Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018), APPROX/RANDOM 2018, August 20-22, 2018, Princeton, NJ, USA
APPROX/RANDOM 2018
August 20-22, 2018
Princeton, NJ, USA
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
Eric
Blais
Eric Blais
Klaus
Jansen
Klaus Jansen
José
D. P. Rolim
José D. P. Rolim
David
Steurer
David Steurer
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
116
2018
978-3-95977-085-9
https://www.dagstuhl.de/dagpub/978-3-95977-085-9
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter
Table of Contents
Preface
Conference Organization
0:i-0:xvi
Front Matter
Eric
Blais
Eric Blais
Klaus
Jansen
Klaus Jansen
José
D. P. Rolim
José D. P. Rolim
David
Steurer
David Steurer
10.4230/LIPIcs.APPROX-RANDOM.2018.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Polylogarithmic Approximation Algorithms for Weighted-F-Deletion Problems
Let F be a family of graphs. A canonical vertex deletion problem corresponding to F is defined as follows: given an n-vertex undirected graph G and a weight function w: V(G) - >R^+, find a minimum weight subset S subseteq V(G) such that G-S belongs to F. This is known as Weighted F Vertex Deletion problem. In this paper we devise a recursive scheme to obtain O(log^{O(1)} n)-approximation algorithms for such problems, building upon the classical technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to those problems, where an optimum solution S together with a well-structured set X, form a balanced separator of the input graph. In this paper, we obtain the first O(log^{O(1)} n)-approximation algorithms for the following vertex deletion problems.
- Let {F} be a finite set of graphs containing a planar graph, and F=G(F) be the family of graphs such that every graph H in G(F) excludes all graphs in F as minors. The vertex deletion problem corresponding to F=G(F) is the Weighted Planar F-Minor-Free Deletion (WPF-MFD) problem. We give randomized and deterministic approximation algorithms for WPF-MFD with ratios O(log^{1.5} n) and O(log^2 n), respectively. Previously, only a randomized constant factor approximation algorithm for the unweighted version of the problem was known [FOCS 2012].
- We give an O(log^2 n)-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal graphs. On the way to this algorithm, we also obtain a constant factor approximation algorithm for Multicut on chordal graphs.
- We give an O(log^3 n)-factor approximation algorithm for Weighted Distance Hereditary Vertex Deletion (WDHVD), also known as Weighted Rankwidth-1 Vertex Deletion (WR-1VD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth one.
We believe that our recursive scheme can be applied to obtain O(log^{O(1)} n)-approximation algorithms for many other problems as well.
Approximation Algorithms
Planar- F-Deletion
Separator
Mathematics of computing~Approximation algorithms
1:1-1:15
Regular Paper
This research has received funding from the European Research Council under ERC grant no. 306992 PARAPPROX, ERC grant no. 715744 PaPaALG, and ERC grant no. 725978 SYSTEMATICGRAPH.
https://arxiv.org/abs/1707.04908
Akanksha
Agrawal
Akanksha Agrawal
Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary
Daniel
Lokshtanov
Daniel Lokshtanov
University of Bergen, Norway
Pranabendu
Misra
Pranabendu Misra
University of Bergen, Norway
Saket
Saurabh
Saket Saurabh
Institute of Mathematical Sciences, HBNI, Chennai, India, University of Bergen, Norway, and UMI ReLax
Meirav
Zehavi
Meirav Zehavi
Ben-Gurion University, Beersheba, Israel
10.4230/LIPIcs.APPROX-RANDOM.2018.1
Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Feedback vertex set inspired kernel for chordal vertex deletion. In Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1383-1398, 2017.
Akanksha Agrawal, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi. Polylogarithmic approximation algorithms for weighted-dollarbackslashmathcalFdollar-deletion problems. arXiv preprint arXiv:1707.04908, 2017.
Vineet Bafna, Piotr Berman, and Toshihiro Fujito. A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM Journal on Discrete Mathematics, 12(3):289-297, 1999.
Nikhil Bansal, Daniel Reichman, and Seeun William Umboh. LP-based robust algorithms for noisy minor-free and bounded treewidth graphs. In Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1964-1979, 2017.
Reuven Bar-Yehuda and Shimon Even. A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2(2):198-203, 1981.
Reuven Bar-Yehuda, Dan Geiger, Joseph Naor, and Ron M. Roth. Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and Bayesian inference. SIAM Journal on Computing, 27(4):942-959, 1998.
Richard B. Borie, R. Gary Parker, and Craig A. Tovey. Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica, 7(5&6):555-581, 1992.
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Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
M Farber. On diameters and radii of bridged graphs. Discrete Mathematics, 73:249-260, 1989.
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Samuel Fiorini, Gwenaël Joret, and Ugo Pietropaoli. Hitting diamonds and growing cacti. In Proceedings of the 14th Conference on Integer Programming and Combinatorial Optimization (IPCO), volume 6080, pages 191-204, 2010.
Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and kernelization. SIAM Journal on Discrete Mathematics, 30(1):383-410, 2016.
Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, and Saket Saurabh. Planar f-deletion: Approximation, kernelization and optimal fpt algorithms. In Proceedings of IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS), pages 470-479, 2012.
Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Bidimensionality and EPTAS. In Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 748-759, 2011.
Fedor V. Fomin, Daniel Lokshtanov, and Saket Saurabh. Bidimensionality and geometric graphs. In Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1563-1575, 2012.
Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Bidimensionality and kernels. In Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 503-510, 2010.
Naveen Garg, Vijay V. Vazirani, and Mihalis Yannakakis. Approximate max-flow min-(multi)cut theorems and their applications. SIAM Journal on Computing, 25(2):235-251, 1996.
Daniel Golovin, Viswanath Nagarajan, and Mohit Singh. Approximating the k-multicut problem. In ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 621-630, 2006.
Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.
Anupam Gupta, Euiwoong Lee, Jason Li, Pasin Manurangsi, and Michał Włodarczyk. Losing treewidth by separating subsets. arXiv preprint arXiv:1804.01366, 2018.
Peter L Hammer and Frédéric Maffray. Completely separable graphs. Discrete applied mathematics, 27(1):85-99, 1990.
Petr Hlinený, Sang-il Oum, Detlef Seese, and Georg Gottlob. Width parameters beyond tree-width and their applications. The Computer Journal, 51(3):326-362, 2008.
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Bart M. P. Jansen and Marcin Pilipczuk. Approximation and kernelization for chordal vertex deletion. In Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1399-1418, 2017.
E. J. Kim and O. Kwon. A Polynomial Kernel for Distance-Hereditary Vertex Deletion. ArXiv e-prints, 2016. URL: http://arxiv.org/abs/1610.07229.
http://arxiv.org/abs/1610.07229
T Leighton and S Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM, 46:787–-832, 1999.
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G. L. Nemhauser and L. E. Trotter, Jr. Properties of vertex packing and independence system polyhedra. Mathematical Programming, 6:48-61, 1974.
Sang-il Oum. Rank-width and vertex-minors. Journal of Combinatorial Theory, Series B, 95(1):79-100, 2005.
Sang-il Oum. Approximating rank-width and clique-width quickly. ACM Transactions on Algorithms, 5(1), 2008.
Sang-il Oum. Rank-width: Algorithmic and structural results. CoRR, abs/1601.03800, 2016.
Sang-il Oum and Paul D. Seymour. Approximating clique-width and branch-width. Journal of Combinatorial Theory, Series B, 96(4):514-528, 2006.
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Mihalis Yannakakis. The effect of a connectivity requirement on the complexity of maximum subgraph problems. Journal of the ACM, 26(4):618-630, 1979.
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Akanksha Agrawal, Daniel Losktanov, Pranabendu Misra, Saket Saurabh, and Meirav Zehavi
Creative Commons Attribution 3.0 Unported license
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Improved Approximation Bounds for the Minimum Constraint Removal Problem
In the minimum constraint removal problem, we are given a set of geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable, and (perhaps surprisingly) no sub-linear approximations are known even for simple obstacles such as rectangles and disks. The main result of our paper is a new approximation technique that gives O(sqrt{n})-approximation for rectangles, disks as well as rectilinear polygons. The technique also gives O(sqrt{n})-approximation for the minimum color path problem in graphs. We also present some inapproximability results for the geometric constraint removal problem.
Minimum Constraint Removal
Minimum Color Path
Barrier Resilience
Obstacle Removal
Obstacle Free Path
Approximation
Theory of computation~Approximation algorithms analysis
2:1-2:19
Regular Paper
Research of Sayan Bandyapadhyay, Neeraj Kumar and Subhash Suri was supported in part by the NSF grant CCF-1525817. Research of Sayan Bandyapadhyay and Kasturi Varadarajan was supported in part by the NSF grant CCF-1615845.
Sayan
Bandyapadhyay
Sayan Bandyapadhyay
Department of Computer Science, University of Iowa, Iowa City, USA
Neeraj
Kumar
Neeraj Kumar
Department of Computer Science, University of California, Santa Barbara, USA
Subhash
Suri
Subhash Suri
Department of Computer Science, University of California, Santa Barbara, USA
Kasturi
Varadarajan
Kasturi Varadarajan
Department of Computer Science, University of Iowa, Iowa City, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.2
P. Agarwal, N. Kumar, S. Sintos, and S. Suri. Computing shortest paths in the plane with removable obstacles. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018), pages 5:1-5:15, 2018.
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H. J. Broersma, X. Li, G. Woeginger, and S. Zhang. Paths and cycles in colored graphs. Australasian journal of combinatorics, 31(1):299-311, 2005.
D. Y. C. Chan and D. G. Kirkpatrick. Multi-path algorithms for minimum-colour path problems with applications to approximating barrier resilience. Theor. Comput. Sci., 553:74-90, 2014.
T. M. Chan and E. Grant. Exact algorithms and apx-hardness results for geometric packing and covering problems. Computational Geometry, 47(2):112-124, 2014.
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E. Eiben and I. Kanj. How to navigate through obstacles? CoRR, abs/1712.04043, 2017. URL: http://arxiv.org/abs/1712.04043.
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K. Hauser. The minimum constraint removal problem with three robotics applications. In Tenth Workshop on the Algorithmic Foundations of Robotics, WAFR 2012, pages 1-17, 2012.
J. Hershberger, N. Kumar, and S. Suri. Shortest paths in the plane with obstacle violations. In 25th Annual European Symposium on Algorithms, (ESA 2017), pages 49:1-49:14, 2017.
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K-C. R. Tseng and D. G. Kirkpatrick. On barrier resilience of sensor networks. In 7th ALGOSENSORS 2011, pages 130-144, 2011.
S. Yuan, S. Varma, and J. P. Jue. Minimum-color path problems for reliability in mesh networks. In 24th INFOCOM 2005, volume 4, pages 2658-2669. IEEE, 2005.
Sayan Bandyapadhyay, Neeraj Kumar, Subhash Suri, and Kasturi Varadarajan
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
A Tight 4/3 Approximation for Capacitated Vehicle Routing in Trees
Given a set of clients with demands, the Capacitated Vehicle Routing problem is to find a set of tours that collectively cover all client demand, such that the capacity of each vehicle is not exceeded and such that the sum of the tour lengths is minimized. In this paper, we provide a 4/3-approximation algorithm for Capacitated Vehicle Routing on trees, improving over the previous best-known approximation ratio of (sqrt{41}-1)/4 by Asano et al.[Asano et al., 2001], while using the same lower bound. Asano et al. show that there exist instances whose optimal cost is 4/3 times this lower bound. Notably, our 4/3 approximation ratio is therefore tight for this lower bound, achieving the best-possible performance.
Approximation algorithms
Graph algorithms
Capacitated vehicle routing
Theory of computation~Routing and network design problems
3:1-3:15
Regular Paper
Amariah
Becker
Amariah Becker
Brown University Department of Computer Science, Providence, RI, USA
Research funded by NSF grant CCF-14-09520
10.4230/LIPIcs.APPROX-RANDOM.2018.3
Kemal Altinkemer and Bezalel Gavish. Heuristics for unequal weight delivery problems with a fixed error guarantee. Operations Research Letters, 6(4):149-158, 1987.
Tetsuo Asano, Naoki Katoh, and Kazuhiro Kawashima. A new approximation algorithm for the capacitated vehicle routing problem on a tree. Journal of Combinatorial Optimization, 5(2):213-231, 2001.
Tetsuo Asano, Naoki Katoh, Hisao Tamaki, and Takeshi Tokuyama. Covering points in the plane by k-tours: towards a polynomial time approximation scheme for general k. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pages 275-283. ACM, 1997.
Amariah Becker, Philip N Klein, and David Saulpic. Polynomial-time approximation schemes for k-center and bounded-capacity vehicle routing in metrics with bounded highway dimension. arXiv preprint arXiv:1707.08270, 2017.
Amariah Becker, Philip N Klein, and David Saulpic. A quasi-polynomial-time approximation scheme for vehicle routing on planar and bounded-genus graphs. In LIPIcs-Leibniz International Proceedings in Informatics, volume 87. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017.
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Christos H Papadimitriou and Mihalis Yannakakis. The traveling salesman problem with distances one and two. Mathematics of Operations Research, 18(1):1-11, 1993.
Amariah Becker
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Low Rank Approximation in the Presence of Outliers
We consider the problem of principal component analysis (PCA) in the presence of outliers. Given a matrix A (d x n) and parameters k, m, the goal is to remove a set of at most m columns of A (outliers), so as to minimize the rank-k approximation error of the remaining matrix (inliers). While much of the work on this problem has focused on recovery of the rank-k subspace under assumptions on the inliers and outliers, we focus on the approximation problem. Our main result shows that sampling-based methods developed in the outlier-free case give non-trivial guarantees even in the presence of outliers. Using this insight, we develop a simple algorithm that has bi-criteria guarantees. Further, unlike similar formulations for clustering, we show that bi-criteria guarantees are unavoidable for the problem, under appropriate complexity assumptions.
Low rank approximation
PCA
Robustness to outliers
Theory of computation~Approximation algorithms analysis
4:1-4:16
Regular Paper
The first author is partially supported by a Google Faculty Award.
Aditya
Bhaskara
Aditya Bhaskara
School of Computing, University of Utah, Salt Lake City, UT, USA, http://www.cs.utah.edu/~bhaskara/
Srivatsan
Kumar
Srivatsan Kumar
School of Computing, University of Utah, Salt Lake City, UT, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.4
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S. Charles Brubaker. Robust PCA and clustering in noisy mixtures. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009, pages 1078-1087, 2009.
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Moses Charikar, Samir Khuller, David M. Mount, and Giri Narasimhan. Algorithms for facility location problems with outliers. In Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, January 7-9, 2001, Washington, DC, USA., pages 642-651, 2001. URL: http://dl.acm.org/citation.cfm?id=365411.365555.
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Flavio Chierichetti, Sreenivas Gollapudi, Ravi Kumar, Silvio Lattanzi, Rina Panigrahy, and David P. Woodruff. Algorithms for lp low-rank approximation. In Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017, pages 806-814, 2017. URL: http://proceedings.mlr.press/v70/chierichetti17a.html.
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I. Diakonikolas, G. Kamath, D. M. Kane, J. Li, A. Moitra, and A. Stewart. Robust estimators in high dimensions without the computational intractability. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 655-664, Oct 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.85.
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Ilias Diakonikolas, Daniel M. Kane, and Alistair Stewart. List-decodable robust mean estimation and learning mixtures of spherical gaussians. CoRR, abs/1711.07211, 2017. URL: http://arxiv.org/abs/1711.07211.
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Uriel Feige and Joe Kilian. Heuristics for semirandom graph problems. Journal of Computing and System Sciences, 63:639-671, 2001.
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Aditya Bhaskara and Srivatsan Kumar
Creative Commons Attribution 3.0 Unported license
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Greedy Bipartite Matching in Random Type Poisson Arrival Model
We introduce a new random input model for bipartite matching which we call the Random Type Poisson Arrival Model. Just like in the known i.i.d. model (introduced by Feldman et al. [Feldman et al., 2009]), online nodes have types in our model. In contrast to the adversarial types studied in the known i.i.d. model, following the random graphs studied in Mastin and Jaillet [A. Mastin, 2013], in our model each type graph is generated randomly by including each offline node in the neighborhood of an online node with probability c/n independently. In our model, nodes of the same type appear consecutively in the input and the number of times each type node appears is distributed according to the Poisson distribution with parameter 1. We analyze the performance of the simple greedy algorithm under this input model. The performance is controlled by the parameter c and we are able to exactly characterize the competitive ratio for the regimes c = o(1) and c = omega(1). We also provide a precise bound on the expected size of the matching in the remaining regime of constant c. We compare our results to the previous work of Mastin and Jaillet who analyzed the simple greedy algorithm in the G_{n,n,p} model where each online node type occurs exactly once. We essentially show that the approach of Mastin and Jaillet can be extended to work for the Random Type Poisson Arrival Model, although several nontrivial technical challenges need to be overcome. Intuitively, one can view the Random Type Poisson Arrival Model as the G_{n,n,p} model with less randomness; that is, instead of each online node having a new type, each online node has a chance of repeating the previous type.
bipartite matching
stochastic input models
online algorithms
greedy algorithms
Theory of computation~Graph algorithms analysis
Theory of computation~Online algorithms
5:1-5:15
Regular Paper
Research is supported by NSERC.
https://arxiv.org/abs/1805.00578
Allan
Borodin
Allan Borodin
University of Toronto, 10 Kings College Road, Toronto, Canada
Christodoulos
Karavasilis
Christodoulos Karavasilis
University of Toronto, 10 Kings College Road, Toronto, Canada
Denis
Pankratov
Denis Pankratov
University of Toronto, 10 Kings College Road, Toronto, Canada
10.4230/LIPIcs.APPROX-RANDOM.2018.5
P. Jaillet A. Mastin. Greedy online bipartite matching on random graphs, 2013. URL: https://arxiv.org/abs/1307.2536v1.
https://arxiv.org/abs/1307.2536v1
M. Zadimoghaddam B. Haeupler, V.S. Mirrokni. Online stochastic weighted matching: Improved approximation algorithms, WINE 2011.
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Allan Borodin, Christodoulos Karavasilis, and Denis Pankratov. An experimental study of algorithms for online bipartite matching, Unpublished work in progress, 2018.
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Allan Borodin, Christodoulos Karavasilis, and Denis Pankratov
Creative Commons Attribution 3.0 Unported license
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Semi-Direct Sum Theorem and Nearest Neighbor under l_infty
We introduce semi-direct sum theorem as a framework for proving asymmetric communication lower bounds for the functions of the form V_{i=1}^n f(x,y_i). Utilizing tools developed in proving direct sum theorem for information complexity, we show that if the function is of the form V_{i=1}^n f(x,y_i) where Alice is given x and Bob is given y_i's, it suffices to prove a lower bound for a single f(x,y_i). This opens a new avenue of attack other than the conventional combinatorial technique (i.e. "richness lemma" from [Miltersen et al., 1995]) for proving randomized lower bounds for asymmetric communication for functions of such form.
As the main technical result and an application of semi-direct sum framework, we prove an information lower bound on c-approximate Nearest Neighbor (ANN) under l_infty which implies that the algorithm of [Indyk, 2001] for c-approximate Nearest Neighbor under l_infty is optimal even under randomization for both decision tree and cell probe data structure model (under certain parameter assumption for the latter). In particular, this shows that randomization cannot improve [Indyk, 2001] under decision tree model. Previously only a deterministic lower bound was known by [Andoni et al., 2008] and randomized lower bound for cell probe model by [Kapralov and Panigrahy, 2012]. We suspect further applications of our framework in exhibiting randomized asymmetric communication lower bounds for big data applications.
Asymmetric Communication Lower Bound
Data Structure Lower Bound
Nearest Neighbor Search
Theory of computation~Communication complexity
6:1-6:17
Regular Paper
Mark
Braverman
Mark Braverman
Department of Computer Science, Princeton University, 35 Olden St. Princeton NJ 08540, USA
Research supported in part by an NSF Awards, DMS-1128155, CCF- 1525342, and CCF-1149888, a Packard Fellowship in Science and Engineering, and the Simons Collaboration on Algorithms and Geometry. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Young Kun
Ko
Young Kun Ko
Department of Computer Science, Princeton University, 35 Olden St. Princeton NJ 08540, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.6
Alexandr Andoni, Dorian Croitoru, and Mihai Patrascu. Hardness of nearest neighbor under l-infinity. In Foundations of Computer Science, 2008. FOCS'08. IEEE 49th Annual IEEE Symposium on, pages 424-433. IEEE, 2008.
Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. SIAM Journal on Computing, 42(3):1327-1363, 2013.
Mark Braverman. Interactive information complexity. SIAM Journal on Computing, 44(6):1698-1739, 2015. URL: http://dx.doi.org/10.1137/130938517.
http://dx.doi.org/10.1137/130938517
Mark Braverman, Ankit Garg, Denis Pankratov, and Omri Weinstein. From information to exact communication. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 151-160. ACM, 2013.
Mark Braverman and Anup Rao. Information equals amortized communication. IEEE Transactions on Information Theory, 60(10):6058-6069, 2014.
Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, and Andrew Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In Foundations of Computer Science, 2001. Proceedings. 42nd IEEE Symposium on, pages 270-278. IEEE, 2001.
Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012.
Anirban Dasgupta, Ravi Kumar, and D. Sivakumar. Sparse and lopsided set disjointness via information theory. In Anupam Gupta, Klaus Jansen, José Rolim, and Rocco Servedio, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 517-528, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg.
