchronizing word, then it has a synchronizing word of length at most (n-1)^2. Berlinkov recently made a breakthrough in the probabilistic analysis of synchronization: he proved that, for the uniform distribution on deterministic automata with n states, an automaton admits a synchronizing word with high probability. In this article, we are interested in the typical length of the smallest synchronizing word, when such a word exists: we prove that a random automaton admits a synchronizing word of length O(n log^{3}n) with high probability. As a consequence, this proves that most automata satisfy the Cerny conjecture.
random automata
synchronization
the Černý conjecture
43:1-43:12
Regular Paper
Cyril
Nicaud
Cyril Nicaud
10.4230/LIPIcs.APPROX-RANDOM.2016.43
David Aldous, Grégory Miermont, and Jim Pitman. Brownian bridge asymptotics for random p-mappings. Electron. J. Probab, 9:37-56, 2004.
Mikhail V. Berlinkov. On the probability of being synchronizable. In Sathish Govindarajan and Anil Maheshwari, editors, Algorithms and Discrete Applied Mathematics - Second International Conference, CALDAM 2016, Thiruvananthapuram, India, February 18-20, 2016, Proceedings, volume 9602 of Lecture Notes in Computer Science, pages 73-84. Springer, 2016. URL: http://dx.doi.org/10.1007/978-3-319-29221-2_7.
http://dx.doi.org/10.1007/978-3-319-29221-2_7
Mikhail V. Berlinkov and Marek Szykula. Algebraic synchronization criterion and computing reset words. In Giuseppe F. Italiano, Giovanni Pighizzini, and Donald Sannella, editors, Mathematical Foundations of Computer Science 2015 - 40th International Symposium, MFCS 2015, Milan, Italy, August 24-28, 2015, Proceedings, Part I, volume 9234 of Lecture Notes in Computer Science, pages 103-115. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-48057-1_8.
http://dx.doi.org/10.1007/978-3-662-48057-1_8
Peter J Cameron. Dixon’s theorem and random synchronization. Discrete Mathematics, 313(11):1233-1236, 2013.
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A Direct-Sum Theorem for Read-Once Branching Programs
We study a direct-sum question for read-once branching programs. If M(f) denotes the minimum average memory required to compute a function f(x_1,x_2, ..., x_n) how much memory is required to compute f on k independent inputs that arrive in parallel? We show that when the inputs are sampled independently from some domain X and M(f) = Omega(n), then computing the value of f on k streams requires average memory at least Omega(k * M(f)/n).
Our results are obtained by defining new ways to measure the information complexity of read-once branching programs. We define two such measures: the transitional and cumulative information content. We prove that any read-once branching program with transitional information content I can be simulated using average memory O(n(I+1)). On the other hand, if every read-once branching program with cumulative information content I can be simulated with average memory O(I+1), then computing f on k inputs requires average memory at least Omega(k * (M(f)-1)).
Direct-sum
Information complexity
Streaming Algorithms
44:1-44:15
Regular Paper
Anup
Rao
Anup Rao
Makrand
Sinha
Makrand Sinha
10.4230/LIPIcs.APPROX-RANDOM.2016.44
Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. Journal of Computer and System Sciences, 58(1):137-147, 1999.
Chrisil Arackaparambil, Joshua Brody, and Amit Chakrabarti. Functional monitoring without monotonicity. In Susanne Albers, Alberto Marchetti-Spaccamela, Yossi Matias, Sotiris Nikoletseas, and Wolfgang Thomas, editors, Automata, Languages and Programming, volume 5555 of Lecture Notes in Computer Science, pages 95-106. Springer Berlin Heidelberg, 2009.
Brian Babcock and Chris Olston. Distributed top-k monitoring. In Proceedings of the 2003 ACM SIGMOD International Conference on Management of Data, SIGMOD'03, pages 28-39, New York, NY, USA, 2003. ACM.
Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An Information Statistics Approach to Data Stream and Communication Complexity. In FOCS, pages 209-218, 2002.
Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. SIAM J. Comput., 42(3):1327-1363, 2013.
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Mark Braverman, Faith Ellen, Rotem Oshman, Toniann Pitassi, and Vinod Vaikuntanathan. A tight bound for set disjointness in the message-passing model. In FOCS, pages 668-677, 2013.
Mark Braverman and Ankit Garg. Public vs Private Coin in Bounded-Round Information. In ICALP, pages 502-513, 2014.