Piotr Indyk. On approximate nearest neighbors under l∞ norm. Journal of Computer and System Sciences, 63(4):627-638, 2001.
Michael Kapralov and Rina Panigrahy. Nns lower bounds via metric expansion for l∞ and emd. In International Colloquium on Automata, Languages, and Programming, pages 545-556. Springer, 2012.
Peter Bro Miltersen. Lower bounds for union-split-find related problems on random access machines. In Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, pages 625-634. ACM, 1994.
Peter Bro Miltersen, Noam Nisan, Shmuel Safra, and Avi Wigderson. On data structures and asymmetric communication complexity. In Proceedings of the twenty-seventh annual ACM symposium on Theory of computing, pages 103-111. ACM, 1995.
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Sivaramakrishnan Natarajan Ramamoorthy and Anup Rao. How to compress asymmetric communication. In Proceedings of the 30th Conference on Computational Complexity, pages 102-123. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015.
Mark Braverman and Young Kun Ko
Creative Commons Attribution 3.0 Unported license
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Nearly Optimal Distinct Elements and Heavy Hitters on Sliding Windows
We study the distinct elements and l_p-heavy hitters problems in the sliding window model, where only the most recent n elements in the data stream form the underlying set. We first introduce the composable histogram, a simple twist on the exponential (Datar et al., SODA 2002) and smooth histograms (Braverman and Ostrovsky, FOCS 2007) that may be of independent interest. We then show that the composable histogram{} along with a careful combination of existing techniques to track either the identity or frequency of a few specific items suffices to obtain algorithms for both distinct elements and l_p-heavy hitters that are nearly optimal in both n and epsilon.
Applying our new composable histogram framework, we provide an algorithm that outputs a (1+epsilon)-approximation to the number of distinct elements in the sliding window model and uses O{1/(epsilon^2) log n log (1/epsilon)log log n+ (1/epsilon) log^2 n} bits of space. For l_p-heavy hitters, we provide an algorithm using space O{(1/epsilon^p) log^2 n (log^2 log n+log 1/epsilon)} for 0<p <=2, improving upon the best-known algorithm for l_2-heavy hitters (Braverman et al., COCOON 2014), which has space complexity O{1/epsilon^4 log^3 n}. We also show complementing nearly optimal lower bounds of Omega ((1/epsilon) log^2 n+(1/epsilon^2) log n) for distinct elements and Omega ((1/epsilon^p) log^2 n) for l_p-heavy hitters, both tight up to O{log log n} and O{log 1/epsilon} factors.
Streaming algorithms
sliding windows
heavy hitters
distinct elements
Theory of computation~Streaming, sublinear and near linear time algorithms
7:1-7:22
Regular Paper
https://arxiv.org/abs/1805.00212
Vladimir
Braverman
Vladimir Braverman
Department of Computer Science, Johns Hopkins University, Baltimore, MD, USA
This material is based upon work supported in part by the National Science Foundation under Grants No. 1447639, 1650041, and 1652257, Cisco faculty award, and by the ONR Award N00014-18-1-2364.
Elena
Grigorescu
Elena Grigorescu
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Research supported in part by NSF CCF-1649515.
Harry
Lang
Harry Lang
Department of Mathematics, Johns Hopkins University, Baltimore, MD, USA
This material is based upon work supported by the Franco-American Fulbright Commission. The author thanks INRIA (l’Institut national de recherche en informatique et en automatique) for hosting him during the writing of this paper.
David P.
Woodruff
David P. Woodruff
School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
D. Woodruff would like to acknowledge the support by the National Science Foundation under Grant No. CCF-1815840.
Samson
Zhou
Samson Zhou
Department of Computer Science, Purdue University, West Lafayette, IN, USA
Research supported in part by NSF CCF-1649515.
10.4230/LIPIcs.APPROX-RANDOM.2018.7
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http://arxiv.org/abs/1805.00212
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Vladimir Braverman, Elena Grigorescu, Harry Lang, David P. Woodruff, and Samson Zhou
Creative Commons Attribution 3.0 Unported license
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Survivable Network Design for Group Connectivity in Low-Treewidth Graphs
In the Group Steiner Tree problem (GST), we are given a (edge or vertex)-weighted graph G=(V,E) on n vertices, together with a root vertex r and a collection of groups {S_i}_{i in [h]}: S_i subseteq V(G). The goal is to find a minimum-cost subgraph H that connects the root to every group. We consider a fault-tolerant variant of GST, which we call Restricted (Rooted) Group SNDP. In this setting, each group S_i has a demand k_i in [k], k in N, and we wish to find a minimum-cost subgraph H subseteq G such that, for each group S_i, there is a vertex in the group that is connected to the root via k_i (vertex or edge) disjoint paths.
While GST admits O(log^2 n log h) approximation, its higher connectivity variants are known to be Label-Cover hard, and for the vertex-weighted version, the hardness holds even when k=2 (it is widely believed that there is no subpolynomial approximation for the Label-Cover problem [Bellare et al., STOC 1993]). More precisely, the problem admits no 2^{log^{1-epsilon}n}-approximation unless NP subseteq DTIME(n^{polylog(n)}). Previously, positive results were known only for the edge-weighted version when k=2 [Gupta et al., SODA 2010; Khandekar et al., Theor. Comput. Sci., 2012] and for a relaxed variant where k_i disjoint paths from r may end at different vertices in a group [Chalermsook et al., SODA 2015], for which the authors gave a bicriteria approximation. For k >= 3, there is no non-trivial approximation algorithm known for edge-weighted Restricted Group SNDP, except for the special case of the relaxed variant on trees (folklore).
Our main result is an O(log n log h) approximation algorithm for Restricted Group SNDP that runs in time n^{f(k, w)}, where w is the treewidth of the input graph. Our algorithm works for both edge and vertex weighted variants, and the approximation ratio nearly matches the lower bound when k and w are constants. The key to achieving this result is a non-trivial extension of a framework introduced in [Chalermsook et al., SODA 2017]. This framework first embeds all feasible solutions to the problem into a dynamic program (DP) table. However, finding the optimal solution in the DP table remains intractable. We formulate a linear program relaxation for the DP and obtain an approximate solution via randomized rounding. This framework also allows us to systematically construct DP tables for high-connectivity problems. As a result, we present new exact algorithms for several variants of survivable network design problems in low-treewidth graphs.
Approximation Algorithms
Hardness of Approximation
Survivable Network Design
Group Steiner Tree
Theory of computation~Routing and network design problems
8:1-8:19
Regular Paper
https://arxiv.org/abs/1802.10403
Parinya
Chalermsook
Parinya Chalermsook
Department of Computer Science, Aalto University, Espoo, Finland
Syamantak
Das
Syamantak Das
Indraprastha Institute of Information Technology Delhi, Delhi, India
https://orcid.org/0000-0002-4393-8678
Guy
Even
Guy Even
Tel-Aviv University, Tel-Aviv, Israel
Bundit
Laekhanukit
Bundit Laekhanukit
Max-Planck-Institut für Informatik, Saarücken, Germany and, Institute for Theoretical Computer Science, Shanghai University of Finance and Economics, Shanghai, China
https://orcid.org/0000-0002-4476-8914
ISF grant # 621/12, I-CORE grant # 4/11, NSF grant #CCF-1740425
Daniel
Vaz
Daniel Vaz
Max-Planck-Institut für Informatik, Germany & Graduate School of Computer Science, Saarland University, Saarücken, Germany
https://orcid.org/0000-0003-2224-2185
10.4230/LIPIcs.APPROX-RANDOM.2018.8
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Parinya Chalermsook, Syamantak Das, Guy Even, Bundit Laekhanukit, and Daniel Vaz
Creative Commons Attribution 3.0 Unported license
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Perturbation Resilient Clustering for k-Center and Related Problems via LP Relaxations
We consider clustering in the perturbation resilience model that has been studied since the work of Bilu and Linial [Yonatan Bilu and Nathan Linial, 2010] and Awasthi, Blum and Sheffet [Awasthi et al., 2012]. A clustering instance I is said to be alpha-perturbation resilient if the optimal solution does not change when the pairwise distances are modified by a factor of alpha and the perturbed distances satisfy the metric property - this is the metric perturbation resilience property introduced in [Angelidakis et al., 2017] and a weaker requirement than prior models. We make two high-level contributions.
- We show that the natural LP relaxation of k-center and asymmetric k-center is integral for 2-perturbation resilient instances. We belive that demonstrating the goodness of standard LP relaxations complements existing results [Maria{-}Florina Balcan et al., 2016; Angelidakis et al., 2017] that are based on new algorithms designed for the perturbation model.
- We define a simple new model of perturbation resilience for clustering with outliers. Using this model we show that the unified MST and dynamic programming based algorithm proposed in [Angelidakis et al., 2017] exactly solves the clustering with outliers problem for several common center based objectives (like k-center, k-means, k-median) when the instances is 2-perturbation resilient. We further show that a natural LP relxation is integral for 2-perturbation resilient instances of k-center with outliers.
Clustering
Perturbation Resilience
LP Integrality
Outliers
Beyond Worst Case Analysis
Theory of computation~Facility location and clustering
9:1-9:16
Regular Paper
Work on this paper supported in part by NSF grants CCF-1319376 and CCF-1526799.
https://arxiv.org/abs/1806.04202
Chandra
Chekuri
Chandra Chekuri
Department of Computer Science, University of Illinois, Urbana-Champaign, IL 61801, USA
Shalmoli
Gupta
Shalmoli Gupta
Department of Computer Science, University of Illinois, Urbana-Champaign, IL 61801, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.9
Charu C. Aggarwal. Outlier Analysis. Springer Publishing Company, Incorporated, 2013.
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Haris Angelidakis, Konstantin Makarychev, and Yury Makarychev. Algorithms for stable and perturbation-resilient problems. In STOC, pages 438-451, 2017.
Aaron Archer. Two O(log^* K)-approximation algorithms for the asymmetric k-center problem. In IPCO, pages 1-14, 2001.
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Pranjal Awasthi, Avrim Blum, and Or Sheffet. Center-based clustering under perturbation stability. Inf. Process. Lett., 112(1-2):49-54, 2012. URL: http://dx.doi.org/10.1016/j.ipl.2011.10.006.
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http://arxiv.org/abs/1502.03316
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Maria-Florina Balcan, Avrim Blum, and Anupam Gupta. Approximate clustering without the approximation. In SODA, pages 1068-1077, 2009.
Maria-Florina Balcan, Nika Haghtalab, and Colin White. k-center clustering under perturbation resilience. In ICALP, pages 68:1-68:14, 2016.
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Maria-Florina Balcan and Colin White. Clustering under local stability: Bridging the gap between worst-case and beyond worst-case analysis. CoRR, abs/1705.07157, 2017. URL: http://arxiv.org/abs/1705.07157.
http://arxiv.org/abs/1705.07157
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http://arxiv.org/abs/1602.08254
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Chandra Chekuri and Shalmoli Gupta
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Sherali-Adams Integrality Gaps Matching the Log-Density Threshold
The log-density method is a powerful algorithmic framework which in recent years has given rise to the best-known approximations for a variety of problems, including Densest-k-Subgraph and Small Set Bipartite Vertex Expansion. These approximations have been conjectured to be optimal based on various instantiations of a general conjecture: that it is hard to distinguish a fully random combinatorial structure from one which contains a similar planted sub-structure with the same "log-density".
We bolster this conjecture by showing that in a random hypergraph with edge probability n^{-alpha}, Omega(log n) rounds of Sherali-Adams cannot rule out the existence of a k-subhypergraph with edge density k^{-alpha-o(1)}, for any k and alpha. This holds even when the bound on the objective function is lifted. This gives strong integrality gaps which exactly match the gap in the above distinguishing problems, as well as the best-known approximations, for Densest k-Subgraph, Smallest p-Edge Subgraph, their hypergraph extensions, and Small Set Bipartite Vertex Expansion (or equivalently, Minimum p-Union). Previously, such integrality gaps were known only for Densest k-Subgraph for one specific parameter setting.
Approximation algorithms
integrality gaps
lift-and-project
log-density
Densest k-Subgraph
Theory of computation~Approximation algorithms analysis
10:1-10:19
Regular Paper
https://arxiv.org/abs/1804.07842
Eden
Chlamtác
Eden Chlamtác
Ben-Gurion University, Beer Sheva, Israel
Partially supported by ISF grant 1002/14.
Pasin
Manurangsi
Pasin Manurangsi
University of California, Berkeley, USA
Supported by NSF under Grants No. CCF 1655215 and CCF 1815434.
10.4230/LIPIcs.APPROX-RANDOM.2018.10
Noga Alon, Sanjeev Arora, Rajsekar Manokaran, Dana Moshkovitz, and Omri Weinstein. Inapproximabilty of densest k-subgraph from average case hardness. Unpublished Manuscript, 2011.
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Eden Chlamtáč and Pasin Manurangsi. Tight approximations in the semirandom log-density framework via label reduction. In preparation.
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E. Chlamtáč and P. Manurangsi
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Lower Bounds for Approximating Graph Parameters via Communication Complexity
In a celebrated work, Blais, Brody, and Matulef [Blais et al., 2012] developed a technique for proving property testing lower bounds via reductions from communication complexity. Their work focused on testing properties of functions, and yielded new lower bounds as well as simplified analyses of known lower bounds. Here, we take a further step in generalizing the methodology of [Blais et al., 2012] to analyze the query complexity of graph parameter estimation problems. In particular, our technique decouples the lower bound arguments from the representation of the graph, allowing it to work with any query type.
We illustrate our technique by providing new simpler proofs of previously known tight lower bounds for the query complexity of several graph problems: estimating the number of edges in a graph, sampling edges from an almost-uniform distribution, estimating the number of triangles (and more generally, r-cliques) in a graph, and estimating the moments of the degree distribution of a graph. We also prove new lower bounds for estimating the edge connectivity of a graph and estimating the number of instances of any fixed subgraph in a graph. We show that the lower bounds for estimating the number of triangles and edge connectivity also hold in a strictly stronger computational model that allows access to uniformly random edge samples.
sublinear graph parameter estimation
lower bounds
communication complexity
Theory of computation~Lower bounds and information complexity
11:1-11:18
Regular Paper
https://arxiv.org/abs/1709.04262
Talya
Eden
Talya Eden
School of Electrical Engineering, Tel Aviv University, Tel Aviv 6997801, Israel
This research was partially supported by a grant from the Blavatnik fund. The author is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship, and for the Weinstein Institute for their support.
Will
Rosenbaum
Will Rosenbaum
School of Electrical Engineering, Tel Aviv University, Tel Aviv 6997801, Israel
https://orcid.org/0000-0002-7723-9090
10.4230/LIPIcs.APPROX-RANDOM.2018.11
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Talya Eden and William B. Rosenbaum
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Communication Complexity of Correlated Equilibrium with Small Support
We define a two-player N x N game called the 2-cycle game, that has a unique pure Nash equilibrium which is also the only correlated equilibrium of the game. In this game, every 1/poly(N)-approximate correlated equilibrium is concentrated on the pure Nash equilibrium. We show that the randomized communication complexity of finding any 1/poly(N)-approximate correlated equilibrium of the game is Omega(N). For small approximation values, our lower bound answers an open question of Babichenko and Rubinstein (STOC 2017).
Correlated equilibrium
Nash equilibrium
Communication complexity
Theory of computation~Communication complexity
Theory of computation~Exact and approximate computation of equilibria
12:1-12:16
Regular Paper
Anat
Ganor
Anat Ganor
Tel Aviv University, Tel Aviv, Israel
The research leading to these results has received funding from the Israel Science Foundation (grant number 552/16) and the I-CORE Program of the planning and budgeting committee and The Israel Science Foundation (grant number 4/11).
Karthik
C. S.
Karthik C. S.
Weizmann Institute of Science, Rehovot, Israel
This work was supported by Irit Dinur’s ERC-CoG grant 772839 and ISF-UGC grant 1399/14.
10.4230/LIPIcs.APPROX-RANDOM.2018.12
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http://dx.doi.org/10.1561/0400000076
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Anat Ganor and Karthik C. S.
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On Minrank and the Lovász Theta Function
Two classical upper bounds on the Shannon capacity of graphs are the theta-function due to Lovász and the minrank parameter due to Haemers. We provide several explicit constructions of n-vertex graphs with a constant theta-function and minrank at least n^delta for a constant delta>0 (over various prime order fields). This implies a limitation on the theta-function-based algorithmic approach to approximating the minrank parameter of graphs. The proofs involve linear spaces of multivariate polynomials and the method of higher incidence matrices.
Minrank
Theta Function
Shannon capacity
Multivariate polynomials
Higher incidence matrices
Mathematics of computing~Information theory
13:1-13:15
Regular Paper
Ishay
Haviv
Ishay Haviv
School of Computer Science, The Academic College of Tel Aviv-Yaffo, Tel Aviv 61083, Israel
10.4230/LIPIcs.APPROX-RANDOM.2018.13
Rudolf Ahlswede, Ning Cai, Shuo-Yen Robert Li, and Raymond W. Yeung. Network information flow. IEEE Trans. Inform. Theory, 46(4):1204-1216, 2000.
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László Babai and Peter Frankl. Linear Algebra Methods in Combinatorics With Applications to Geometry and Computer Science. Wiley-Interscience Series in Discrete Mathematics and Optimization. The University of Chicago, second edition, 1992.
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Anna Blasiak, Robert Kleinberg, and Eyal Lubetzky. Broadcasting with side information: Bounding and approximating the broadcast rate. IEEE Trans. Information Theory, 59(9):5811-5823, 2013.
Aart Blokhuis. On the Sperner capacity of the cyclic triangle. Journal of Algebraic Combinatorics, 2(2):123-124, 1993.
Moses Charikar. On semidefinite programming relaxations for graph coloring and vertex cover. In Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 616-620, 2002.
Eden Chlamtáč and Ishay Haviv. Linear index coding via semidefinite programming. Combinatorics, Probability & Computing, 23(2):223-247, 2014. Preliminary version in SODA'12.
Amin Coja-Oghlan. The Lovász number of random graphs. Combinatorics, Probability & Computing, 14(4):439-465, 2005. Preliminary version in RANDOM'03.
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Salim El Rouayheb, Alex Sprintson, and Costas Georghiades. On the relation between the index coding and the network coding problems. In proceedings of IEEE International Symposium on Information Theory, pages 1823-1827. IEEE Press, 2008.
Uriel Feige. Randomized graph products, chromatic numbers, and the Lovász ϑ-function. Combinatorica, 17(1):79-90, 1997. Preliminary version in STOC'95.
Uriel Feige, Michael Langberg, and Gideon Schechtman. Graphs with tiny vector chromatic numbers and huge chromatic numbers. SIAM J. Comput., 33(6):1338-1368, 2004. Preliminary version in FOCS'02.
Peter Frankl and Vojtěch Rödl. Forbidden intersections. Trans. Amer. Math. Soc., 300(1):259-286, 1987.
Peter Frankl and Richard M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357-368, 1981.
Alexander Golovnev, Oded Regev, and Omri Weinstein. The minrank of random graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2017, pages 46:1-46:13, 2017.
Martin Grötschel, László Lovász, and Alexander Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2):169-197, 1981.
Willem Haemers. On some problems of Lovász concerning the Shannon capacity of a graph. IEEE Trans. Inform. Theory, 25(2):231-232, 1979.
Willem Haemers. An upper bound for the Shannon capacity of a graph. In Algebraic methods in graph theory, Vol. I, II (Szeged, 1978), volume 25 of Colloq. Math. Soc. János Bolyai, pages 267-272. North-Holland, Amsterdam, 1981.
Ishay Haviv and Michael Langberg. On linear index coding for random graphs. In IEEE International Symposium on Information Theory, pages 2231-2235, 2012.
Ishay Haviv and Michael Langberg. H-wise independence. In Innovations in Theoretical Computer Science (ITCS'13), pages 541-552, 2013.
Syed A. Jafar. Topological interference management through index coding. IEEE Transactions on Information Theory, 60(1):529-568, 2014.
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Ishay Haviv
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Online Makespan Minimization: The Power of Restart
We consider the online makespan minimization problem on identical machines. Chen and Vestjens (ORL 1997) show that the largest processing time first (LPT) algorithm is 1.5-competitive. For the special case of two machines, Noga and Seiden (TCS 2001) introduce the SLEEPY algorithm that achieves a competitive ratio of (5 - sqrt{5})/2 ~~ 1.382, matching the lower bound by Chen and Vestjens (ORL 1997). Furthermore, Noga and Seiden note that in many applications one can kill a job and restart it later, and they leave an open problem whether algorithms with restart can obtain better competitive ratios.
We resolve this long-standing open problem on the positive end. Our algorithm has a natural rule for killing a processing job: a newly-arrived job replaces the smallest processing job if 1) the new job is larger than other pending jobs, 2) the new job is much larger than the processing one, and 3) the processed portion is small relative to the size of the new job. With appropriate choice of parameters, we show that our algorithm improves the 1.5 competitive ratio for the general case, and the 1.382 competitive ratio for the two-machine case.
Online Scheduling
Makespan Minimization
Identical Machines
Theory of computation~Scheduling algorithms
Theory of computation~Approximation algorithms analysis
Theory of computation~Online algorithms
14:1-14:19
Regular Paper
https://arxiv.org/abs/1806.02207
Zhiyi
Huang
Zhiyi Huang
Department of Computer Sicence, The University of Hong Kong, Hong Kong
Partially supported by the Hong Kong RGC under the grant HKU17202115E.
Ning
Kang
Ning Kang
Department of Computer Sicence, The University of Hong Kong, Hong Kong
Zhihao Gavin
Tang
Zhihao Gavin Tang
Department of Computer Sicence, The University of Hong Kong, Hong Kong
Xiaowei
Wu
Xiaowei Wu
Department of Computing, The Hong Kong Polytechnic University, Hong Kong
Part of the work was done when the author was a postdoc at the University of Hong Kong.
Yuhao
Zhang
Yuhao Zhang
Department of Computer Sicence, The University of Hong Kong, Hong Kong
10.4230/LIPIcs.APPROX-RANDOM.2018.14
Susanne Albers. Better bounds for online scheduling. SIAM Journal on Computing, 29(2):459-473, 1999.
Nir Avrahami and Yossi Azar. Minimizing total flow time and total completion time with immediate dispatching. Algorithmica, 47(3):253-268, 2007.
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Piotr Berman, Moses Charikar, and Marek Karpinski. On-line load balancing for related machines. Journal of Algorithms, 35(1):108-121, 2000.
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Bo Chen, André van Vliet, and Gerhard J. Woeginger. New lower and upper bounds for on-line scheduling. Operations Research Letters, 16(4):221-230, 1994.
Bo Chen and Arjen P. A. Vestjens. Scheduling on identical machines: How good is LPT in an on-line setting? Operations Research Letters, 21(4):165-169, 1997.
Marek Chrobak, Wojciech Jawor, Jirí Sgall, and Tomás Tichý. Online scheduling of equal-length jobs: Randomization and restarts help. SIAM Journal on Computing, 36(6):1709-1728, 2007.
György Dósa and Leah Epstein. Online scheduling with a buffer on related machines. Journal of Combinatorial Optimization, 20(2):161-179, 2010.
György Dósa and Leah Epstein. Preemptive online scheduling with reordering. SIAM Journal on Discrete Mathematics, 25(1):21-49, 2011.
Tomás Ebenlendr, Wojciech Jawor, and Jirí Sgall. Preemptive online scheduling: Optimal algorithms for all speeds. Algorithmica, 53(4):504-522, 2009.
Matthias Englert, Deniz Özmen, and Matthias Westermann. The power of reordering for online minimum makespan scheduling. SIAM Journal on Computing, 43(3):1220-1237, 2014.
Leah Epstein, John Noga, Steven S. Seiden, Jirí Sgall, and Gerhard J. Woeginger. Randomized online scheduling on two uniform machines. In SODA, pages 317-326. ACM/SIAM, 1999.
Leah Epstein and Jirí Sgall. A lower bound for on-line scheduling on uniformly related machines. Operations Research Letters, 26(1):17-22, 2000.
Rudolf Fleischer and Michaela Wahl. On-line scheduling revisited. Journal of Scheduling, 3(6):343-353, 2000.
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Han Hoogeveen, Chris N Potts, and Gerhard J Woeginger. On-line scheduling on a single machine: maximizing the number of early jobs. Operations Research Letters, 27(5):193-197, 2000.
Zhiyi Huang, Ning Kang, Zhihao Gavin Tang, Xiaowei Wu, and Yuhao Zhang. Online makespan minimization: The power of restart. CoRR (to appear in APPROX 2018), abs/1806.02207, 2018.
J. F. Rudin III. Improved Bound for the Online Scheduling Problem. PhD thesis, University of Texas at Dallas, 2001.
J. F. Rudin III and R. Chandrasekaran. Improved bounds for the online scheduling problem. SIAM Journal on Computing, 32(3):717-735, 2003. URL: http://arxiv.org/abs/http://dx.doi.org/10.1137/S0097539702403438.
http://arxiv.org/abs/http://dx.doi.org/10.1137/S0097539702403438
David R. Karger, Steven J. Phillips, and Eric Torng. A better algorithm for an ancient scheduling problem. Journal of Algorithms, 20(2):400-430, 1996.
Hans Kellerer, Vladimir Kotov, Maria Grazia Speranza, and Zsolt Tuza. Semi on-line algorithms for the partition problem. Operations Research Letters, 21(5):235-242, 1997.
Shisheng Li, Yinghua Zhou, Guangzhong Sun, and Guoliang Chen. Study on parallel machine scheduling problem with buffer. In IMSCCS, pages 278-273. IEEE Computer Society, 2007.
John Noga and Steven S. Seiden. An optimal online algorithm for scheduling two machines with release times. Theoretical Computer Science, 268(1):133-143, 2001.
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Guochuan Zhang. A simple semi on-line algorithm for p2//c_maxwith a buffer. Information Processing Letters, 61(3):145-148, 1997.
Zhiyi Huang, Ning Kang, Zhihao Tang, Xiaowei Wu, and Yuhao Zhang
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On Sketching the q to p Norms
We initiate the study of data dimensionality reduction, or sketching, for the q -> p norms. Given an n x d matrix A, the q -> p norm, denoted |A |_{q -> p} = sup_{x in R^d \ 0} |Ax |_p / |x |_q, is a natural generalization of several matrix and vector norms studied in the data stream and sketching models, with applications to datamining, hardness of approximation, and oblivious routing. We say a distribution S on random matrices L in R^{nd} - > R^k is a (k,alpha)-sketching family if from L(A), one can approximate |A |_{q -> p} up to a factor alpha with constant probability. We provide upper and lower bounds on the sketching dimension k for every p, q in [1, infty], and in a number of cases our bounds are tight. While we mostly focus on constant alpha, we also consider large approximation factors alpha, as well as other variants of the problem such as when A has low rank.
Dimensionality Reduction
Norms
Sketching
Streaming
Theory of computation~Numeric approximation algorithms
15:1-15:20
Regular Paper
D. Woodruff would like to acknowledge the support by the National Science Foundation under Grant No. CCF-1815840.
https://arxiv.org/abs/1806.06429
Aditya
Krishnan
Aditya Krishnan
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Sidhanth
Mohanty
Sidhanth Mohanty
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
David P.
Woodruff
David P. Woodruff
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.15
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T. S. Jayram. On the information complexity of cascaded norms with small domains. In 2013 IEEE Information Theory Workshop, ITW 2013, Sevilla, Spain, September 9-13, 2013, pages 1-5, 2013.
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Aditya Krishnan, Sidhanth Mohanty, and David P. Woodruff
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Flow-time Optimization for Concurrent Open-Shop and Precedence Constrained Scheduling Models
Scheduling a set of jobs over a collection of machines is a fundamental problem that needs to be solved millions of times a day in various computing platforms: in operating systems, in large data clusters, and in data centers. Along with makespan, flow-time, which measures the length of time a job spends in a system before it completes, is arguably the most important metric to measure the performance of a scheduling algorithm. In recent years, there has been a remarkable progress in understanding flow-time based objective functions in diverse settings such as unrelated machines scheduling, broadcast scheduling, multi-dimensional scheduling, to name a few.
Yet, our understanding of the flow-time objective is limited mostly to the scenarios where jobs have no dependencies. On the other hand, in almost all real world applications, think of MapReduce settings for example, jobs have dependencies that need to be respected while making scheduling decisions. In this paper, we take first steps towards understanding this complex problem. In particular, we consider two classical scheduling problems that capture dependencies across jobs: 1) concurrent open-shop scheduling (COSSP) and 2) precedence constrained scheduling. Our main motivation to study these problems specifically comes from their relevance to two scheduling problems that have gained importance in the context of data centers: co-flow scheduling and DAG scheduling. We design almost optimal approximation algorithms for COSSP and PCSP, and show hardness results.
Approximation
Weighted Flow Time
Concurrent Open Shop
Precedence Constraints
Theory of computation~Scheduling algorithms
16:1-16:21
Regular Paper
https://arxiv.org/abs/1807.02553
Janardhan
Kulkarni
Janardhan Kulkarni
Microsoft Research, Redmond, WA, USA
Shi
Li
Shi Li
Department of Computer Science and Engineering, University at Buffalo, Buffalo, NY, USA
The work of the author is in part supported by NSF grants CCF-1566356 and CCF-1717134.
10.4230/LIPIcs.APPROX-RANDOM.2018.16
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Janardhan Kulkarni and Shi Li
Creative Commons Attribution 3.0 Unported license
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Sublinear-Time Quadratic Minimization via Spectral Decomposition of Matrices
We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix with a polylogarithmic number of blocks, such that in each block all the entries are the same, and the pseudorandom part has a small spectral norm, achieving better error bound than the existing decomposition theorem of Frieze and Kannan (FOCS'96). As an additional application of the decomposition theorem, we give a sublinear-time approximation algorithm for computing the top singular values of a matrix.
Qudratic function minimization
Approximation Algorithms
Matrix spectral decomposition
Graph limits
Theory of computation~Sketching and sampling
Theory of computation~Probabilistic computation
17:1-17:19
Regular Paper
Amit
Levi
Amit Levi
University of Waterloo, Canada
https://orcid.org/0000-0002-8530-5182
Research supported by NSERC Discovery grant and the David R. Cheriton Graduate Scholarship. Part of this work was done while the author was visiting NII Tokyo.
Yuichi
Yoshida
Yuichi Yoshida
National Institute of Informatics, Tokyo, Japan
https://orcid.org/0000-0001-8919-8479
Research supported by JSPS KAKENHI Grant Number JP17H04676 and JST ERATO Grant Number JPMJER1201.
10.4230/LIPIcs.APPROX-RANDOM.2018.17
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Amit Levi and Yuichi Yoshida
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https://creativecommons.org/licenses/by/3.0/legalcode
Deterministic Heavy Hitters with Sublinear Query Time
We study the classic problem of finding l_1 heavy hitters in the streaming model. In the general turnstile model, we give the first deterministic sublinear-time sketching algorithm which takes a linear sketch of length O(epsilon^{-2} log n * log^*(epsilon^{-1})), which is only a factor of log^*(epsilon^{-1}) more than the best existing polynomial-time sketching algorithm (Nelson et al., RANDOM '12). Our approach is based on an iterative procedure, where most unrecovered heavy hitters are identified in each iteration. Although this technique has been extensively employed in the related problem of sparse recovery, this is the first time, to the best of our knowledge, that it has been used in the context of heavy hitters. Along the way we also obtain a sublinear time algorithm for the closely related problem of the l_1/l_1 compressed sensing, matching the space usage of previous (super-)linear time algorithms. In the strict turnstile model, we show that the runtime can be improved and the sketching matrix can be made strongly explicit with O(epsilon^{-2}log^3 n/log^3(1/epsilon)) rows.
heavy hitters
turnstile model
sketching algorithm
strongly explicit
Theory of computation~Streaming, sublinear and near linear time algorithms
18:1-18:18
Regular Paper
https://arxiv.org/abs/1712.01971
Yi
Li
Yi Li
Nanyang Technological University, Singapore
Vasileios
Nakos
Vasileios Nakos
Harvard University, USA
Supported in part by NSF grant IIS-1447471
10.4230/LIPIcs.APPROX-RANDOM.2018.18
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Yi Li and Vasileios Nakos
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https://creativecommons.org/licenses/by/3.0/legalcode
On Low-Risk Heavy Hitters and Sparse Recovery Schemes
We study the heavy hitters and related sparse recovery problems in the low failure probability regime. This regime is not well-understood, and the main previous work on this is by Gilbert et al. (ICALP'13). We recognize an error in their analysis, improve their results, and contribute new sparse recovery algorithms, as well as provide upper and lower bounds for the heavy hitters problem with low failure probability. Our results are summarized as follows:
1) (Heavy Hitters) We study three natural variants for finding heavy hitters in the strict turnstile model, where the variant depends on the quality of the desired output. For the weakest variant, we give a randomized algorithm improving the failure probability analysis of the ubiquitous Count-Min data structure. We also give a new lower bound for deterministic schemes, resolving a question about this variant posed in Question 4 in the IITK Workshop on Algorithms for Data Streams (2006). Under the strongest and well-studied l_{infty}/ l_2 variant, we show that the classical Count-Sketch data structure is optimal for very low failure probabilities, which was previously unknown.
2) (Sparse Recovery Algorithms) For non-adaptive sparse-recovery, we give sublinear-time algorithms with low-failure probability, which improve upon Gilbert et al. (ICALP'13). In the adaptive case, we improve the failure probability from a constant by Indyk et al. (FOCS '11) to e^{-k^{0.99}}, where k is the sparsity parameter.
3) (Optimal Average-Case Sparse Recovery Bounds) We give matching upper and lower bounds in all parameters, including the failure probability, for the measurement complexity of the l_2/l_2 sparse recovery problem in the spiked-covariance model, completely settling its complexity in this model.
heavy hitters
sparse recovery
turnstile model
spike covariance model
lower bounds
Theory of computation~Streaming, sublinear and near linear time algorithms
19:1-19:13
Regular Paper
https://arxiv.org/abs/1709.02919
Yi
Li
Yi Li
Nanyang Technological University, Singapore
Vasileios
Nakos
Vasileios Nakos
Harvard University, USA
Supported in part by NSF grant IIS-1447471
David P.
Woodruff
David P. Woodruff
Carnegie Mellon University, USA
Supported in part by NSF grant CCF-1815840
10.4230/LIPIcs.APPROX-RANDOM.2018.19
Zeyuan Allen Zhu, Rati Gelashvili, and Ilya P. Razenshteyn. Restricted isometry property for general p-norms. IEEE Trans. Information Theory, 62(10):5839-5854, 2016.
Vladimir Braverman, Gereon Frahling, Harry Lang, Christian Sohler, and Lin F. Yang. Clustering high dimensional dynamic data streams. In Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017, pages 576-585, 2017.
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Sumit Ganguly. Data stream algorithms via expander graphs. In International Symposium on Algorithms and Computation, pages 52-63. Springer, 2008.
Anna C Gilbert, Yi Li, Ely Porat, and Martin J Strauss. Approximate sparse recovery: optimizing time and measurements. SIAM Journal on Computing, 41(2):436-453, 2012.
Anna C Gilbert, Hung Q Ngo, Ely Porat, Atri Rudra, and Martin J Strauss. 𝓁₂/𝓁₂-foreach sparse recovery with low risk. In International Colloquium on Automata, Languages, and Programming, pages 461-472. Springer, 2013.
Robert D. Gordon. Values of mills' ratio of area to bounding ordinate and of the normal probability integral for large values of the argument. Ann. Math. Statist., 12(3):364-366, 09 1941.
Rishi Gupta, Piotr Indyk, Eric Price, and Yaron Rachlin. Compressive sensing with local geometric features. Int. J. Comput. Geometry Appl., 22(4):365, 2012.
Venkatesan Guruswami, Christopher Umans, and Salil Vadhan. Unbalanced expanders and randomness extractors from parvaresh-vardy codes. Journal of the ACM (JACM), 56(4):20, 2009.
Piotr Indyk. Algorithms for dynamic geometric problems over data streams. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 373-380, 2004.
Piotr Indyk, Eric Price, and David P Woodruff. On the power of adaptivity in sparse recovery. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 285-294. IEEE, 2011.
Hossein Jowhari, Mert Sağlam, and Gábor Tardos. Tight bounds for lp samplers, finding duplicates in streams, and related problems. In Proceedings of the thirtieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, pages 49-58. ACM, 2011.
Kasper Green Larsen, Jelani Nelson, Huy L Nguyen, and Mikkel Thorup. Heavy hitters via cluster-preserving clustering. In Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, pages 61-70. IEEE, 2016.
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Vasileios Nakos, Xiaofei Shi, David P. Woodruff, and Hongyang Zhang. Improved algorithms for adaptive compressed sensing. In ICALP, 2018.
Jelani Nelson, Huy L. Nguyên, and David P. Woodruff. On deterministic sketching and streaming for sparse recovery and norm estimation. 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 627-638, 2012.
E. Price and D. P. Woodruff. (1 + ε)-approximate sparse recovery. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 295-304, Oct 2011.
Yi Li, Vasileios Nakos, and David P. Woodruff
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut
In this work, we study the trade-off between the running time of approximation algorithms and their approximation guarantees. By leveraging a structure of the "hard" instances of the Arora-Rao-Vazirani lemma [Sanjeev Arora et al., 2009; James R. Lee, 2005], we show that the Sum-of-Squares hierarchy can be adapted to provide "fast", but still exponential time, approximation algorithms for several problems in the regime where they are believed to be NP-hard. Specifically, our framework yields the following algorithms; here n denote the number of vertices of the graph and r can be any positive real number greater than 1 (possibly depending on n).
- A (2 - 1/(O(r)))-approximation algorithm for Vertex Cover that runs in exp (n/(2^{r^2)})n^{O(1)} time.
- An O(r)-approximation algorithms for Uniform Sparsest Cut and Balanced Separator that runs in exp (n/(2^{r^2)})n^{O(1)} time.
Our algorithm for Vertex Cover improves upon Bansal et al.'s algorithm [Nikhil Bansal et al., 2017] which achieves (2 - 1/(O(r)))-approximation in time exp (n/(r^r))n^{O(1)}. For Uniform Sparsest Cut and Balanced Separator, our algorithms improve upon O(r)-approximation exp (n/(2^r))n^{O(1)}-time algorithms that follow from a work of Charikar et al. [Moses Charikar et al., 2010].
Approximation algorithms
Exponential-time algorithms
Vertex Cover
Sparsest Cut
Balanced Separator
Theory of computation~Approximation algorithms analysis
20:1-20:17
Regular Paper
This material is based upon work supported by the National Science under Grants No. CCF 1655215 and CCF 1815434.
https://arxiv.org/abs/1807.09898
Pasin
Manurangsi
Pasin Manurangsi
University of California, Berkeley, USA
Luca
Trevisan
Luca Trevisan
University of California, Berkeley, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.20
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http://dx.doi.org/10.1145/2554797.2554836
P. Manurangsi and L. Trevisan
Creative Commons Attribution 3.0 Unported license
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Deterministic O(1)-Approximation Algorithms to 1-Center Clustering with Outliers
The 1-center clustering with outliers problem asks about identifying a prototypical robust statistic that approximates the location of a cluster of points. Given some constant 0 < alpha < 1 and n points such that alpha n of them are in some (unknown) ball of radius r, the goal is to compute a ball of radius O(r) that also contains alpha n points. This problem can be formulated with the points in a normed vector space such as R^d or in a general metric space.
The problem has a simple randomized solution: a randomly selected point is a correct solution with constant probability, and its correctness can be verified in linear time. However, the deterministic complexity of this problem was not known. In this paper, for any L^p vector space, we show an O(nd)-time solution with a ball of radius O(r) for a fixed alpha > 1/2, and for any normed vector space, we show an O(nd)-time solution with a ball of radius O(r) when alpha > 1/2 as well as an O(nd log^{(k)}(n))-time solution with a ball of radius O(r) for all alpha > 0, k in N, where log^{(k)}(n) represents the kth iterated logarithm, assuming distance computation and vector space operations take O(d) time. For an arbitrary metric space, we show for any C in N an O(n^{1+1/C})-time solution that finds a ball of radius 2Cr, assuming distance computation between any pair of points takes O(1)-time, and show that for any alpha, C, an O(n^{1+1/C})-time solution that finds a ball of radius ((2C-3)(1-alpha)-1)r cannot exist.
Deterministic
Approximation Algorithm
Cluster
Statistic
Theory of computation~Facility location and clustering
Theory of computation~Divide and conquer
21:1-21:19
Regular Paper
Shyam
Narayanan
Shyam Narayanan
Harvard University, Cambridge, Massachusetts, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.21
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Nimrod Megiddo. The weighted euclidean 1-center problem. Mathematics of Operations Research, 8(4):498-504, 1983.
J. Misra and David Gries. Finding repeated elements. Science of Computer Programming, 2:143-152, 1982.
Hamid Zarrabi-Zadeh and Asish Mukhopadhyay. Streaming 1-center with outliers in high dimensions. In Proceedings of the 21st Annual Canadian Conference on Computational Geometry, CCCG '09, pages 83-86, 2009.
Shyam Narayanan
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Robust Online Speed Scaling With Deadline Uncertainty
A speed scaling problem is considered, where time is divided into slots, and jobs with payoff v arrive at the beginning of the slot with associated deadlines d. Each job takes one slot to be processed, and multiple jobs can be processed by the server in each slot with energy cost g(k) for processing k jobs in one slot. The payoff is accrued by the algorithm only if the job is processed by its deadline. We consider a robust version of this speed scaling problem, where a job on its arrival reveals its payoff v, however, the deadline is hidden to the online algorithm, which could potentially be chosen adversarially and known to the optimal offline algorithm. The objective is to derive a robust (to deadlines) and optimal online algorithm that achieves the best competitive ratio. We propose an algorithm (called min-LCR) and show that it is an optimal online algorithm for any convex energy cost function g(.). We do so without actually evaluating the optimal competitive ratio, and give a general proof that works for any convex g, which is rather novel. For the popular choice of energy cost function g(k) = k^alpha, alpha >= 2, we give concrete bounds on the competitive ratio of the algorithm, which ranges between 2.618 and 3 depending on the value of alpha. The best known online algorithm for the same problem, but where deadlines are revealed to the online algorithm has competitive ratio of 2 and a lower bound of sqrt{2}. Thus, importantly, lack of deadline knowledge does not make the problem degenerate, and the effect of deadline information on the optimal competitive ratio is limited.