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Amit Chakrabarti, Subhash Khot, and Xiaodong Sun. Near-optimal lower bounds on the multi-party communication complexity of set disjointness. In 18th Annual IEEE Conference on Computational Complexity (Complexity 2003), 7-10 July 2003, Aarhus, Denmark, pages 107-117, 2003.
Arkadev Chattopadhyay, Jaikumar Radhakrishnan, and Atri Rudra. Topology matters in communication. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 631-640, 2014.
G. Cormode, S. Muthukrishnan, and Wei Zhuang. What’s different: Distributed, continuous monitoring of duplicate-resilient aggregates on data streams. In Data Engineering, 2006. ICDE'06. Proceedings of the 22nd International Conference on, pages 57-57, April 2006.
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Graham Cormode, S. Muthukrishnan, and Ke Yi. Algorithms for distributed functional monitoring. ACM Trans. Algorithms, 7(2):21:1-21:20, March 2011.
Graham Cormode, S. Muthukrishnan, Ke Yi, and Qin Zhang. Optimal sampling from distributed streams. In Proceedings of the Twenty-ninth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS'10, pages 77-86, New York, NY, USA, 2010. ACM.
Thomas M. Cover and Joy A. Thomas. Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing). Wiley-Interscience, 2006.
Pavol Duris and Jose D.P. Rolim. Lower bounds on the multiparty communication complexity. Journal of Computer and System Sciences, 56(1):90-95, 1998.
Funda Ergun and Hossein Jowhari. On distance to monotonicity and longest increasing subsequence of a data stream. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'08, pages 730-736, Philadelphia, PA, USA, 2008. Society for Industrial and Applied Mathematics.
Anna Gál and Parikshit Gopalan. Lower bounds on streaming algorithms for approximating the length of the longest increasing subsequence. SIAM J. Comput., 39(8):3463-3479, August 2010.
Anat Ganor, Gillat Kol, and Ran Raz. Exponential Separation of Information and Communication for Boolean Functions. Electronic Colloquium on Computational Complexity (ECCC), 21:113, 2014.
André Gronemeier. Asymptotically optimal lower bounds on the nih-multi-party information complexity of the and-function and disjointness. In STACS 2009, pages 505-516, 2009.
Sudipto Guha and Zhiyi Huang. Revisiting the direct sum theorem and space lower bounds in random order streams. In Susanne Albers, Alberto Marchetti-Spaccamela, Yossi Matias, Sotiris Nikoletseas, and Wolfgang Thomas, editors, Automata, Languages and Programming, volume 5555 of Lecture Notes in Computer Science, pages 513-524. Springer Berlin Heidelberg, 2009.
Prahladh Harsha, Rahul Jain, David A. McAllester, and Jaikumar Radhakrishnan. The communication complexity of correlation. IEEE Transactions on Information Theory, 56(1):438-449, 2010. URL: http://dx.doi.org/10.1109/TIT.2009.2034824.
http://dx.doi.org/10.1109/TIT.2009.2034824
Zengfeng Huang, Božidar Radunović, Milan Vojnović, and Qin Zhang. Communication complexity of approximate maximum matching in distributed graph data. In STACS, 2015.
Ram Keralapura, Graham Cormode, and Jeyashankher Ramamirtham. Communication-efficient distributed monitoring of thresholded counts. In Proceedings of the 2006 ACM SIGMOD International Conference on Management of Data, SIGMOD'06, pages 289-300, New York, NY, USA, 2006. ACM.
Amit Manjhi, Vladislav Shkapenyuk, Kedar Dhamdhere, and Christopher Olston. Finding (recently) frequent items in distributed data streams. In Proceedings of the 21st International Conference on Data Engineering, ICDE'05, pages 767-778, Washington, DC, USA, 2005. IEEE Computer Society.
Marco Molinaro, David P. Woodruff, and Grigory Yaroslavtsev. Beating the direct sum theorem in communication complexity with implications for sketching. In SODA, pages 1738-1756, 2013.
Jeff M. Phillips, Elad Verbin, and Qin Zhang. Lower bounds for number-in-hand multiparty communication complexity, made easy. In Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'12, pages 486-501. SIAM, 2012.
Izchak Sharfman, Assaf Schuster, and Daniel Keren. A geometric approach to monitoring threshold functions over distributed data streams. ACM Transactions on Database Systems, 32(4), November 2007.
Izchak Sharfman, Assaf Schuster, and Daniel Keren. Shape sensitive geometric monitoring. In Proceedings of the Twenty-seventh ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS'08, pages 301-310, New York, NY, USA, 2008. ACM.
David Woodruff. Optimal space lower bounds for all frequency moments. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'04, pages 167-175, Philadelphia, PA, USA, 2004. Society for Industrial and Applied Mathematics.
David P. Woodruff and Qin Zhang. Tight bounds for distributed functional monitoring. In STOC, pages 941-960, 2012.
David P. Woodruff and Qin Zhang. An optimal lower bound for distinct elements in the message passing model. In SODA, pages 718-733, 2014.
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Explicit List-Decodable Codes with Optimal Rate for Computationally Bounded Channels
A stochastic code is a pair of encoding and decoding procedures where Encoding procedure receives a k bit message m, and a d bit uniform string S. The code is (p,L)-list-decodable against a class C of "channel functions" from n bits to n bits, if for every message m and every channel C in C that induces at most $pn$ errors, applying decoding on the "received word" C(Enc(m,S)) produces a list of at most L messages that contain m with high probability (over the choice of uniform S). Note that both the channel C and the decoding algorithm Dec do not receive the random variable S. The rate of a code is the ratio between the message length and the encoding length, and a code is explicit if Enc, Dec run in time poly(n).
Guruswami and Smith (J. ACM, to appear), showed that for every constants 0 < p < 1/2 and c>1 there are Monte-Carlo explicit constructions of stochastic codes with rate R >= 1-H(p)-epsilon that are (p,L=poly(1/epsilon))-list decodable for size n^c channels. Monte-Carlo, means that the encoding and decoding need to share a public uniformly chosen poly(n^c) bit string Y, and the constructed stochastic code is (p,L)-list decodable with high probability over the choice of Y.
Guruswami and Smith pose an open problem to give fully explicit (that is not Monte-Carlo) explicit codes with the same parameters, under hardness assumptions. In this paper we resolve this open problem, using a minimal assumption: the existence of poly-time computable pseudorandom generators for small circuits, which follows from standard complexity assumptions by Impagliazzo and Wigderson (STOC 97).
Guruswami and Smith also asked to give a fully explicit unconditional constructions with the same parameters against O(log n)-space online channels. (These are channels that have space O(log n) and are allowed to read the input codeword in one pass). We resolve this open problem.
Finally, we consider a tighter notion of explicitness, in which the running time of encoding and list-decoding algorithms does not increase, when increasing the complexity of the channel. We give explicit constructions (with rate approaching 1-H(p) for every p <= p_0 for some p_0>0) for channels that are circuits of size 2^{n^{Omega(1/d)}} and depth d. Here, the running time of encoding and decoding is a fixed polynomial (that does not depend on d).
Our approach builds on the machinery developed by Guruswami and Smith, replacing some probabilistic arguments with explicit constructions. We also present a simplified and general approach that makes the reductions in the proof more efficient, so that we can handle weak classes of channels.
Error Correcting Codes
List Decoding
Pseudorandomness
45:1-45:38
Regular Paper
Ronen
Shaltiel
Ronen Shaltiel
Jad
Silbak
Jad Silbak
10.4230/LIPIcs.APPROX-RANDOM.2016.45
Miklós Ajtai. Σ^1_1-formulae on finite structures. Ann. Pure Appl. Logic, 24(1):1-48, 1983.
M. Braverman. Polylogarithmic independence fools ac^0 circuits. J. ACM, 57(5), 2010. URL: http://dx.doi.org/10.1145/1754399.1754401.
http://dx.doi.org/10.1145/1754399.1754401
A. Garcia and H. Stichtenoth. On the asymptotic behavior of some towers of function fields over finite fields. Journal of Number Theory, 61(2):248-273, 1996.
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Counting Hypergraph Matchings up to Uniqueness Threshold
We study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges. With an activity parameter lambda, each matching M is assigned a weight lambda^{|M|}. The counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph. This problem unifies two extensively studied statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings).