Online Algorithms
Speed Scaling
Greedy Algorithms
Scheduling
Theory of computation~Scheduling algorithms
22:1-22:17
Regular Paper
Goonwanth
Reddy
Goonwanth Reddy
Department of Electrical Engineering, Indian Institute of Technology, Madras, Chennai, India
Rahul
Vaze
Rahul Vaze
School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India
10.4230/LIPIcs.APPROX-RANDOM.2018.22
Susanne Albers, Fabian Müller, and Swen Schmelzer. Speed scaling on parallel processors. Algorithmica, 68(2):404-425, 2014.
Yossi Azar, Nikhil R Devanur, Zhiyi Huang, and Debmalya Panigrahi. Speed scaling in the non-clairvoyant model. In Proceedings of the 27th ACM symposium on Parallelism in Algorithms and Architectures, pages 133-142. ACM, 2015.
Yossi Azar, Bala Kalyanasundaram, Serge Plotkin, Kirk R Pruhs, and Orli Waarts. Online load balancing of temporary tasks. In Workshop on algorithms and data structures, pages 119-130. Springer, 1993.
Nikhil Bansal, Ho-Leung Chan, Tak-Wah Lam, and Lap-Kei Lee. Scheduling for speed bounded processors. In International Colloquium on Automata, Languages, and Programming, pages 409-420. Springer, 2008.
Nikhil Bansal, Ho-Leung Chan, Kirk Pruhs, and Dmitriy Katz. Improved bounds for speed scaling in devices obeying the cube-root rule. Automata, languages and programming, pages 144-155, 2009.
Nikhil Bansal, Tracy Kimbrel, and Kirk Pruhs. Speed scaling to manage energy and temperature. Journal of the ACM (JACM), 54(1):3, 2007.
Nikhil Bansal, Kirk Pruhs, and Cliff Stein. Speed scaling for weighted flow time. SIAM Journal on Computing, 39(4):1294-1308, 2009.
Neal Barcelo, Peter Kling, Michael Nugent, and Kirk Pruhs. Optimal Speed Scaling with a Solar Cell, pages 521-535. Springer International Publishing, Cham, 2016. URL: http://dx.doi.org/10.1007/978-3-319-48749-6_38.
http://dx.doi.org/10.1007/978-3-319-48749-6_38
Sanjoy Baruah, Gilad Koren, Bhubaneswar Mishra, Arvind Raghunathan, Louis Rosier, and Dennis Shasha. On-line scheduling in the presence of overload. In Foundations of Computer Science, 1991. Proceedings., 32nd Annual Symposium on, pages 100-110. IEEE, 1991.
Ho-Leung Chan, Joseph Wun-Tat Chan, Tak-Wah Lam, Lap-Kei Lee, Kin-Sum Mak, and Prudence WH Wong. Optimizing throughput and energy in online deadline scheduling. ACM Transactions on Algorithms (TALG), 6(1):10, 2009.
Ho-Leung Chan, Jeff Edmonds, Tak-Wah Lam, Lap-Kei Lee, Alberto Marchetti-Spaccamela, and Kirk Pruhs. Nonclairvoyant speed scaling for flow and energy. Algorithmica, 61(3):507-517, 2011.
Aaron Coté, Adam Meyerson, Alan Roytman, Michael Shindler, and Brian Tagiku. Energy-efficient online scheduling with deadlines. unpublished manuscript, 2010.
Anupam Gupta, Ravishankar Krishnaswamy, and Kirk Pruhs. Nonclairvoyantly scheduling power-heterogeneous processors. Sustainable Computing: Informatics and Systems, 1(3):248-255, 2011.
Xin Han, Tak-Wah Lam, Lap-Kei Lee, Isaac KK To, and Prudence WH Wong. Deadline scheduling and power management for speed bounded processors. Theoretical Computer Science, 411(40-42):3587-3600, 2010.
Sandy Irani, Sandeep Shukla, and Rajesh Gupta. Algorithms for power savings. ACM Transactions on Algorithms (TALG), 3(4):41, 2007.
Tak-Wah Lam, Lap-Kei Lee, Isaac KK To, and Prudence WH Wong. Speed scaling functions for flow time scheduling based on active job count. In European Symposium on Algorithms, pages 647-659. Springer, 2008.
Marco Riedel. Online request server matching. Theoretical computer science, 268(1):145-160, 2001.
Adam Wierman, Lachlan LH Andrew, and Ao Tang. Power-aware speed scaling in processor sharing systems. In INFOCOM 2009, IEEE, pages 2007-2015. IEEE, 2009.
Frances Yao, Alan Demers, and Scott Shenker. A scheduling model for reduced cpu energy. In Foundations of Computer Science, 1995. Proceedings., 36th Annual Symposium on, pages 374-382. IEEE, 1995.
Goonwanth Reddy and Rahul Vaze
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Multi-Agent Submodular Optimization
Recent years have seen many algorithmic advances in the area of submodular optimization: (SO) min/max~f(S): S in F, where F is a given family of feasible sets over a ground set V and f:2^V - > R is submodular. This progress has been coupled with a wealth of new applications for these models. Our focus is on a more general class of multi-agent submodular optimization (MASO) min/max Sum_{i=1}^{k} f_i(S_i): S_1 u+ S_2 u+ ... u+ S_k in F. Here we use u+ to denote disjoint union and hence this model is attractive where resources are being allocated across k agents, each with its own submodular cost function f_i(). This was introduced in the minimization setting by Goel et al. In this paper we explore the extent to which the approximability of the multi-agent problems are linked to their single-agent versions, referred to informally as the multi-agent gap.
We present different reductions that transform a multi-agent problem into a single-agent one. For minimization, we show that (MASO) has an O(alpha * min{k, log^2 (n)})-approximation whenever (SO) admits an alpha-approximation over the convex formulation. In addition, we discuss the class of "bounded blocker" families where there is a provably tight O(log n) multi-agent gap between (MASO) and (SO). For maximization, we show that monotone (resp. nonmonotone) (MASO) admits an alpha (1-1/e) (resp. alpha * 0.385) approximation whenever monotone (resp. nonmonotone) (SO) admits an alpha-approximation over the multilinear formulation; and the 1-1/e multi-agent gap for monotone objectives is tight. We also discuss several families (such as spanning trees, matroids, and p-systems) that have an (optimal) multi-agent gap of 1. These results substantially expand the family of tractable models for submodular maximization.
submodular optimization
multi-agent
approximation algorithms
Theory of computation~Submodular optimization and polymatroids
23:1-23:20
Regular Paper
https://arxiv.org/abs/1803.03767
Richard
Santiago
Richard Santiago
McGill University, Montreal, Canada
F. Bruce
Shepherd
F. Bruce Shepherd
University of British Columbia, Vancouver, Canada
10.4230/LIPIcs.APPROX-RANDOM.2018.23
Niv Buchbinder and Moran Feldman. Constrained submodular maximization via a non-symmetric technique. arXiv preprint arXiv:1611.03253, 2016.
Niv Buchbinder, Moran Feldman, Joseph Seffi, and Roy Schwartz. A tight linear time (1/2)-approximation for unconstrained submodular maximization. SIAM Journal on Computing, 44(5):1384-1402, 2015.
Gruia Calinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a submodular set function subject to a matroid constraint. In Integer programming and combinatorial optimization, pages 182-196. Springer, 2007.
Gruia Calinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40(6):1740-1766, 2011.
Chandra Chekuri and Alina Ene. Submodular cost allocation problem and applications. In International Colloquium on Automata, Languages, and Programming, pages 354-366. Springer, 2011. Extended version: arXiv preprint arXiv:1105.2040.
Alina Ene and Huy L Nguyen. Constrained submodular maximization: Beyond 1/e. In Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, pages 248-257. IEEE, 2016.
Alina Ene and Jan Vondrák. Hardness of submodular cost allocation: Lattice matching and a simplex coloring conjecture. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014), 28:144-159, 2014.
Uriel Feige, Vahab S Mirrokni, and Jan Vondrak. Maximizing non-monotone submodular functions. SIAM Journal on Computing, 40(4):1133-1153, 2011.
Moran Feldman, Joseph Naor, and Roy Schwartz. A unified continuous greedy algorithm for submodular maximization. In Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pages 570-579. IEEE, 2011.
Marshall L Fisher, George L Nemhauser, and Laurence A Wolsey. An analysis of approximations for maximizing submodular set functions-II. Springer, 1978.
Lisa Fleischer, Michel X Goemans, Vahab S Mirrokni, and Maxim Sviridenko. Tight approximation algorithms for maximum general assignment problems. In Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pages 611-620. Society for Industrial and Applied Mathematics, 2006.
Gagan Goel, Chinmay Karande, Pushkar Tripathi, and Lei Wang. Approximability of combinatorial problems with multi-agent submodular cost functions. In Foundations of Computer Science, 2009. FOCS'09. 50th Annual IEEE Symposium on, pages 755-764. IEEE, 2009.
Michel X. Goemans and VS Ramakrishnan. Minimizing submodular functions over families of sets. Combinatorica, 15(4):499-513, 1995.
Pranava R Goundan and Andreas S Schulz. Revisiting the greedy approach to submodular set function maximization. Optimization online, pages 1-25, 2007.
Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric algorithms and combinatorial optimization, volume 2. Springer Science &Business Media, 2012.
Anupam Gupta, Aaron Roth, Grant Schoenebeck, and Kunal Talwar. Constrained non-monotone submodular maximization: Offline and secretary algorithms. In International Workshop on Internet and Network Economics, pages 246-257. Springer, 2010.
Ara Hayrapetyan, Chaitanya Swamy, and Éva Tardos. Network design for information networks. In Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pages 933-942. Society for Industrial and Applied Mathematics, 2005.
Elad Hazan, Shmuel Safra, and Oded Schwartz. On the complexity of approximating k-set packing. computational complexity, 15(1):20-39, 2006.
Satoru Iwata, Lisa Fleischer, and Satoru Fujishige. A combinatorial strongly polynomial algorithm for minimizing submodular functions. Journal of the ACM (JACM), 48(4):761-777, 2001.
Satoru Iwata and Kiyohito Nagano. Submodular function minimization under covering constraints. In Foundations of Computer Science, 2009. FOCS'09. 50th Annual IEEE Symposium on, pages 671-680. IEEE, 2009.
Rishabh Iyer, Stefanie Jegelka, and Jeff Bilmes. Monotone closure of relaxed constraints in submodular optimization: Connections between minimization and maximization: Extended version, 2014.
Subhash Khot, Richard J Lipton, Evangelos Markakis, and Aranyak Mehta. Inapproximability results for combinatorial auctions with submodular utility functions. In International Workshop on Internet and Network Economics, pages 92-101. Springer, 2005.
Christos Koufogiannakis and Neal E Young. Greedy δ-approximation algorithm for covering with arbitrary constraints and submodular cost. Algorithmica, 66(1):113-152, 2013.
Andreas Krause and Carlos Guestrin. Near-optimal observation selection using submodular functions. In AAAI, volume 7, pages 1650-1654, 2007.
Andreas Krause, Jure Leskovec, Carlos Guestrin, Jeanne VanBriesen, and Christos Faloutsos. Efficient sensor placement optimization for securing large water distribution networks. Journal of Water Resources Planning and Management, 134(6):516-526, November 2008.
KW Krause, MA Goodwin, and RW Smith. Optimal software test planning through automated network analysis. TRW Systems Group, 1973.
Jon Lee, Vahab S Mirrokni, Viswanath Nagarajan, and Maxim Sviridenko. Non-monotone submodular maximization under matroid and knapsack constraints. In Proceedings of the forty-first annual ACM symposium on Theory of computing, pages 323-332. ACM, 2009.
Jon Lee, Maxim Sviridenko, and Jan Vondrák. Submodular maximization over multiple matroids via generalized exchange properties. Mathematics of Operations Research, 35(4):795-806, 2010.
Benny Lehmann, Daniel Lehmann, and Noam Nisan. Combinatorial auctions with decreasing marginal utilities. Games and Economic Behavior, 55(2):270-296, 2006. URL: http://EconPapers.repec.org/RePEc:eee:gamebe:v:55:y:2006:i:2:p:270-296.
http://EconPapers.repec.org/RePEc:eee:gamebe:v:55:y:2006:i:2:p:270-296
László Lovász. Submodular functions and convexity. In Mathematical Programming The State of the Art, pages 235-257. Springer, 1983.
Vahab Mirrokni, Michael Schapira, and Jan Vondrák. Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions. In Proceedings of the 9th ACM conference on Electronic commerce, pages 70-77. ACM, 2008.
George L Nemhauser, Laurence A Wolsey, and Marshall L Fisher. An analysis of approximations for maximizing submodular set functions - i. Mathematical Programming, 14(1):265-294, 1978.
George L Nemhauser and Leonard A Wolsey. Best algorithms for approximating the maximum of a submodular set function. Mathematics of operations research, 3(3):177-188, 1978.
Richard Santiago and F Bruce Shepherd. Multivariate submodular optimization. arXiv preprint arXiv:1612.05222, 2016.
Richard Santiago and F Bruce Shepherd. Multi-agent submodular optimization. arXiv preprint arXiv:1803.03767, 2018.
Alexander Schrijver. A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory, Series B, 80(2):346-355, 2000.
Ajit Singh, Andrew Guillory, and Jeff Bilmes. On bisubmodular maximization. In Artificial Intelligence and Statistics, pages 1055-1063, 2012.
Zoya Svitkina and Lisa Fleischer. Submodular approximation: Sampling-based algorithms and lower bounds. SIAM Journal on Computing, 40(6):1715-1737, 2011.
Zoya Svitkina and ÉVA Tardos. Facility location with hierarchical facility costs. ACM Transactions on Algorithms (TALG), 6(2):37, 2010.
Jan Vondrák. Optimal approximation for the submodular welfare problem in the value oracle model. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 67-74. ACM, 2008.
Jan Vondrák. Symmetry and approximability of submodular maximization problems. SIAM Journal on Computing, 42(1):265-304, 2013.
Jan Vondrák, Chandra Chekuri, and Rico Zenklusen. Submodular function maximization via the multilinear relaxation and contention resolution schemes. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 783-792. ACM, 2011.
Richard Santiago and F. Bruce Shepherd
Creative Commons Attribution 3.0 Unported license
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Generalized Assignment of Time-Sensitive Item Groups
We study the generalized assignment problem with time-sensitive item groups (chi-AGAP). It has central applications in advertisement placement on the Internet, and in virtual network embedding in Cloud data centers. We are given a set of items, partitioned into n groups, and a set of T identical bins (or, time-slots). Each group 1 <= j <= n has a time-window chi_j = [r_j, d_j]subseteq [T] in which it can be packed. Each item i in group j has a size s_i>0 and a non-negative utility u_{it} when packed into bin t in chi_j. A bin can accommodate at most one item from each group and the total size of the items in a bin cannot exceed its capacity. The goal is to find a feasible packing of a subset of the items in the bins such that the total utility from groups that are completely packed is maximized. Our main result is an Omega(1)-approximation algorithm for chi-AGAP. Our approximation technique relies on a non-trivial rounding of a configuration LP, which can be adapted to other common scenarios of resource allocation in Cloud data centers.
Approximation Algorithms
Packing and Covering problems
Generalized Assignment problem
Theory of computation~Packing and covering problems
24:1-24:18
Regular Paper
Kanthi
Sarpatwar
Kanthi Sarpatwar
IBM Research, Yorktown Heights, NY, USA
Baruch
Schieber
Baruch Schieber
IBM Research, Yorktown Heights, NY, USA
Hadas
Shachnai
Hadas Shachnai
Computer Science Department, Technion, Haifa, Israel
10.4230/LIPIcs.APPROX-RANDOM.2018.24
Ron Adany, Moran Feldman, Elad Haramaty, Rohit Khandekar, Baruch Schieber, Roy Schwartz, Hadas Shachnai, and Tami Tamir. All-or-nothing generalized assignment with application to scheduling advertising campaigns. In IPCO, pages 13-24. Springer, 2013.
Micah Adler, Phillip B Gibbons, and Yossi Matias. Scheduling space-sharing for internet advertising. Journal of Scheduling, 5(2):103-119, 2002.
V. Balachandran. An integer generalized transportation model for optimal job assignment in computer networks. Operations Research, 24(4):742-759, 1976.
Nikhil Bansal, Nitish Korula, Viswanath Nagarajan, and Aravind Srinivasan. Solving packing integer programs via randomized rounding with alterations. Theory of Computing, 8(1):533-565, 2012.
Marco Bender, Clemens Thielen, and Stephan Westphal. Packing items into several bins facilitates approximating the separable assignment problem. Information Processing Letters, 115(6-8):570-575, 2015.
Niv Buchbinder and Moran Feldman. Deterministic algorithms for submodular maximization problems. In SODA, pages 392-403, 2016.
C. Chekuri and S. Khanna. A PTAS for the multiple knapsack problem. SIAM J. on Computing, 35(3):713-728, 2006.
Lin Chen and Guochuan Zhang. Packing groups of items into multiple knapsacks. In STACS, pages 28:1-28:13, 2016.
NM Mosharaf Kabir Chowdhury, Muntasir Raihan Rahman, and Raouf Boutaba. Virtual network embedding with coordinated node and link mapping. In INFOCOM, pages 783-791. IEEE, 2009.
Robert G Cromley and Dean M Hanink. Coupling land use allocation models with raster GIS. Journal of Geographical Systems, 1(2):137-153, 1999.
Marek Cygan, Fabrizio Grandoni, and Monaldo Mastrolilli. How to sell hyperedges: The hypermatching assignment problem. In SODA, pages 342-351. SIAM, 2013.
Milind Dawande, Subodha Kumar, and Chelliah Sriskandarajah. Performance bounds of algorithms for scheduling advertisements on a web page. Journal of Scheduling, 6(4):373-394, 2003.
Milind Dawande, Subodha Kumar, and Chelliah Sriskandarajah. Scheduling web advertisements: a note on the minspace problem. Journal of Scheduling, 8(1):97-106, 2005.
Uriel Feige and Jan Vondrák. Approximation algorithms for allocation problems: Improving the factor of 1-1/e. In FOCS, pages 667-676, 2006.
Lisa Fleischer, Michel X Goemans, Vahab S Mirrokni, and Maxim Sviridenko. Tight approximation algorithms for maximum separable assignment problems. Mathematics of Operations Research, 36(3):416-431, 2011.
Ari Freund and Joseph Naor. Approximating the advertisement placement problem. Journal of Scheduling, 7(5):365-374, 2004.
Francesca Guerriero, Giovanna Miglionico, and Filomena Olivito. Managing tv commercials inventory in the italian advertising market. International Journal of Production Research, 54(18):5499-5521, 2016.
Arshia Kaul, Sugandha Aggarwal, Anshu Gupta, Niraj Dayama, Mohan Krishnamoorthy, and PC Jha. Optimal advertising on a two-dimensional web banner. International Journal of System Assurance Engineering and Management, pages 1-6, 2017.
Subodha Kumar, Milind Dawande, and Vijay Mookerjee. Optimal scheduling and placement of internet banner advertisements. IEEE Transactions on Knowledge and Data Engineering, 19(11), 2007.
Amin Ghalami Osgouei, Amir Khorsandi Koohanestani, Hossein Saidi, and Ali Fanian. Online assignment of non-SDN virtual network nodes to a physical SDN. Computer Networks, 129:105-116, 2017.
Mark J Panaggio, Pak-Wing Fok, Ghan S Bhatt, Simon Burhoe, Michael Capps, Christina J Edholm, Fadoua El Moustaid, Tegan Emerson, Star-Lena Estock, Nathan Gold, Ryan Halabi, Madelyn Houser, Peter R Kramer, Hsuan-Wei Lee, Qingxia Li, Weiqiang Li, Dan Lu, Yuzhou Qian, Louis F Rossi, Deborah Shutt, Vicky Chuqiao Yang, and Yingxiang Zhou. Prediction and optimal scheduling of advertisements in linear television. arXiv preprint arXiv:1608.07305, 2016.
Shinjini Pandey, Goutam Dutta, and Harit Joshi. Survey on revenue management in media and broadcasting. Interfaces, 47(3):195-213, 2017.
Adil Razzaq, Peter Sjödin, and Markus Hidell. Minimizing bottleneck nodes of a substrate in virtual network embedding. In NOF, pages 35-40, 2011.
Robert Ricci, Chris Alfeld, and Jay Lepreau. A solver for the network testbed mapping problem. ACM SIGCOMM Computer Communication Review, 33(2):65-81, 2003.
Kanthi K. Sarpatwar, Baruch Schieber, and Hadas Shachnai. Brief announcement: Approximation algorithms for preemptive resource allocation. To appear in SPAA, 2018.
SintecMedia - On Air. http://www.sintecmedia.com/OnAir.html, 2013.
Jan Vondrák. Optimal approximation for the submodular welfare problem in the value oracle model. In STOC, pages 67-74, 2008.
Minlan Yu, Yung Yi, Jennifer Rexford, and Mung Chiang. Rethinking virtual network embedding: substrate support for path splitting and migration. Computer Communication Review, 38(2):17-29, 2008.
Kanthi Sarpatwar, Baruch Schieber, and Hadas Shachnai
Creative Commons Attribution 3.0 Unported license
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On Geodesically Convex Formulations for the Brascamp-Lieb Constant
We consider two non-convex formulations for computing the optimal constant in the Brascamp-Lieb inequality corresponding to a given datum and show that they are geodesically log-concave on the manifold of positive definite matrices endowed with the Riemannian metric corresponding to the Hessian of the log-determinant function. The first formulation is present in the work of Lieb [Lieb, 1990] and the second is new and inspired by the work of Bennett et al. [Bennett et al., 2008]. Recent work of Garg et al. [Ankit Garg et al., 2017] also implies a geodesically log-concave formulation of the Brascamp-Lieb constant through a reduction to the operator scaling problem. However, the dimension of the arising optimization problem in their reduction depends exponentially on the number of bits needed to describe the Brascamp-Lieb datum. The formulations presented here have dimensions that are polynomial in the bit complexity of the input datum.
Geodesic convexity
positive definite cone
geodesics
Brascamp-Lieb constant
Theory of computation~Nonconvex optimization
Theory of computation~Convex optimization
25:1-25:15
Regular Paper
Suvrit
Sra
Suvrit Sra
Massachusetts Institute of Technology (MIT), Cambridge, MA, USA
Nisheeth K.
Vishnoi
Nisheeth K. Vishnoi
École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
Ozan
Yildiz
Ozan Yildiz
École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
10.4230/LIPIcs.APPROX-RANDOM.2018.25
Zeyuan Allen-Zhu, Garg Ankit, Yuanzhi Li, Rafael Oliveira, and Avi Wigderson. Operator scaling via geodesically convex optimization, invariant theory and polynomial identity testing. In Proceedings of the 50th Annual ACM Symposium on Theory of Computing, Los Angeles, CA, USA, June 25-29, 2018. To appear. URL: http://arxiv.org/abs/arXiv:1804.01076.
http://arxiv.org/abs/arXiv:1804.01076
Tsuyoshi Ando. Concavity of certain maps on positive definite matrices and applications to Hadamard products. Linear Algebra and its Applications, 26:203-241, 1979. URL: http://dx.doi.org/10.1016/0024-3795(79)90179-4.
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Jonathan Bennett, Anthony Carbery, Michael Christ, and Terence Tao. The Brascamp-Lieb inequalities: finiteness, structure and extremals. Geometric and Functional Analysis, 17(5):1343-1415, 2008.
Rajendra Bhatia. Positive definite matrices. Princeton University Press, 2009.
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http://dx.doi.org/10.1007/BF02922101
Zeev Dvir, Ankit Garg, Rafael Mendes de Oliveira, and József Solymosi. Rank bounds for design matrices with block entries and geometric applications. Discrete Analysis, 2018:5, Mar 2018. URL: http://dx.doi.org/10.19086/da.3118.
http://dx.doi.org/10.19086/da.3118
Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Breaking the quadratic barrier for 3-LCC’s over the reals. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, Los Angeles, CA, USA, May 31- June 3, 2014, pages 784-793, New York, NY, USA, 2014. URL: http://dx.doi.org/10.1145/2591796.2591818.
http://dx.doi.org/10.1145/2591796.2591818
Ankit Garg, Leonid Gurvits, Rafael Mendes de Oliveira, and Avi Wigderson. Algorithmic and optimization aspects of Brascamp-Lieb inequalities, via operator scaling. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing, Montreal, QC, Canada, June 19-23, 2017, pages 397-409, 2017. URL: http://dx.doi.org/10.1145/3055399.3055458.
http://dx.doi.org/10.1145/3055399.3055458
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Jingbo Liu, Thomas A. Courtade, Paul W. Cuff, and Sergio Verdú. Smoothing Brascamp-Lieb inequalities and strong converses for common randomness generation. In IEEE International Symposium on Information Theory, Barcelona, Spain, July 10-15, 2016, pages 1043-1047. IEEE, 2016. URL: http://dx.doi.org/10.1109/ISIT.2016.7541458.
http://dx.doi.org/10.1109/ISIT.2016.7541458
Jingbo Liu, Thomas A. Courtade, Paul W. Cuff, and Sergio Verdú. Information-theoretic perspectives on Brascamp-Lieb inequality and its reverse. CoRR, abs/1702.06260, 2017. URL: http://arxiv.org/abs/1702.06260.
http://arxiv.org/abs/1702.06260
Kaare Brandt Petersen, Michael Syskind Pedersen, et al. The matrix cookbook. Technical University of Denmark, 7(15):510, 2008.
W. Pusz and S.L. Woronowicz. Functional calculus for sesquilinear forms and the purification map. Reports on Mathematical Physics, 8(2):159-170, 1975. URL: http://dx.doi.org/10.1016/0034-4877(75)90061-0.
http://dx.doi.org/10.1016/0034-4877(75)90061-0
Tamás Rapcsák. Geodesic convex functions, pages 61-86. Springer US, Boston, MA, 1997. URL: http://dx.doi.org/10.1007/978-1-4615-6357-0_6.
http://dx.doi.org/10.1007/978-1-4615-6357-0_6
Hirohiko Shima. The geometry of Hessian structures. World Scientific Publishing, 2007.
Carl Ludwig Siegel. Symplectic geometry. American Journal of Mathematics, 65(1):1-86, 1943. URL: http://www.jstor.org/stable/2371774.
http://www.jstor.org/stable/2371774
Mohit Singh and Nisheeth K Vishnoi. Entropy, optimization and counting. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 50-59. ACM, 2014.
Suvrit Sra and Reshad Hosseini. Conic geometric optimization on the manifold of positive definite matrices. SIAM Journal on Optimization, 25(1):713-739, 2015.
Damian Straszak and Nisheeth K. Vishnoi. Computing maximum entropy distributions everywhere. ArXiv e-prints, 2017. URL: http://arxiv.org/abs/1711.02036.
http://arxiv.org/abs/1711.02036
Nisheeth K Vishnoi. Geodesic convex optimization: Differentiation on manifolds, geodesics, and convexity, Jun 2018. URL: https://nisheethvishnoi.files.wordpress.com/2018/06/geodesicconvexity.pdf.
https://nisheethvishnoi.files.wordpress.com/2018/06/geodesicconvexity.pdf
Suvrit Sra, Nisheeth K. Vishnoi, and Ozan Yıldız
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Tensor Rank is Hard to Approximate
We prove that approximating the rank of a 3-tensor to within a factor of 1 + 1/1852 - delta, for any delta > 0, is NP-hard over any field. We do this via reduction from bounded occurrence 2-SAT.
tensor rank
high rank tensor
slice elimination
approximation algorithm
hardness of approximation
Theory of computation~Problems, reductions and completeness
26:1-26:9
Regular Paper
Joseph
Swernofsky
Joseph Swernofsky
Kungliga Tekniska Högskolan, Lindstedtsvägen 3, Stockholm SE-100 44, Sweden
Supported by the Knut and Alice Wallenberg Foundation.
10.4230/LIPIcs.APPROX-RANDOM.2018.26
Boris Alexeev, Michael A. Forbes, and Jacob Tsimerman. Tensor rank: Some lower and upper bounds. CoRR, abs/1102.0072, 2011. URL: http://arxiv.org/abs/1102.0072.
http://arxiv.org/abs/1102.0072
Piotr Berman and Marek Karpinski. Efficient amplifiers and bounded degree optimization. Electronic Colloquium on Computational Complexity (ECCC), 8(53), 2001. URL: http://eccc.hpi-web.de/eccc-reports/2001/TR01-053/index.html.
http://eccc.hpi-web.de/eccc-reports/2001/TR01-053/index.html
Markus Bläser. On the complexity of the multiplication of matrices of small formats. J. Complexity, 19(1):43-60, 2003. URL: http://dx.doi.org/10.1016/S0885-064X(02)00007-9.
http://dx.doi.org/10.1016/S0885-064X(02)00007-9
Markus Bläser. Explicit tensors. In Perspectives in Computational Complexity, pages 117-130. Springer, 2014.
Markus Bläser, Christian Ikenmeyer, Gorav Jindal, and Vladimir Lysikov. Generalized matrix completion and algebraic natural proofs. Electronic Colloquium on Computational Complexity (ECCC), 25:64, 2018. URL: https://eccc.weizmann.ac.il/report/2018/064.
https://eccc.weizmann.ac.il/report/2018/064
Johan Håstad. Tensor rank is np-complete. J. Algorithms, 11(4):644-654, 1990. URL: http://dx.doi.org/10.1016/0196-6774(90)90014-6.
http://dx.doi.org/10.1016/0196-6774(90)90014-6
Tamara G. Kolda and Brett W. Bader. Tensor decompositions and applications. SIAM Review, 51(3):455-500, 2009. URL: http://dx.doi.org/10.1137/07070111X.
http://dx.doi.org/10.1137/07070111X
Joseph B Kruskal. Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear algebra and its applications, 18(2):95-138, 1977.
Julian D Laderman. A noncommutative algorithm for multiplying 3× 3 matrices using 23 multiplications. Bulletin of the American Mathematical Society, 82(1):126-128, 1976.
J. M. Landsberg and Mateusz Michalek. Abelian tensors. CoRR, abs/1504.03732, 2015. URL: http://arxiv.org/abs/1504.03732.
http://arxiv.org/abs/1504.03732
Viñtor Yakovlevich Pan. Methods of computing values of polynomials. Russian Mathematical Surveys, 21(1):105-136, 1966.
Ran Raz. Tensor-rank and lower bounds for arithmetic formulas. Electronic Colloquium on Computational Complexity (ECCC), 17:2, 2010. URL: http://eccc.hpi-web.de/report/2010/002.
http://eccc.hpi-web.de/report/2010/002
Marcus Schaefer and Daniel Stefankovic. The complexity of tensor rank. CoRR, abs/1612.04338, 2016. URL: http://arxiv.org/abs/1612.04338.
http://arxiv.org/abs/1612.04338
Yaroslav Shitov. How hard is the tensor rank? arXiv preprint arXiv:1611.01559, 2016.
Amir Shpilka. Lower bounds for matrix product. CoRR, cs.CC/0201001, 2002. URL: http://arxiv.org/abs/cs.CC/0201001.
http://arxiv.org/abs/cs.CC/0201001
Zhao Song, David P. Woodruff, and Peilin Zhong. Relative error tensor low rank approximation. CoRR, abs/1704.08246, 2017. URL: http://arxiv.org/abs/1704.08246.
http://arxiv.org/abs/1704.08246
Volker Strassen. Gaussian elimination is not optimal. Numerische mathematik, 13(4):354-356, 1969.
Nengkun Yu, Eric Chitambar, Cheng Guo, and Runyao Duan. Tensor rank of the tripartite state |w⟩^⊗ n. Physical Review A, 81(1):014301, 2010.
Joseph Swernofsky
Creative Commons Attribution 3.0 Unported license
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An O(1)-Approximation Algorithm for Dynamic Weighted Vertex Cover with Soft Capacity
This study considers the soft capacitated vertex cover problem in a dynamic setting. This problem generalizes the dynamic model of the vertex cover problem, which has been intensively studied in recent years. Given a dynamically changing vertex-weighted graph G=(V,E), which allows edge insertions and edge deletions, the goal is to design a data structure that maintains an approximate minimum vertex cover while satisfying the capacity constraint of each vertex. That is, when picking a copy of a vertex v in the cover, the number of v's incident edges covered by the copy is up to a given capacity of v. We extend Bhattacharya et al.'s work [SODA'15 and ICALP'15] to obtain a deterministic primal-dual algorithm for maintaining a constant-factor approximate minimum capacitated vertex cover with O(log n / epsilon) amortized update time, where n is the number of vertices in the graph. The algorithm can be extended to (1) a more general model in which each edge is associated with a non-uniform and unsplittable demand, and (2) the more general capacitated set cover problem.
approximation algorithm
dynamic algorithm
primal-dual
vertex cover
Theory of computation~Dynamic graph algorithms
27:1-27:14
Regular Paper
https://arxiv.org/abs/1802.05623
Hao-Ting
Wei
Hao-Ting Wei
Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu 30013, Taiwan
Wing-Kai
Hon
Wing-Kai Hon
Department of Computer Science, National Tsing Hua University, Hsinchu 30013, Taiwan
Paul
Horn
Paul Horn
Department of Mathematics, University of Denver, Denver, USA
Supported by NSA Young Investigator Grant H98230-15-1-0258, and Simons Collaboration Grant #525039.
Chung-Shou
Liao
Chung-Shou Liao
Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu 30013, Taiwan
Supported by MOST Taiwan under Grants MOST105-2628-E-007-010-MY3 and MOST105-2221-E-007-085-MY3.
Kunihiko
Sadakane
Kunihiko Sadakane
Department of Mathematical Informatics, The University of Tokyo, Tokyo, Japan
10.4230/LIPIcs.APPROX-RANDOM.2018.27
Hao-Ting Wei, Wing-Kai Hon, Paul Horn, Chung-Shou Liao, and Kunihiko Sadakane
Creative Commons Attribution 3.0 Unported license
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Fixed-Parameter Approximation Schemes for Weighted Flowtime
Given a set of n jobs with integral release dates, processing times and weights, it is a natural and important scheduling problem to compute a schedule that minimizes the sum of the weighted flow times of the jobs. There are strong lower bounds for the possible approximation ratios. In the non-preemptive case, even on a single machine the best known result is a O(sqrt{n})-approximation which is best possible. In the preemptive case on m identical machines there is a O(log min{n/m,P})-approximation (where P denotes the maximum job size) which is also best possible.
We study the problem in the parametrized setting where our parameter k is an upper bound on the maximum (integral) processing time and weight of a job, a standard parameter for scheduling problems. We present a (1+epsilon)-approximation algorithm for the preemptive and the non-preemptive case of minimizing weighted flow time on m machines with a running time of f(k,epsilon,m)* n^{O(1)}, i.e., our combined parameters are k,epsilon, and m. Key to our results is to distinguish time intervals according to whether in the optimal solution the pending jobs have large or small total weight. Depending on this we employ dynamic programming, linear programming, greedy routines, or combinations of the latter to compute the schedule for each respective interval.
Scheduling
fixed-parameter algorithms
approximation algorithms
approximation schemes
Theory of computation~Scheduling algorithms
28:1-28:19
Regular Paper
Andreas
Wiese
Andreas Wiese
Department of Industrial Engineering and Center for Mathematical Modeling, Universidad de Chile, Chile
This work was partially supported by the Millennium Nucleus Information and Coordination in Networks ICM/FIC RC130003 and the grant Fondecyt Regular 1170223.
10.4230/LIPIcs.APPROX-RANDOM.2018.28
Foto N. Afrati, Evripidis Bampis, Chandra Chekuri, David R. Karger, Claire Kenyon, Sanjeev Khanna, Ioannis Milis, Maurice Queyranne, Martin Skutella, Clifford Stein, and Maxim Sviridenko. Approximation schemes for minimizing average weighted completion time with release dates. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pages 32-44. IEEE, 1999.
Yossi Azar and Noam Touitou. Improved online algorithm for weighted flow time. CoRR, abs/1712.10273, 2017. URL: http://arxiv.org/abs/1712.10273.
http://arxiv.org/abs/1712.10273
Nikhil Bansal. Minimizing flow time on a constant number of machines with preemption. Oper. Res. Lett., 33(3):267-273, 2005. URL: http://dx.doi.org/10.1016/j.orl.2004.07.008.
http://dx.doi.org/10.1016/j.orl.2004.07.008
Nikhil Bansal and Kedar Dhamdhere. Minimizing weighted flow time. ACM Trans. Algorithms, 3(4), 2007. URL: http://dx.doi.org/10.1145/1290672.1290676.
http://dx.doi.org/10.1145/1290672.1290676
Nikhil Bansal and Kirk Pruhs. The geometry of scheduling. SIAM Journal on Computing, 43(5):1684-1698, 2014.
Jatin Batra, Naveen Garg, and Amit Kumar. Constant factor approximation algorithm for weighted flow time on a single machine in pseudo-polynomial time. CoRR, abs/1802.07439, 2018. URL: http://arxiv.org/abs/1802.07439.
http://arxiv.org/abs/1802.07439
Chandra Chekuri and Sanjeev Khanna. A ptas for minimizing weighted completion time on uniformly related machines. In Fernando Orejas, Paul G. Spirakis, and Jan van Leeuwen, editors, Automata, Languages and Programming, pages 848-861, Berlin, Heidelberg, 2001. Springer Berlin Heidelberg.
Chandra Chekuri and Sanjeev Khanna. Approximation schemes for preemptive weighted flow time. In Proceedings of the thiry-fourth annual ACM Symposium on Theory of computing, pages 297-305. ACM, 2002.
Chandra Chekuri, Sanjeev Khanna, and An Zhu. Algorithms for minimizing weighted flow time. In Proceedings of the thirty-third annual ACM Symposium on Theory of computing, pages 84-93. ACM, 2001.
Leah Epstein and Asaf Levin. Minimum total weighted completion time: Faster approximation schemes. CoRR, abs/1404.1059, 2014. URL: http://arxiv.org/abs/1404.1059.
http://arxiv.org/abs/1404.1059
Naveen Garg, Amit Kumar, and V. N. Muralidhara. Minimizing total flow-time: The unrelated case. In Proceedings of the 19th International Symposium on Algorithms and Computation (ISAAC 2008), pages 424-435. Springer, 2008. URL: http://dx.doi.org/10.1007/978-3-540-92182-0_39.
http://dx.doi.org/10.1007/978-3-540-92182-0_39
Fabrizio Grandoni, Stefan Kratsch, and Andreas Wiese. Parameterized approximation schemes for independent set of rectangles and geometric knapsack, 2017. Unpublished.
Hans Kellerer, Thomas Tautenhahn, and Gerhard Woeginger. Approximability and nonapproximability results for minimizing total flow time on a single machine. SIAM Journal on Computing, 28(4):1155-1166, 1999.
Dusan Knop and Martin Koutecký. Scheduling meets n-fold integer programming. CoRR, abs/1603.02611, 2016. URL: http://arxiv.org/abs/1603.02611.
http://arxiv.org/abs/1603.02611
Stefano Leonardi and Danny Raz. Approximating total flow time on parallel machines. In Proceedings of the twenty-ninth annual ACM Symposium on Theory of Computing, pages 110-119. ACM, 1997.
Joseph Y-T. Leung. Handbook of scheduling: algorithms, models, and performance analysis. CRC Press, 2004.
Dániel Marx. Parameterized complexity and approximation algorithms. The Computer Journal, 51(1):60-78, 2008.
Matthias Mnich and Andreas Wiese. Scheduling and fixed-parameter tractability. Math. Program., 154(1-2):533-562, 2015.
Andreas Wiese. A (1+epsilon)-approximation for unsplittable flow on a path in fixed-parameter running time. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, pages 67:1-67:13, 2017.
Andreas Wiese
Creative Commons Attribution 3.0 Unported license
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List-Decoding Homomorphism Codes with Arbitrary Codomains
The codewords of the homomorphism code aHom(G,H) are the affine homomorphisms between two finite groups, G and H, generalizing Hadamard codes. Following the work of Goldreich-Levin (1989), Grigorescu et al. (2006), Dinur et al. (2008), and Guo and Sudan (2014), we further expand the range of groups for which local list-decoding is possible up to mindist, the minimum distance of the code. In particular, for the first time, we do not require either G or H to be solvable. Specifically, we demonstrate a poly(1/epsilon) bound on the list size, i. e., on the number of codewords within distance (mindist-epsilon) from any received word, when G is either abelian or an alternating group, and H is an arbitrary (finite or infinite) group. We conjecture that a similar bound holds for all finite simple groups as domains; the alternating groups serve as the first test case.
The abelian vs. arbitrary result permits us to adapt previous techniques to obtain efficient local list-decoding for this case. We also obtain efficient local list-decoding for the permutation representations of alternating groups (the codomain is a symmetric group) under the restriction that the domain G=A_n is paired with codomain H=S_m satisfying m < 2^{n-1}/sqrt{n}.
The limitations on the codomain in the latter case arise from severe technical difficulties stemming from the need to solve the homomorphism extension (HomExt) problem in certain cases; these are addressed in a separate paper (Wuu 2018).
We introduce an intermediate "semi-algorithmic" model we call Certificate List-Decoding that bypasses the HomExt bottleneck and works in the alternating vs. arbitrary setting. A certificate list-decoder produces partial homomorphisms that uniquely extend to the homomorphisms in the list. A homomorphism extender applied to a list of certificates yields the desired list.
Error-correcting codes
Local algorithms
Local list-decoding
Finite groups
Homomorphism codes
Mathematics of computing~Coding theory
Mathematics of computing~Probabilistic algorithms
29:1-29:18
Regular Paper
https://arxiv.org/abs/1806.02969
László
Babai
László Babai
University of Chicago, Chicago IL, USA
https://orcid.org/0000-0002-2058-685X
Partially supported by NSF Grants CCF 1423309 and CCF 1718902. The views expressed in the paper are those of the authors and do not necessarily reflect the views of the NSF.
Timothy J. F.
Black
Timothy J. F. Black
University of Chicago, Chicago IL, USA
https://orcid.org/0000-0003-2469-9867
Partially supported by L. Babai’s cited NSF grants.
Angela
Wuu
Angela Wuu
University of Chicago, Chicago IL, USA
Partially supported by L. Babai’s cited NSF grants.