For this model, the critical activity lambda_c= (d^d)/(k (d-1)^{d+1}) is the threshold for the uniqueness of Gibbs measures on the infinite (d+1)-uniform (k+1)-regular hypertree. Consider hypergraphs of maximum degree at most k+1 and maximum size of hyperedges at most d+1. We show that when lambda < lambda_c, there is an FPTAS for computing the partition function; and when lambda = lambda_c, there is a PTAS for computing the log-partition function. These algorithms are based on the decay of correlation (strong spatial mixing) property of Gibbs distributions. When lambda > 2lambda_c, there is no PRAS for the partition function or the log-partition function unless NP=RP.
Towards obtaining a sharp transition of computational complexity of approximate counting, we study the local convergence from a sequence of finite hypergraphs to the infinite lattice with specified symmetry. We show a surprising connection between the local convergence and the reversibility of a natural random walk. This leads us to a barrier for the hardness result: The non-uniqueness of infinite Gibbs measure is not realizable by any finite gadgets.
approximate counting; phase transition; spatial mixing
46:1-46:29
Regular Paper
Renjie
Song
Renjie Song
Yitong
Yin
Yitong Yin
Jinman
Zhao
Jinman Zhao
10.4230/LIPIcs.APPROX-RANDOM.2016.46
Mohsen Bayati, David Gamarnik, Dimitriy Katz, Chandra Nair, and Prasad Tetali. Simple deterministic approximation algorithms for counting matchings. In Proceedings of the 39th ACM Symposium on Theory of Computing (STOC), pages 122-127, 2007.
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Andrzej Dudek, Marek Karpinski, Andrzej Ruciński, and Edyta Szymańska. Approximate counting of matchings in (3, 3)-hypergraphs. In SWAT, pages 380-391, 2014.
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Andreas Galanis and Leslie Ann Goldberg. The complexity of approximately counting in 2-spin systems on k-uniform bounded-degree hypergraphs. arXiv preprint arXiv:1505.06146, 2015.
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David Gamarnik and Dmitriy Katz. Correlation decay and deterministic FPTAS for counting colorings of a graph. Journal of Discrete Algorithms, 12:29-47, 2012.
David Gamarnik, Dmitriy Katz, and Sidhant Misra. Strong spatial mixing of list coloring of graphs. Random Structures &Algorithms, 2013.
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Alistair Sinclair, Piyush Srivastava, and Yitong Yin. Spatial mixing and approximation algorithms for graphs with bounded connective constant. In Proceedings of the 54th IEEE Symposium on Foundations of Computer Science (FOCS), pages 300-309, 2013.
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Sampling in Potts Model on Sparse Random Graphs
We study the problem of sampling almost uniform proper q-colorings in sparse Erdos-Renyi random graphs G(n,d/n), a research initiated by Dyer, Flaxman, Frieze and Vigoda [Dyer et al., RANDOM STRUCT ALGOR, 2006]. We obtain a fully polynomial time almost uniform sampler (FPAUS) for the problem provided q>3d+4, improving the current best bound q>5.5d [Efthymiou, SODA, 2014].
Our sampling algorithm works for more generalized models and broader family of sparse graphs. It is an efficient sampler (in the same sense of FPAUS) for anti-ferromagnetic Potts model with activity 0<=b<1 on G(n,d/n) provided q>3(1-b)d+4. We further identify a family of sparse graphs to which all these results can be extended. This family of graphs is characterized by the notion of contraction function, which is a new measure of the average degree in graphs.
Potts model
Sampling
Random Graph
Approximation Algorithm
47:1-47:22
Regular Paper
Yitong
Yin
Yitong Yin
Chihao
Zhang
Chihao Zhang
10.4230/LIPIcs.APPROX-RANDOM.2016.47
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David Gamarnik, Dmitriy Katz, and Sidhant Misra. Strong spatial mixing of list coloring of graphs. Random Structures &Algorithms, 46(4):599-613, 2015.
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Michael Molloy. The Glauber dynamics on colorings of a graph with high girth and maximum degree. SIAM Journal on Computing, 33(3):721-737, 2004.
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Alistair Sinclair, Piyush Srivastava, Daniel Štefankovič, and Yitong Yin. Spatial mixing and the connective constant: Optimal bounds. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'15), pages 1549-1563. SIAM, 2015.
Alistair Sinclair, Piyush Srivastava, and Yitong Yin. Spatial mixing and approximation algorithms for graphs with bounded connective constant. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS'13), pages 300-309. IEEE, 2013.
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