10.4230/LIPIcs.APPROX-RANDOM.2018.29
László Babai. On the length of subgroup chains in the symmetric group. Communications in Algebra, 14(9):1729-1736, 1986. URL: http://dx.doi.org/10.1080/00927878608823393.
http://dx.doi.org/10.1080/00927878608823393
László Babai. Local expansion of vertex-transitive graphs and random generation in finite groups. In 23rd STOC, pages 164-174. ACM, 1991. URL: http://dx.doi.org/10.1145/103418.103440.
http://dx.doi.org/10.1145/103418.103440
László Babai, Robert Beals, and Ákos Seress. Polynomial-time theory of matrix groups. In 41st STOC, pages 55-64. ACM, 2009. URL: http://dx.doi.org/10.1145/1536414.1536425.
http://dx.doi.org/10.1145/1536414.1536425
László Babai, Timothy J. F. Black, and Angela Wuu. List-decoding homomorphism codes with arbitrary codomains. arXiv, 2018. (full version of this paper). URL: http://arxiv.org/abs/1806.02969.
http://arxiv.org/abs/1806.02969
László Babai and Endre Szemerédi. On the complexity of matrix group problems I. In 25th FOCS, pages 229-240. IEEE Computer Soc., 1984. URL: http://dx.doi.org/10.1109/SFCS.1984.715919.
http://dx.doi.org/10.1109/SFCS.1984.715919
László Babai. The probability of generating the symmetric group. J. Combinat. Theory, Series A, 52(1):148-153, 1989. URL: http://dx.doi.org/10.1016/0097-3165(89)90068-X.
http://dx.doi.org/10.1016/0097-3165(89)90068-X
Robert Beals, Charles R. Leedham-Green, Alice C. Niemeyer, Cheryl E. Praeger, and Ákos Seress. Constructive recognition of finite alternating and symmetric groups acting as matrix groups on their natural permutation modules. J. Algebra, 292(1):4-46, 2005. URL: http://dx.doi.org/10.1016/j.jalgebra.2005.01.035.
http://dx.doi.org/10.1016/j.jalgebra.2005.01.035
Abhishek Bhowmick and Shachar Lovett. The list decoding radius of Reed-Muller codes over small fields. In 47th STOC, pages 277-285. ACM, 2015. URL: http://dx.doi.org/10.1145/2746539.2746543.
http://dx.doi.org/10.1145/2746539.2746543
Timothy Black, Alan Guo, Madhu Sudan, and Angela Wuu. List decoding nilpotent groups. In preparation, 2018.
Irit Dinur, Elena Grigorescu, Swastik Kopparty, and Madhu Sudan. Decodability of group homomorphisms beyond the Johnson bound. In 40th STOC, pages 275-284. ACM, 2008. URL: http://dx.doi.org/10.1145/1374376.1374418.
http://dx.doi.org/10.1145/1374376.1374418
John D. Dixon and Brian Mortimer. Permutation Groups. Graduate Texts in Math. Springer New York, 1996.
Oded Goldreich and Leonid A. Levin. A hard-core predicate for all one-way functions. In 21st STOC, pages 25-32. ACM, 1989. URL: http://dx.doi.org/10.1145/73007.73010.
http://dx.doi.org/10.1145/73007.73010
Parikshit Gopalan, Adam R. Klivans, and David Zuckerman. List-decoding Reed-Muller codes over small fields. In 40th STOC, pages 265-274. ACM, 2008. URL: http://dx.doi.org/10.1145/1374376.1374417.
http://dx.doi.org/10.1145/1374376.1374417
Elena Grigorescu, Swastik Kopparty, and Madhu Sudan. Local decoding and testing for homomorphisms. In APPROX-RANDOM, volume 4110 of Lecture Notes in Computer Science, pages 375-385. Springer, 2006. URL: http://dx.doi.org/10.1007/11830924_35.
http://dx.doi.org/10.1007/11830924_35
Alan Guo. Group homomorphisms as error correcting codes. Electronic J. Combinatorics, 22(1):P1.4, 2015.
Alan Guo and Madhu Sudan. List decoding group homomorphisms between supersolvable groups. In APPROX/RANDOM, volume 28 of LIPIcs, pages 737-747. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2014. URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.737.
http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.737
William Cary Huffman and Vera Pless. Fundamentals of Error-Correcting Codes. Cambridge Univ. Press, 2003.
I. Martin Isaacs. Finite Group Theory. Graduate studies in mathematics. Amer. Math. Soc., 2008.
Derek J. S. Robinson. A Course in the Theory of Groups. Springer, 2nd edition, 1995.
Ákos Seress. Permutation Group Algorithms. Cambridge Tracts in Math. Cambridge Univ. Press, 2003.
Angela Wuu. Homomorphism extension. arXiv, 2018. URL: http://arxiv.org/abs/1802.08656.
http://arxiv.org/abs/1802.08656
László Babai, Timothy J. F. Black, and Angela Wuu
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Optimal Deterministic Extractors for Generalized Santha-Vazirani Sources
Let F be a finite alphabet and D be a finite set of distributions over F. A Generalized Santha-Vazirani (GSV) source of type (F, D), introduced by Beigi, Etesami and Gohari (ICALP 2015, SICOMP 2017), is a random sequence (F_1, ..., F_n) in F^n, where F_i is a sample from some distribution d in D whose choice may depend on F_1, ..., F_{i-1}.
We show that all GSV source types (F, D) fall into one of three categories: (1) non-extractable; (2) extractable with error n^{-Theta(1)}; (3) extractable with error 2^{-Omega(n)}.
We provide essentially randomness-optimal extraction algorithms for extractable sources. Our algorithm for category (2) sources extracts one bit with error epsilon from n = poly(1/epsilon) samples in time linear in n. Our algorithm for category (3) sources extracts m bits with error epsilon from n = O(m + log 1/epsilon) samples in time min{O(m2^m * n),n^{O(|F|)}}.
We also give algorithms for classifying a GSV source type (F, D): Membership in category (1) can be decided in NP, while membership in category (3) is polynomial-time decidable.
feasibility of randomness extraction
extractor lower bounds
martingales
Theory of computation~Expander graphs and randomness extractors
Mathematics of computing~Probability and statistics
Mathematics of computing~Information theory
30:1-30:15
Regular Paper
Salman
Beigi
Salman Beigi
Institute for Research in Fundamental Sciences, Tehran, Iran
Andrej
Bogdanov
Andrej Bogdanov
Chinese University of Hong Kong
Omid
Etesami
Omid Etesami
Institute for Research in Fundamental Sciences, Tehran, Iran
Siyao
Guo
Siyao Guo
Northeastern University, Boston, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.30
Salman Beigi, Andrej Bogdanov, Omid Etesami, and Siyao Guo. Complete classification of generalized Santha-Vazirani sources. Technical Report TR17-136, Electronic Colloquium on Computational Complexity, 2017.
Salman Beigi, Omid Etesami, and Amin Gohari. Deterministic randomness extraction from generalized and distributed santha-vazirani sources. In International Colloquium on Automata, Languages, and Programming, pages 143-154. Springer, 2015.
Salman Beigi, Omid Etesami, and Amin Gohari. Deterministic randomness extraction from generalized and distributed santha-vazirani sources. SIAM Journal on Computing, 46(1):1-36, 2017.
Iddo Bentov, Ariel Gabizon, and David Zuckerman. Bitcoin beacon. arXiv preprint arXiv:1605.04559, 2016.
Jean Bourgain. More on the sum-product phenomenon in prime fields and its applications. International Journal of Number Theory, 1(01):1-32, 2005.
Jean Bourgain. On the construction of affine extractors. GAFA Geometric And Functional Analysis, 17(1):33-57, 2007.
Eshan Chattopadhyay and David Zuckerman. Explicit two-source extractors and resilient functions. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 670-683, 2016. URL: http://dx.doi.org/10.1145/2897518.2897528.
http://dx.doi.org/10.1145/2897518.2897528
Benny Chor and Oded Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM Journal on Computing, 17(2):230-261, 1988.
Yevgeniy Dodis, Shien Jin Ong, Manoj Prabhakaran, and Amit Sahai. On the (im)possibility of cryptography with imperfect randomness. In 45th Symposium on Foundations of Computer Science (FOCS 2004), 17-19 October 2004, Rome, Italy, Proceedings, pages 196-205, 2004. URL: http://dx.doi.org/10.1109/FOCS.2004.44.
http://dx.doi.org/10.1109/FOCS.2004.44
Zeev Dvir. Extractors for varieties. Computational complexity, 21(4):515-572, 2012.
Zeev Dvir, Ariel Gabizon, and Avi Wigderson. Extractors and rank extractors for polynomial sources. Computational Complexity, 18(1):1-58, 2009.
Ariel Gabizon. Deterministic extractors for affine sources over large fields. In Deterministic Extraction from Weak Random Sources, pages 33-53. Springer, 2011.
Saeed Mahloujifar, Dimitrios I Diochnos, and Mohammad Mahmoody. Learning under p-tampering attacks. arXiv preprint arXiv:1711.03707, 2017.
Saeed Mahloujifar and Mohammad Mahmoody. Blockwise p-tampering attacks on cryptographic primitives, extractors, and learners. In Theory of Cryptography Conference, pages 245-279. Springer, 2017.
Jaikumar Radhakrishnan and Amnon Ta-Shma. Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM J. Discrete Math., 13(1):2-24, 2000. URL: http://dx.doi.org/10.1137/S0895480197329508.
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Salil P Vadhan. Pseudorandomness, volume 56. Now, 2012.
Salman Beigi, Andrej Bogdanov, Omid Etesami, and Siyao Guo
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Adaptive Lower Bound for Testing Monotonicity on the Line
In the property testing model, the task is to distinguish objects possessing some property from the objects that are far from it. One of such properties is monotonicity, when the objects are functions from one poset to another. This is an active area of research. In this paper we study query complexity of epsilon-testing monotonicity of a function f : [n]->[r]. All our lower bounds are for adaptive two-sided testers.
- We prove a nearly tight lower bound for this problem in terms of r. The bound is Omega((log r)/(log log r)) when epsilon = 1/2. No previous satisfactory lower bound in terms of r was known.
- We completely characterise query complexity of this problem in terms of n for smaller values of epsilon. The complexity is Theta(epsilon^{-1} log (epsilon n)). Apart from giving the lower bound, this improves on the best known upper bound.
Finally, we give an alternative proof of the Omega(epsilon^{-1}d log n - epsilon^{-1}log epsilon^{-1}) lower bound for testing monotonicity on the hypergrid [n]^d due to Chakrabarty and Seshadhri (RANDOM'13).
property testing
monotonicity on the line
monotonicity on the hypergrid
Theory of computation~Lower bounds and information complexity
31:1-31:10
Regular Paper
This research is supported by the ERDF project number 1.1.1.2/I/16/113.
Aleksandrs
Belovs
Aleksandrs Belovs
Faculty of Computing, University of Latvia, Raina bulvaris 19, Riga, Latvia.
supported by the ERDF project number 1.1.1.2/I/16/113
10.4230/LIPIcs.APPROX-RANDOM.2018.31
Aleksandrs Belovs and Eric Blais. Quantum algorithm for monotonicity testing on the hypercube. Theory of Computing, 11(16):403-412, 2015.
Aleksandrs Belovs and Eric Blais. A polynomial lower bound for testing monotonicity. In Proc. of 48th ACM STOC, pages 1021-1032, 2016.
Eric Blais, Joshua Brody, and Kevin Matulef. Property testing lower bounds via communication complexity. Computational Complexity, 21(2):311-358, 2012.
Eric Blais, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Lower bounds for testing properties of functions over hypergrid domains. In Proc. of 29th IEEE CCC, pages 309-320, 2014.
Deeparnab Chakrabarty and C. Seshadhri. An optimal lower bound for monotonicity testing over hypergrids. Theory of Computing, 10:453-464, 2014.
Deeparnab Chakrabarty and Comandur Seshadhri. A o(n) monotonicity tester for Boolean functions over the hypercube. In Proc. of 45th ACM STOC, pages 411-418, 2013.
Deeparnab Chakrabarty and Comandur Seshadhri. Optimal bounds for monotonicity and Lipschitz testing over hypercubes and hypergrids. In Proc. of 45th ACM STOC, pages 419-428, 2013.
Xi Chen, Anindya De, Rocco A. Servedio, and Li-Yang Tan. Boolean function monotonicity testing requires (almost) n^1/2 non-adaptive queries. In Proc. of 47th ACM STOC, pages 519-528, 2015.
Xi Chen, Rocco A. Servedio, and Li-Yang Tan. New algorithms and lower bounds for monotonicity testing. In Proc. of 55th IEEE FOCS, pages 286-295, 2014.
Xi Chen, Erik Waingarten, and Jinyu Xie. Beyond Talagrand functions: new lower bounds for testing monotonicity and unateness. In Proc. of 49th ACM STOC, pages 523-536, 2017.
Yevgeniy Dodis, Oded Goldreich, Eric Lehman, Sofya Raskhodnikova, Dana Ron, and Alex Samorodnitsky. Improved testing algorithms for monotonicity. In Proc. of 3rd, pages 97-108. Springer, 1999.
Funda" Ergün, Sampath" "Kannan, S Ravi" "Kumar, Ronitt" "Rubinfeld, and Mahesh "Viswanathan. Spot-checkers. Journal of Computer and System Sciences, 60(3):717-751, 2000.
Eldar Fischer. On the strength of comparisons in property testing. Information and Computation, 189(1):107-116, 2004.
Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. Monotonicity testing over general poset domains. In Proc. of 34th ACM STOC, pages 474-483, 2002.
Oded Goldreich, Shafi Goldwasser, Eric Lehman, Dana Ron, and Alex Samorodnitsky. Testing monotonicity. Combinatorica, 20(3):301-337, 2000.
Oded Goldreich, Shari Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM, 45(4):653-750, 1998.
Subhash Khot, Dor Minzer, and Muli Safra. On monotonicity testing and Boolean isoperimetric type theorems. In Proc. of 56th IEEE FOCS, pages 52-58, 2015.
Ramesh Krishnan S Pallavoor, Sofya Raskhodnikova, and Nithin Varma. Parameterized property testing of functions. In Proc. of 8th ACM ITCS, volume 67 of LIPIcs, page 12. Dagstuhl, 2017.
Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252-271, 1996.
Aleksandrs Belovs
Creative Commons Attribution 3.0 Unported license
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Swendsen-Wang Dynamics for General Graphs in the Tree Uniqueness Region
The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph G=(V,E). The dynamics is a "global" Markov chain which is conjectured to converge to equilibrium in O(|V|^{1/4}) steps for any graph G at any (inverse) temperature beta. It was recently proved by Guo and Jerrum (2017) that the Swendsen-Wang dynamics has polynomial mixing time on any graph at all temperatures, yet there are few results providing o(|V|) upper bounds on its convergence time.
We prove fast convergence of the Swendsen-Wang dynamics on general graphs in the tree uniqueness region of the ferromagnetic Ising model. In particular, when beta < beta_c(d) where beta_c(d) denotes the uniqueness/non-uniqueness threshold on infinite d-regular trees, we prove that the relaxation time (i.e., the inverse spectral gap) of the Swendsen-Wang dynamics is Theta(1) on any graph of maximum degree d >= 3. Our proof utilizes a version of the Swendsen-Wang dynamics which only updates isolated vertices. We establish that this variant of the Swendsen-Wang dynamics has mixing time O(log{|V|}) and relaxation time Theta(1) on any graph of maximum degree d for all beta < beta_c(d). We believe that this Markov chain may be of independent interest, as it is a monotone Swendsen-Wang type chain. As part of our proofs, we provide modest extensions of the technology of Mossel and Sly (2013) for analyzing mixing times and of the censoring result of Peres and Winkler (2013). Both of these results are for the Glauber dynamics, and we extend them here to general monotone Markov chains. This class of dynamics includes for example the heat-bath block dynamics, for which we obtain new tight mixing time bounds.
Swendsen-Wang dynamics
mixing time
relaxation time
spatial mixing
censoring
Theory of computation~Random walks and Markov chains
Theory of computation~Design and analysis of algorithms
32:1-32:18
Regular Paper
Research supported in part by NSF grants CCF-1617306 and CCF-1563838.
https://arxiv.org/pdf/1806.04602.pdf
Antonio
Blanca
Antonio Blanca
School of Computer Science, Georgia Institute of Technology, Atlanta GA 30332, USA
Zongchen
Chen
Zongchen Chen
School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332, USA
Eric
Vigoda
Eric Vigoda
School of Computer Science, Georgia Institute of Technology, Atlanta GA 30332, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.32
A. Blanca, P. Caputo, A. Sinclair, and E. Vigoda. Spatial Mixing and Non-local Markov chains. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1965-1980, 2018.
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A. Galanis, D. Štefankovič, and E. Vigoda. Swendsen-Wang algorithm on the mean-field Potts model. In Proceedings of the 19th International Workshop on Randomization and Computation (RANDOM), pages 815-828, 2015.
D. Gamarnik and D. Katz. Correlation decay and deterministic FPTAS for counting list-colorings of a graph. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1245-1254, 2007.
S. Geman and C. Graffigne. Markov random field image models and their applications to computer vision. In Proceedings of the International Congress of Mathematicians, pages 1496-1517, 1986.
R. Gheissari, E. Lubetzky, and Y. Peres. Exponentially slow mixing in the mean-field Swendsen-Wang dynamics. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1981-1988, 2018.
W. R. Gilks, S. Richardson, and D. J. Spiegelhalter. Markov chain Monte Carlo in practice. London: Chapman and Hall, 1996.
V. K. Gore and M. R. Jerrum. The Swendsen-Wang process does not always mix rapidly. Journal of Statistical Physics, 97(1-2):67-86, 1999.
G. R. Grimmett. The Random-Cluster Model, volume 333. Springer-Verlag, 2009.
H. Guo and M. Jerrum. Random cluster dynamics for the Ising model is rapidly mixing. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1818-1827, 2017.
T. P. Hayes and A. Sinclair. A general lower bound for mixing of single-site dynamics on graphs. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 511-520, 2005.
A. E. Holroyd. Some Circumstances Where Extra Updates Can Delay Mixing. Journal of Statistical Physics, 145(6):1649-1652, 2011.
E. Ising. Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik, 31(1):253-258, 1925.
M. Jerrum and A. Sinclair. Polynomial-time approximation algorithms for the Ising model. SIAM Journal on Computing, 22(5):1087-1116, 1993.
M. Jerrum, A. Sinclair, and E. Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACM, 51(4):671-697, 2004.
M. R. Jerrum, L. G. Valiant, and V. V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theoretical Computer Science, 43(2-3):169-188, 1986.
R. Kannan, L. Lovász, and M. Simonovits. Random walks and an O^*(n⁵) volume algorithm for convex bodies. Random Structures &Algorithms, 11(1):1-50, 1997.
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D. A. Levin and Y. Peres. Markov Chains and Mixing Times, 2nd edition, volume 107. American Mathematical Society, 2017.
L. Li, P. Lu, and Y. Yin. Correlation decay up to uniqueness in spin systems. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 67-84, 2013.
Y. Long, A. Nachmias, W. Ning, and Y. Peres. A power law of order 1/4 for critical mean-field Swendsen-Wang dynamics. Memoirs of the American Mathematical Society, 232(1092):84 pages, 2014.
F. Martinelli and E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case. Communications in Mathematical Physics, 161(3):447-486, 1994.
F. Martinelli and E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case. Communications in Mathematical Physics, 161(3):458-514, 1994.
A. Montanari and A. Saberi. The spread of innovations in social networks. Proceedings of the National Academy of Sciences, 107(47):20196-20201, 2010.
E. Mossel and A. Sly. Exact thresholds for Ising-Gibbs samplers on general graphs. The Annals of Probability, 41(1):294-328, 2013.
Y. Peres. Personal communication, 2016.
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A. Sinclair, P. Srivastava, D. Štefankovič, and Y. Yin. Spatial mixing and the connective constant: Optimal bounds. Probability Theory and Related Fields, 168(1-2):153-197, 2017.
A. Sinclair, P. Srivastava, and M. Thurley. Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. Journal of Statistical Physics, 155(4):666-686, 2014.
A. Sly. Computational transition at the uniqueness threshold. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 287-296, 2010.
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Antonio Blanca, Zongchen Chen, and Eric Vigoda
Creative Commons Attribution 3.0 Unported license
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Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs
We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q >= 3 and Delta >= 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Delta-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases.
The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and Delta in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value.
In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters p,q on random Delta-regular graphs for all values of q >= 1 and p<p_c(q,Delta), where p_c(q,Delta) corresponds to a uniqueness threshold for the model on the Delta-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Delta-regular graphs.
sampling
Potts model
random regular graphs
phase transitions
Theory of computation~Randomness, geometry and discrete structures
Theory of computation~Design and analysis of algorithms
33:1-33:15
Regular Paper
https://arxiv.org/abs/1804.08111
Antonio
Blanca
Antonio Blanca
School of Computer Science, Georgia Institute of Technology, Atlanta GA 30332, USA
Research supported in part by NSF grants CCF-1617306 and CCF-1563838.
Andreas
Galanis
Andreas Galanis
Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD, UK
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.
Leslie Ann
Goldberg
Leslie Ann Goldberg
Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD, UK
Daniel
Stefankovic
Daniel Stefankovic
Department of Computer Science, University of Rochester, Rochester, NY 14627, USA
Research supported in part by NSF grant CCF-1563757.
Eric
Vigoda
Eric Vigoda
School of Computer Science, Georgia Institute of Technology, Atlanta GA 30332, USA
Kuan
Yang
Kuan Yang
Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD, UK
10.4230/LIPIcs.APPROX-RANDOM.2018.33
K. S. Alexander. Mixing properties and exponential decay for lattice systems in finite volumes. Ann. Probab., 32(1A):441-487, 2004.
V. Beffara and H. Duminil-Copin. The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1. Probability Theory and Related Fields, 153(3):511-542, 2012.
M. Bordewich, C. Greenhill, and V. Patel. Mixing of the Glauber dynamics for the ferromagnetic Potts model. Random Structures &Algorithms, 48(1):21-52, 2016.
G. R. Brightwell and P. Winkler. Random colorings of a Cayley tree. Contemporary combinatorics, 10:247-276, 2002.
A. Dembo, A. Montanari, A. Sly, and N. Sun. The replica symmetric solution for Potts models on d-regular graphs. Communications in Mathematical Physics, 327(2):551-575, 2014.
C. Efthymiou. A simple algorithm for random colouring G(n, d/n) using (2 + ε)d colours. In Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '12, pages 272-280, 2012.
C. Efthymiou. A simple algorithm for sampling colorings of G(n,d/n) up to the Gibbs uniqueness threshold. SIAM Journal on Computing, 45(6):2087-2116, 2016.
C. Efthymiou, T. P. Hayes, D. Štefankovič, and E. Vigoda. Sampling random colorings of sparse random graphs. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '18, pages 1759-1771, 2018.
A. Galanis, L. A. Goldberg, and K. Yang. Uniqueness of the 3-state antiferromagnetic Potts model on the tree. arXiv/1804.03514, 2018.
A. Galanis, D. Štefankovič, and E. Vigoda. Inapproximability for antiferromagnetic spin systems in the tree nonuniqueness region. J. ACM, 62(6):50:1-50:60, 2015.
A. Galanis, D. Štefankovič, E. Vigoda, and L. Yang. Ferromagnetic Potts model: Refined #BIS-hardness and related results. SIAM Journal on Computing, 45(6):2004-2065, 2016.
A. Gerschenfeld and A. Montanari. Reconstruction for models on random graphs. In Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS '07, pages 194-204, 2007.
G. Grimmett. The Random-Cluster Model. Springer, 2006.
O. Häggström. The random-cluster model on a homogeneous tree. Probability Theory and Related Fields, 104(2):231-253, 1996.
J. Jonasson. The random cluster model on a general graph and a phase transition characterization of nonamenability. Stochastic Processes and their Applications, 79(2):335-354, 1999.
J. Jonasson. Uniqueness of uniform random colorings of regular trees. Statistics &Probability Letters, 57(3):243-248, 2002.
F. Martinelli and E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case. Comm. Math. Phys., 161(3):447-486, 1994.
F. Martinelli and E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case. Communications in Mathematical Physics, 161(3):487-514, 1994.
F. Martinelli, E. Olivieri, and R. H. Schonmann. For 2-D lattice spin systems weak mixing implies strong mixing. Communications in Mathematical Physics, 165(1):33-47, 1994.
M. Mézard and A. Montanari. Information, Physics, and Computation. Oxford University Press, 2009.
E. Mossel and A. Sly. Rapid mixing of Gibbs sampling on graphs that are sparse on average. Random Structures &Algorithms, 35(2):250-270, 2009.
E. Mossel and A. Sly. Exact thresholds for Ising-Gibbs samplers on general graphs. Ann. Probab., 41(1):294-328, 2013.
E. Mossel, D. Weitz, and N. Wormald. On the hardness of sampling independent sets beyond the tree threshold. Probability Theory and Related Fields, 143(3):401-439, 2009.
A. Sinclair, P. Srivastava, D. Štefankovič, and Y. Yin. Spatial mixing and the connective constant: optimal bounds. Probability Theory and Related Fields, 168(1):153-197, 2017.
A. Sinclair, P. Srivastava, and M. Thurley. Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. Journal of Statistical Physics, 155(4):666-686, 2014.
A. Sinclair, P. Srivastava, and Y. Yin. Spatial mixing and approximation algorithms for graphs with bounded connective constant. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, pages 300-309, 2013.
L. E. Thomas. Bound on the mass gap for finite volume stochastic Ising models at low temperature. Communications in Mathematical Physics, 126(1):1-11, 1989.
Y. Yin and C. Zhang. Sampling in Potts model on sparse random graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2016, pages 47:1-47:22, 2016.
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Antonio Blanca, Andreas Galanis, Leslie Ann Goldberg, Daniel Štefankovič, Eric Vigoda, and Kuan Yang
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Polar Codes with Exponentially Small Error at Finite Block Length
We show that the entire class of polar codes (up to a natural necessary condition) converge to capacity at block lengths polynomial in the gap to capacity, while simultaneously achieving failure probabilities that are exponentially small in the block length (i.e., decoding fails with probability exp(-N^{Omega(1)}) for codes of length N). Previously this combination was known only for one specific family within the class of polar codes, whereas we establish this whenever the polar code exhibits a condition necessary for any polarization.
Our results adapt and strengthen a local analysis of polar codes due to the authors with Nakkiran and Rudra [Proc. STOC 2018]. Their analysis related the time-local behavior of a martingale to its global convergence, and this allowed them to prove that the broad class of polar codes converge to capacity at polynomial block lengths. Their analysis easily adapts to show exponentially small failure probabilities, provided the associated martingale, the "Arikan martingale", exhibits a corresponding strong local effect. The main contribution of this work is a much stronger local analysis of the Arikan martingale. This leads to the general result claimed above.
In addition to our general result, we also show, for the first time, polar codes that achieve failure probability exp(-N^{beta}) for any beta < 1 while converging to capacity at block length polynomial in the gap to capacity. Finally we also show that the "local" approach can be combined with any analysis of failure probability of an arbitrary polar code to get essentially the same failure probability while achieving block length polynomial in the gap to capacity.
Polar codes
error exponent
rate of polarization
Mathematics of computing~Coding theory
34:1-34:17
Regular Paper
Jaroslaw
Blasiok
Jaroslaw Blasiok
Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, 33 Oxford Street, Cambridge, MA 02138, USA
Supported by ONR grant N00014-15-1-2388.
Venkatesan
Guruswami
Venkatesan Guruswami
Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
This work was done when the author was visiting the Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138. Research supported in part by NSF grants CCF-1422045 and CCF-1563742.
Madhu
Sudan
Madhu Sudan
Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, 33 Oxford Street, Cambridge, MA 02138, USA.
Work supported in part by a Simons Investigator Award and NSF Awards CCF 1565641 and CCF 1715187.
10.4230/LIPIcs.APPROX-RANDOM.2018.34
Erdal Arıkan. Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE Transactions on Information Theory, pages 3051-3073, July 2009.
Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, Atri Rudra, and Madhu Sudan. General strong polarization. CoRR, abs/1802.02718, 2018. URL: http://arxiv.org/abs/1802.02718.
http://arxiv.org/abs/1802.02718
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Jarosław Błasiok, Venkatesan Guruswami, and Madhu Sudan
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Approximate Degree and the Complexity of Depth Three Circuits
Threshold weight, margin complexity, and Majority-of-Threshold circuit size are basic complexity measures of Boolean functions that arise in learning theory, communication complexity, and circuit complexity. Each of these measures might exhibit a chasm at depth three: namely, all polynomial size Boolean circuits of depth two have polynomial complexity under the measure, but there may exist Boolean circuits of depth three that have essentially maximal complexity exp(Theta(n)). However, existing techniques are far from showing this: for all three measures, the best lower bound for depth three circuits is exp(Omega(n^{2/5})). Moreover, prior methods exclusively study block-composed functions. Such methods appear intrinsically unable to prove lower bounds better than exp(Omega(sqrt{n})) even for depth four circuits, and have yet to prove lower bounds better than exp(Omega(sqrt{n})) for circuits of any constant depth.
We take a step toward showing that all of these complexity measures indeed exhibit a chasm at depth three. Specifically, for any arbitrarily small constant delta > 0, we exhibit a depth three circuit of polynomial size (in fact, an O(log n)-decision list) of complexity exp(Omega(n^{1/2-delta})) under each of these measures.
Our methods go beyond the block-composed functions studied in prior work, and hence may not be subject to the same barriers. Accordingly, we suggest natural candidate functions that may exhibit stronger bounds.
approximate degree
communication complexity
learning theory
polynomial approximation
threshold circuits
Theory of computation
35:1-35:18
Regular Paper
https://eccc.weizmann.ac.il/report/2016/121/
Mark
Bun
Mark Bun
Princeton University, Princeton, NJ, USA
This work was done while the author was at Harvard University and visiting Yale University, supported by an NDSEG Fellowship and NSF grant CNS-1237235.
Justin
Thaler
Justin Thaler
Georgetown University, Washington, DC, USA
Parts of this work were performed while the author was at Yahoo Research.
10.4230/LIPIcs.APPROX-RANDOM.2018.35
Scott Aaronson and Yaoyun Shi. Quantum lower bounds for the collision and the element distinctness problems. J. ACM, 51(4):595-605, 2004. URL: http://dx.doi.org/10.1145/1008731.1008735.
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http://dx.doi.org/10.1145/800057.808710
Mark Bun and Justin Thaler
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Speeding up Switch Markov Chains for Sampling Bipartite Graphs with Given Degree Sequence
We consider the well-studied problem of uniformly sampling (bipartite) graphs with a given degree sequence, or equivalently, the uniform sampling of binary matrices with fixed row and column sums. In particular, we focus on Markov Chain Monte Carlo (MCMC) approaches, which proceed by making small changes that preserve the degree sequence to a given graph. Such Markov chains converge to the uniform distribution, but the challenge is to show that they do so quickly, i.e., that they are rapidly mixing.
The standard example of this Markov chain approach for sampling bipartite graphs is the switch algorithm, that proceeds by locally switching two edges while preserving the degree sequence. The Curveball algorithm is a variation on this approach in which essentially multiple switches (trades) are performed simultaneously, with the goal of speeding up switch-based algorithms. Even though the Curveball algorithm is expected to mix faster than switch-based algorithms for many degree sequences, nothing is currently known about its mixing time. On the other hand, the switch algorithm has been proven to be rapidly mixing for several classes of degree sequences.
In this work we present the first results regarding the mixing time of the Curveball algorithm. We give a theoretical comparison between the switch and Curveball algorithms in terms of their underlying Markov chains. As our main result, we show that the Curveball chain is rapidly mixing whenever a switch-based chain is rapidly mixing. We do this using a novel state space graph decomposition of the switch chain into Johnson graphs. This decomposition is of independent interest.
Binary matrix
graph sampling
Curveball
switch
Markov chain decomposition
Johnson graph
Theory of computation~Random walks and Markov chains
36:1-36:18
Regular Paper
This work is supported by NWO Gravitation Project NETWORKS, Grant Number 024.002.003.
https://arxiv.org/abs/1709.07290
Corrie Jacobien
Carstens
Corrie Jacobien Carstens
Korteweg-de Vries Institute for Mathematics, Amsterdam, The Netherlands
Pieter
Kleer
Pieter Kleer
Centrum Wiskunde & Informatica (CWI), Amsterdam, The Netherlands
10.4230/LIPIcs.APPROX-RANDOM.2018.36
Annabell Berger and Matthias Müller-Hannemann. Uniform sampling of digraphs with a fixed degree sequence. In Graph Theoretic Concepts in Computer Science: 36th International Workshop, WG 2010, Zarós, Crete, Greece, June 28-30, 2010 Revised Papers, pages 220-231, 2010.
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http://arxiv.org/abs/1609.05137
Corrie Jacobien Carstens and Pieter Kleer. Comparing the switch and curveball Markov chains for sampling binary matrices with fixed marginals. CoRR, abs/1709.07290, 2017. URL: http://arxiv.org/abs/1709.07290.
http://arxiv.org/abs/1709.07290
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Corrie Jacobien Carstens and Pieter Kleer
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Randomness Extraction in AC0 and with Small Locality
Randomness extractors, which extract high quality (almost-uniform) random bits from biased random sources, are important objects both in theory and in practice. While there have been significant progress in obtaining near optimal constructions of randomness extractors in various settings, the computational complexity of randomness extractors is still much less studied. In particular, it is not clear whether randomness extractors with good parameters can be computed in several interesting complexity classes that are much weaker than P.
In this paper we study randomness extractors in the following two models of computation: (1) constant-depth circuits (AC^0), and (2) the local computation model. Previous work in these models, such as [Viola, 2005], [Goldreich et al., 2015] and [Bogdanov and Guo, 2013], only achieve constructions with weak parameters. In this work we give explicit constructions of randomness extractors with much better parameters. Our results on AC^0 extractors refute a conjecture in [Goldreich et al., 2015] and answer several open problems there. We also provide a lower bound on the error of extractors in AC^0, which together with the entropy lower bound in [Viola, 2005; Goldreich et al., 2015] almost completely characterizes extractors in this class. Our results on local extractors also significantly improve the seed length in [Bogdanov and Guo, 2013]. As an application, we use our AC^0 extractors to study pseudorandom generators in AC^0, and show that we can construct both cryptographic pseudorandom generators (under reasonable computational assumptions) and unconditional pseudorandom generators for space bounded computation with very good parameters.
Our constructions combine several previous techniques in randomness extractors, as well as introduce new techniques to reduce or preserve the complexity of extractors, which may be of independent interest. These include (1) a general way to reduce the error of strong seeded extractors while preserving the AC^0 property and small locality, and (2) a seeded randomness condenser with small locality.
Randomness Extraction
AC0
Locality
Pseudorandom Generator
Theory of computation~Expander graphs and randomness extractors
Theory of computation~Pseudorandomness and derandomization
37:1-37:20
Regular Paper
https://arxiv.org/abs/1602.01530
Kuan
Cheng
Kuan Cheng
Department of Computer Science, Johns Hopkins University.
Supported in part by NSF award CCF-1617713.
Xin
Li
Xin Li
Department of Computer Science, Johns Hopkins University.
Supported in part by NSF award CCF-1617713.
10.4230/LIPIcs.APPROX-RANDOM.2018.37
M. Ajtai and N. Linial. The influence of large coalitions. Combinatorica, 13(2):129-145, 1993.
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Kuan Cheng and Xin Li
Creative Commons Attribution 3.0 Unported license
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Boolean Function Analysis on High-Dimensional Expanders
We initiate the study of Boolean function analysis on high-dimensional expanders. We describe an analog of the Fourier expansion and of the Fourier levels on simplicial complexes, and generalize the FKN theorem to high-dimensional expanders.
Our results demonstrate that a high-dimensional expanding complex X can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing |X(k)|=O(n) points in comparison to binom{n}{k+1} points in the (k+1)-slice (which consists of all n-bit strings with exactly k+1 ones).
high dimensional expanders
Boolean function analysis
sparse model
Theory of computation~Randomness, geometry and discrete structures
38:1-38:20
Regular Paper
A full version of the paper appears in the http://arxiv.org/abs/1804.08155} [Yotam Dikstein et al., 2018].
Yotam
Dikstein
Yotam Dikstein
Weizmann Institute of Science, ISRAEL
The research of the first and second authors was supported in part by Irit Dinur’s ERC-CoG grant 772839.
Irit
Dinur
Irit Dinur
Weizmann Institute of Science, ISRAEL
Yuval
Filmus
Yuval Filmus
Technion - Israel Institute of Technology, ISRAEL
Taub Fellow - supported by the Taub Foundations. The research was funded by ISF grant 1337/16.
Prahladh
Harsha
Prahladh Harsha
Tata Institute of Fundamental Research, INDIA
Research supported in part by UGC - ISF grant. Part of the work was done when the author was visiting the Weizmann Institute of Science.
10.4230/LIPIcs.APPROX-RANDOM.2018.38
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Yotam Dikstein, Irit Dinur, Yuval Filmus, and Prahladh Harsha. Boolean function analysis on high-dimensional expanders. (manuscript), 2018. URL: http://arxiv.org/abs/1804.08155.
http://arxiv.org/abs/1804.08155
Irit Dinur, Yuval Filmus, and Prahladh Harsha. Low degree almost Boolean functions are sparse juntas. (manuscript), 2017. URL: http://arxiv.org/abs/1711.09428.
http://arxiv.org/abs/1711.09428
Irit Dinur and Tali Kaufman. High dimensional expanders imply agreement expanders. In Proc. 58th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 974-985, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.94.
http://dx.doi.org/10.1109/FOCS.2017.94
Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. On non-optimally expanding sets in Grassmann graphs. In Proc. 50th ACM Symp. on Theory of Computing (STOC), 2018. (To appear).
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David Ellis, Yuval Filmus, and Ehud Friedgut. A stability result for balanced dictatorships in S_n. Random Structures Algorithms, 46(3):494-530, 2015. URL: http://dx.doi.org/10.1002/rsa.20515.
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Yuval Filmus. Friedgut-Kalai-Naor theorem for slices of the Boolean cube. Chic. J. Theoret. Comput. Sci., 2016(14), 2016. URL: http://dx.doi.org/10.4086/cjtcs.2016.014.
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Yuval Filmus. An orthogonal basis for functions over a slice of the Boolean hypercube. Electron. J. Combin., 23(1):P1.23, 2016. URL: http://arxiv.org/abs/1406.0142.
http://arxiv.org/abs/1406.0142
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Yotam Dikstein, Irit Dinur, Yuval Filmus, and Prahladh Harsha
Creative Commons Attribution 3.0 Unported license
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Percolation of Lipschitz Surface and Tight Bounds on the Spread of Information Among Mobile Agents
We consider the problem of spread of information among mobile agents on the torus. The agents are initially distributed as a Poisson point process on the torus, and move as independent simple random walks. Two agents can share information whenever they are at the same vertex of the torus. We study the so-called flooding time: the amount of time it takes for information to be known by all agents. We establish a tight upper bound on the flooding time, and introduce a technique which we believe can be applicable to analyze other processes involving mobile agents.
Lipschitz surface
spread of information
flooding time
moving agents
Mathematics of computing~Probability and statistics
39:1-39:17
Regular Paper
https://arxiv.org/abs/1702.08748
Peter
Gracar
Peter Gracar
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Köln, Germany
Alexandre
Stauffer
Alexandre Stauffer
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom
Supported by a Marie Curie Career Integration Grant PCIG13-GA-2013-618588 DSRELIS, and an EPSRC Early Career Fellowship.
10.4230/LIPIcs.APPROX-RANDOM.2018.39
Martin T. Barlow. Random walks on supercritical percolation clusters. Annals of Probability, 32(4):3024-3084, 2004. URL: http://dx.doi.org/10.1214/009117904000000748.
http://dx.doi.org/10.1214/009117904000000748
Martin T. Barlow and Ben M. Hambly. Parabolic Harnack inequality and local limit theorem for percolation clusters. Electronic Journal of Probability, 14:1-26, 2009. URL: http://dx.doi.org/10.1214/EJP.v14-587.
http://dx.doi.org/10.1214/EJP.v14-587
Elisabetta Candellero and Augusto Teixeira. Percolation and isoperimetry on transitive graphs. arXiv, 2015. URL: http://arxiv.org/abs/1507.07765.
http://arxiv.org/abs/1507.07765
Andrea Clementi, Angelo Monti, Francesco Pasquale, and Riccardo Silvestri. Information Spreading in Stationary Markovian Evolving Graphs. IEEE Transactions on Parallel and Distributed Systems, 22(9):1425-1432, 2011. URL: http://dx.doi.org/10.1109/TPDS.2011.33.
http://dx.doi.org/10.1109/TPDS.2011.33
Andrea E. F. Clementi, Francesco Pasquale, and Riccardo Silvestri. MANETS: High mobility can make up for low transmission power. IEEE/ACM Transactions on Networking, 21(2):610-620, 2009. URL: http://dx.doi.org/10.1109/TNET.2012.2204407.
http://dx.doi.org/10.1109/TNET.2012.2204407
Peter Gracar and Alexandre Stauffer. Multi-scale Lipschitz percolation of increasing events for Poisson random walks. arXiv, 2017. URL: http://arxiv.org/abs/1702.08748.
http://arxiv.org/abs/1702.08748
Peter Gracar and Alexandre Stauffer. Random walks in random conductances: decoupling and spread of infection. arXiv, 2017. URL: http://arxiv.org/abs/1701.08021.
http://arxiv.org/abs/1701.08021
Harry Kesten and Vladas Sidoravicius. The spread of a rumor or infection in a moving population. Annals of Probability, 33(6):2402-2462, 2005. URL: http://dx.doi.org/10.1214/009117905000000413.
http://dx.doi.org/10.1214/009117905000000413
Henry Lam, Zhenming Liu, Michael Mitzenmacher, Xiaorui Sun, and Yajun Wang. Information Dissemination via Random Walks in d-Dimensional Space. arXiv, 2011. URL: http://arxiv.org/abs/1104.5268.
http://arxiv.org/abs/1104.5268
T. M. Liggett, R. H. Schonmann, and A. M. Stacey. Domination by product measures. Ann. Probab., 25(1):71-95, 01 1997. URL: http://dx.doi.org/10.1214/aop/1024404279.
http://dx.doi.org/10.1214/aop/1024404279
Yuval Peres, Alistair Sinclair, Perla Sousi, and Alexandre Stauffer. Mobile geometric graphs: Detection, coverage and percolation. Probability Theory and Related Fields, 156(1-2):273-305, 2013. URL: http://dx.doi.org/10.1007/s00440-012-0428-1.
http://dx.doi.org/10.1007/s00440-012-0428-1
Alberto Pettarin, Andrea Pietracaprina, Geppino Pucci, and Eli Upfal. Tight bounds on information dissemination in sparse mobile networks. In Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, PODC '11, pages 355-362, New York, NY, USA, 2011. ACM. URL: http://dx.doi.org/10.1145/1993806.1993882.
http://dx.doi.org/10.1145/1993806.1993882
Vladas Sidoravicius and Alain-Sol Sznitman. Percolation for the vacant set of random interlacements. Communications on Pure and Applied Mathematics, 62(6):831-858, 2009. URL: http://dx.doi.org/10.1002/cpa.20267.
http://dx.doi.org/10.1002/cpa.20267
Alexandre Stauffer. Space-time percolation and detection by mobile nodes. Annals of Applied Probability, 25(5):2416-2461, 2015. URL: http://dx.doi.org/10.1214/14-AAP1052.
http://dx.doi.org/10.1214/14-AAP1052
Alain-Sol Sznitman. Decoupling inequalities and interlacement percolation on G×ℤ. Inventiones mathematicae, 187(3):645-706, 2012. URL: http://dx.doi.org/10.1007/s00222-011-0340-9.
http://dx.doi.org/10.1007/s00222-011-0340-9
Peter Gracar and Alexandre Stauffer
Creative Commons Attribution 3.0 Unported license
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Flipping out with Many Flips: Hardness of Testing k-Monotonicity
A function f:{0,1}^n - > {0,1} is said to be k-monotone if it flips between 0 and 1 at most k times on every ascending chain. Such functions represent a natural generalization of (1-)monotone functions, and have been recently studied in circuit complexity, PAC learning, and cryptography. Our work is part of a renewed focus in understanding testability of properties characterized by freeness of arbitrary order patterns as a generalization of monotonicity. Recently, Canonne et al. (ITCS 2017) initiate the study of k-monotone functions in the area of property testing, and Newman et al. (SODA 2017) study testability of families characterized by freeness from order patterns on real-valued functions over the line [n] domain.
We study k-monotone functions in the more relaxed parametrized property testing model, introduced by Parnas et al. (JCSS, 72(6), 2006). In this process we resolve a problem left open in previous work. Specifically, our results include the following.
1) Testing 2-monotonicity on the hypercube non-adaptively with one-sided error requires an exponential in sqrt{n} number of queries. This behavior shows a stark contrast with testing (1-)monotonicity, which only needs O~(sqrt{n}) queries (Khot et al. (FOCS 2015)). Furthermore, even the apparently easier task of distinguishing 2-monotone functions from functions that are far from being n^{.01}-monotone also requires an exponential number of queries.
2) On the hypercube [n]^d domain, there exists a testing algorithm that makes a constant number of queries and distinguishes functions that are k-monotone from functions that are far from being O(kd^2) -monotone. Such a dependency is likely necessary, given the lower bound above for the hypercube.
Property Testing
Boolean Functions
k-Monotonicity
Lower Bounds
Theory of computation~Sketching and sampling
Theory of computation~Lower bounds and information complexity
40:1-40:17
Regular Paper
Elena
Grigorescu
Elena Grigorescu
Purdue University, West Lafayette, IN, USA, https://www.cs.purdue.edu/homes/egrigore/
Supported by NSF CCF-1649515
Akash
Kumar
Akash Kumar
Purdue University, West Lafayette, IN, USA, https://www.cs.purdue.edu/homes/akumar
Supported by NSF CCF-1649515 and NSF CCF-1618918
Karl
Wimmer
Karl Wimmer
Duquesne University, Pittsburgh, PA, USA, http://www.mathcs.duq.edu/~wimmer/
10.4230/LIPIcs.APPROX-RANDOM.2018.40
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Elena Grigorescu, Akash Kumar, and Karl Wimmer
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How Long Can Optimal Locally Repairable Codes Be?
A locally repairable code (LRC) with locality r allows for the recovery of any erased codeword symbol using only r other codeword symbols. A Singleton-type bound dictates the best possible trade-off between the dimension and distance of LRCs - an LRC attaining this trade-off is deemed optimal. Such optimal LRCs have been constructed over alphabets growing linearly in the block length. Unlike the classical Singleton bound, however, it was not known if such a linear growth in the alphabet size is necessary, or for that matter even if the alphabet needs to grow at all with the block length. Indeed, for small code distances 3,4, arbitrarily long optimal LRCs were known over fixed alphabets.
Here, we prove that for distances d >=slant 5, the code length n of an optimal LRC over an alphabet of size q must be at most roughly O(d q^3). For the case d=5, our upper bound is O(q^2). We complement these bounds by showing the existence of optimal LRCs of length Omega_{d,r}(q^{1+1/floor[(d-3)/2]}) when d <=slant r+2. Our bounds match when d=5, pinning down n=Theta(q^2) as the asymptotically largest length of an optimal LRC for this case.
Locally Repairable Code
Singleton Bound
Theory of computation~Error-correcting codes
41:1-41:11
Regular Paper
https://arxiv.org/abs/1807.01064
Venkatesan
Guruswami
Venkatesan Guruswami
Computer Science Department, Carnegie Mellon University, Pittsburgh, USA.
https://orcid.org/0000-0001-7926-3396
This work was done when the author was visiting the School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, and the Center of Mathematical Sciences and Applications, Harvard University. Research supported in part by NSF CCF-1563742.
Chaoping
Xing
Chaoping Xing
School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore.
https://orcid.org/0000-0002-1257-1033
Chen
Yuan
Chen Yuan
Centrum Wiskunde & Informatica, Amsterdam, Netherlands.
https://orcid.org/0000-0002-3730-8397
Most of this work was done when the author was with the School of Physical and Mathematical Science, Nanyang Technological University, Singapore. Research supported in part by ERC H2020 grant No.74079 (ALGSTRONGCRYPTO).
10.4230/LIPIcs.APPROX-RANDOM.2018.41
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I. Tamo and A. Barg. A family of optimal locally recoverable codes. IEEE Trans. Inform.Theory, 60:4661-4676, 2014.
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Z. Zhang, J. Xu, and M. Liu. Constructions of optimal locally repairable codes over small fields. SCIENTIA SINICA Mathematica, 47(11):1607-1614, 2017.
Venkatesan Guruswami, Chaoping Xing, and Chen Yuan
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On Minrank and Forbidden Subgraphs
The minrank over a field F of a graph G on the vertex set {1,2,...,n} is the minimum possible rank of a matrix M in F^{n x n} such that M_{i,i} != 0 for every i, and M_{i,j}=0 for every distinct non-adjacent vertices i and j in G. For an integer n, a graph H, and a field F, let g(n,H,F) denote the maximum possible minrank over F of an n-vertex graph whose complement contains no copy of H. In this paper we study this quantity for various graphs H and fields F. For finite fields, we prove by a probabilistic argument a general lower bound on g(n,H,F), which yields a nearly tight bound of Omega(sqrt{n}/log n) for the triangle H=K_3. For the real field, we prove by an explicit construction that for every non-bipartite graph H, g(n,H,R) >= n^delta for some delta = delta(H)>0. As a by-product of this construction, we disprove a conjecture of Codenotti, Pudlák, and Resta. The results are motivated by questions in information theory, circuit complexity, and geometry.
Minrank
Forbidden subgraphs
Shannon capacity
Circuit Complexity
Mathematics of computing~Information theory
42:1-42:14
Regular Paper
Ishay
Haviv
Ishay Haviv
School of Computer Science, The Academic College of Tel Aviv-Yaffo, Tel Aviv 61083, Israel
10.4230/LIPIcs.APPROX-RANDOM.2018.42
Miklós Ajtai, János Komlós, and Endre Szemerédi. A note on Ramsey numbers. J. Comb. Theory, Ser. A, 29(3):354-360, 1980.
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Ishay Haviv
Creative Commons Attribution 3.0 Unported license
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Preserving Randomness for Adaptive Algorithms
Suppose Est is a randomized estimation algorithm that uses n random bits and outputs values in R^d. We show how to execute Est on k adaptively chosen inputs using only n + O(k log(d + 1)) random bits instead of the trivial nk (at the cost of mild increases in the error and failure probability). Our algorithm combines a variant of the INW pseudorandom generator [Impagliazzo et al., 1994] with a new scheme for shifting and rounding the outputs of Est. We prove that modifying the outputs of Est is necessary in this setting, and furthermore, our algorithm's randomness complexity is near-optimal in the case d <= O(1). As an application, we give a randomness-efficient version of the Goldreich-Levin algorithm; our algorithm finds all Fourier coefficients with absolute value at least theta of a function F: {0, 1}^n -> {-1, 1} using O(n log n) * poly(1/theta) queries to F and O(n) random bits (independent of theta), improving previous work by Bshouty et al. [Bshouty et al., 2004].
pseudorandomness
adaptivity
estimation
Theory of computation~Pseudorandomness and derandomization
43:1-43:19
Regular Paper
https://arxiv.org/abs/1611.00783
William M.
Hoza
William M. Hoza
Department of Computer Science, University of Texas at Austin, Austin, TX, USA
https://orcid.org/0000-0001-5162-9181
Supported by the NSF GRFP under Grant DGE-1610403 and by a Harrington Fellowship from the University of Texas at Austin.
Adam R.
Klivans
Adam R. Klivans
Department of Computer Science, University of Texas at Austin, Austin, TX, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.43
R. Armoni. On the derandomization of space-bounded computations. In Randomization and Approximation Techniques in Computer Science, pages 47-59. Springer, 1998.
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http://dx.doi.org/10.1002/(SICI)1098-2418(199712)11:4<315::AID-RSA3>3.0.CO;2-1
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S. P. Vadhan. Pseudorandomness, volume 56. Now, 2012.
William M. Hoza and Adam R. Klivans
Creative Commons Attribution 3.0 Unported license
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Commutative Algorithms Approximate the LLL-distribution
Following the groundbreaking Moser-Tardos algorithm for the Lovász Local Lemma (LLL), a series of works have exploited a key ingredient of the original analysis, the witness tree lemma, in order to: derive deterministic, parallel and distributed algorithms for the LLL, to estimate the entropy of the output distribution, to partially avoid bad events, to deal with super-polynomially many bad events, and even to devise new algorithmic frameworks. Meanwhile, a parallel line of work has established tools for analyzing stochastic local search algorithms motivated by the LLL that do not fall within the Moser-Tardos framework. Unfortunately, the aforementioned results do not transfer to these more general settings. Mainly, this is because the witness tree lemma, provably, does not longer hold. Here we prove that for commutative algorithms, a class recently introduced by Kolmogorov and which captures the vast majority of LLL applications, the witness tree lemma does hold. Armed with this fact, we extend the main result of Haeupler, Saha, and Srinivasan to commutative algorithms, establishing that the output of such algorithms well-approximates the LLL-distribution, i.e., the distribution obtained by conditioning on all bad events being avoided, and give several new applications. For example, we show that the recent algorithm of Molloy for list coloring number of sparse, triangle-free graphs can output exponential many list colorings of the input graph.
Lovasz Local Lemma
Local Search
Commutativity
LLL-distribution
Coloring Triangle-free Graphs
Mathematics of computing~Probabilistic algorithms
44:1-44:20
Regular Paper
https://arxiv.org/abs/1704.02796
Fotis
Iliopoulos
Fotis Iliopoulos
University of California Berkeley, USA
https://orcid.org/0000-0002-1825-0097
Research supported by NSF grant CCF-1514434 and the Onassis Foundation
10.4230/LIPIcs.APPROX-RANDOM.2018.44
Dimitris Achlioptas and Fotis Iliopoulos. Focused stochastic local search and the Lovász local lemma. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 2024-2038, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch141.
http://dx.doi.org/10.1137/1.9781611974331.ch141
Dimitris Achlioptas and Fotis Iliopoulos. Random walks that find perfect objects and the Lovász local lemma. J. ACM, 63(3):22:1-22:29, 2016. URL: http://dx.doi.org/10.1145/2818352.
http://dx.doi.org/10.1145/2818352
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http://arxiv.org/abs/1805.02026
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http://dx.doi.org/10.1002/rsa.3240020403
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http://dx.doi.org/10.1002/rsa.3240020402
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http://dx.doi.org/10.1017/S0963548311000253
Karthekeyan Chandrasekaran, Navin Goyal, and Bernhard Haeupler. Deterministic algorithms for the Lovász local lemma. SIAM J. Comput., 42(6):2132-2155, 2013. URL: http://dx.doi.org/10.1137/100799642.
http://dx.doi.org/10.1137/100799642
Antares Chen, David G. Harris, and Aravind Srinivasan. Partial resampling to approximate covering integer programs. 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 1984-2003. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch139.
http://dx.doi.org/10.1137/1.9781611974331.ch139
Benny Chor and Oded Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput., 17(2):230-261, 1988. URL: http://dx.doi.org/10.1137/0217015.
http://dx.doi.org/10.1137/0217015
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http://dx.doi.org/10.1145/2611462.2611465
Artur Czumaj and Christian Scheideler. Coloring non-uniform hypergraphs: a new algorithmic approach to the general Lovász local lemma. In Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, CA, 2000), pages 30-39, 2000.
Paul Erdős and László Lovász. Problems and results on 3-chromatic hypergraphs and some related questions. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. II, pages 609-627. Colloq. Math. Soc. János Bolyai, Vol. 10. North-Holland, Amsterdam, 1975.
Bernhard Haeupler and David G Harris. Parallel algorithms and concentration bounds for the Lovász local lemma via witness-DAGs. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1170-1187. SIAM, 2017.
Bernhard Haeupler, Barna Saha, and Aravind Srinivasan. New constructive aspects of the Lovász local lemma. J. ACM, 58(6):Art. 28, 28, 2011. URL: http://dx.doi.org/10.1145/2049697.2049702.
http://dx.doi.org/10.1145/2049697.2049702
David G. Harris. New bounds for the Moser-Tardos distribution: Beyond the Lovasz local lemma. CoRR, abs/1610.09653, 2016. URL: http://arxiv.org/abs/1610.09653.
http://arxiv.org/abs/1610.09653
David G. Harris. Oblivious resampling oracles and parallel algorithms for the lopsided Lovász local lemma. CoRR, abs/1702.02547, 2017. URL: http://arxiv.org/abs/1702.02547.
http://arxiv.org/abs/1702.02547
David G. Harris and Aravind Srinivasan. The Moser-Tardos framework with partial resampling. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 469-478. IEEE Computer Society, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.57.
http://dx.doi.org/10.1109/FOCS.2013.57
David G. Harris and Aravind Srinivasan. A constructive algorithm for the Lovász local lemma on permutations. 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 907-925. SIAM, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.68.
http://dx.doi.org/10.1137/1.9781611973402.68
David G. Harris and Aravind Srinivasan. Algorithmic and enumerative aspects of the Moser-Tardos distribution. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 2004-2023, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch140.
http://dx.doi.org/10.1137/1.9781611974331.ch140
Nicholas J. A. Harvey and Jan Vondrák. An algorithmic proof of the Lovász local lemma via resampling oracles. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 1327-1346. IEEE Computer Society, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.85.
http://dx.doi.org/10.1109/FOCS.2015.85
Kashyap Babu Rao Kolipaka and Mario Szegedy. Moser and Tardos meet Lovász. In STOC, pages 235-244. ACM, 2011. URL: http://dx.doi.org/10.1145/1993636.1993669.
http://dx.doi.org/10.1145/1993636.1993669
Kashyap Babu Rao Kolipaka, Mario Szegedy, and Yixin Xu. A sharper local lemma with improved applications. In Anupam Gupta, Klaus Jansen, José D. P. Rolim, and Rocco A. Servedio, editors, 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, volume 7408 of Lecture Notes in Computer Science, pages 603-614. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-32512-0_51.
http://dx.doi.org/10.1007/978-3-642-32512-0_51
Vladimir Kolmogorov. Commutativity in the algorithmic Lovász local lemma. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 780-787. IEEE Computer Society, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.88.
http://dx.doi.org/10.1109/FOCS.2016.88
Linyuan Lu and Laszlo A Szekely. A new asymptotic enumeration technique: the Lovász local lemma. arXiv preprint arXiv:0905.3983, 2009.
Michael Molloy. The list chromatic number of graphs with small clique number. arXiv preprint arXiv:1701.09133, 2017.
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Robin A. Moser. A constructive proof of the Lovász local lemma. In STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing, pages 343-350. ACM, New York, 2009.
Robin A. Moser and Gábor Tardos. A constructive proof of the general Lovász local lemma. J. ACM, 57(2):Art. 11, 15, 2010. URL: http://dx.doi.org/10.1145/1667053.1667060.
http://dx.doi.org/10.1145/1667053.1667060
Christos H. Papadimitriou. On selecting a satisfying truth assignment. In FOCS, pages 163-169. IEEE Computer Society, 1991. URL: http://dx.doi.org/10.1109/SFCS.1991.185365.
http://dx.doi.org/10.1109/SFCS.1991.185365
Wesley Pegden. An extension of the Moser-Tardos algorithmic local lemma. SIAM J. Discrete Math., 28(2):911-917, 2014. URL: http://dx.doi.org/10.1137/110828290.
http://dx.doi.org/10.1137/110828290
J.B. Shearer. On a problem of Spencer. Combinatorica, 5(3):241-245, 1985. URL: http://dx.doi.org/10.1007/BF02579368.
http://dx.doi.org/10.1007/BF02579368
Aravind Srinivasan. Improved algorithmic versions of the Lovász local lemma. In Shang-Hua Teng, editor, SODA, pages 611-620. SIAM, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347150.
http://dl.acm.org/citation.cfm?id=1347082.1347150
Fotis Iliopoulos
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Cover Time of a Biased Random Walk on a Random Regular Graph of Odd Degree
We consider a random walk process, introduced by Orenshtein and Shinkar [Tal Orenshtein and Igor Shinkar, 2014], which prefers to visit previously unvisited edges, on the random r-regular graph G_r for any odd r >= 3. We show that this random walk process has asymptotic vertex and edge cover times 1/(r-2)n log n and r/(2(r-2))n log n, respectively, generalizing the result from [Cooper et al., to appear] from r = 3 to any larger odd r. This completes the study of the vertex cover time for fixed r >= 3, with [Petra Berenbrink et al., 2015] having previously shown that G_r has vertex cover time asymptotic to rn/2 when r >= 4 is even.
Random walk
random regular graph
cover time
Theory of computation~Random walks and Markov chains
45:1-45:14
Regular Paper
Tony
Johansson
Tony Johansson
Department of Mathematics, Uppsala University, Uppsala, Sweden
https://orcid.org/0000-0002-9264-3462
Supported in part by the Knut and Alice Wallenberg Foundation.
10.4230/LIPIcs.APPROX-RANDOM.2018.45
Miklós Ajtai, János Komlós, and Endre Szemerédi. Deterministic simulation in LOGSPACE. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 132-140. ACM, 1987. URL: http://dx.doi.org/10.1145/28395.28410.
http://dx.doi.org/10.1145/28395.28410
David Aldous and James Allen Fill. Reversible markov chains and random walks on graphs, 2002. Unfinished monograph, recompiled 2014, available at URL: http://www.stat.berkeley.edu/~aldous/RWG/book.html.
http://www.stat.berkeley.edu/~aldous/RWG/book.html
Petra Berenbrink, Colin Cooper, and Tom Friedetzky. Random walks which prefer unvisited edges: Exploring high girth even degree expanders in linear time. Random Struct. Algorithms, 46(1):36-54, 2015. URL: http://dx.doi.org/10.1002/rsa.20504.
http://dx.doi.org/10.1002/rsa.20504
Béla Bollobás. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Eur. J. Comb., 1(4):311-316, 1980. URL: http://dx.doi.org/10.1016/S0195-6698(80)80030-8.
http://dx.doi.org/10.1016/S0195-6698(80)80030-8
Colin Cooper and Alan M. Frieze. The cover time of random regular graphs. SIAM J. Discrete Math., 18(4):728-740, 2005. URL: http://dx.doi.org/10.1137/S0895480103428478.
http://dx.doi.org/10.1137/S0895480103428478
Colin Cooper and Alan M. Frieze. Vacant sets and vacant nets: Component structures induced by a random walk. SIAM J. Discrete Math., 30(1):166-205, 2016. URL: http://dx.doi.org/10.1137/14097937X.
http://dx.doi.org/10.1137/14097937X
Colin Cooper, Alan M. Frieze, and Tony Johansson. The cover time of a biased random walk on a random cubic graph. To appear in Proceedings of AofA 2018, preprint available at URL: https://arxiv.org/abs/1801.00760.
https://arxiv.org/abs/1801.00760
Joel Friedman. A proof of Alon’s second eigenvalue conjecture. In Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing, STOC '03, pages 720-724, New York, NY, USA, 2003. ACM. URL: http://dx.doi.org/10.1145/780542.780646.
http://dx.doi.org/10.1145/780542.780646
Alan M. Frieze and Michal Karonski. Introduction to Random Graphs. Cambridge University Press, Cambridge, UK, 2015. URL: http://dx.doi.org/10.1017/CBO9781316339831.011.
http://dx.doi.org/10.1017/CBO9781316339831.011
Tal Orenshtein and Igor Shinkar. Greedy random walk. Combinatorics, Probability & Computing, 23(2):269-289, 2014. URL: http://dx.doi.org/10.1017/S0963548313000552.
http://dx.doi.org/10.1017/S0963548313000552
Tony Johansson
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Satisfiability and Derandomization for Small Polynomial Threshold Circuits
A polynomial threshold function (PTF) is defined as the sign of a polynomial p : {0,1}^n ->R. A PTF circuit is a Boolean circuit whose gates are PTFs. We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth.
- Satisfiability (#SAT). We give the first zero-error randomized algorithm faster than exhaustive search that counts the number of satisfying assignments of a given constant-depth circuit with a super-linear number of wires whose gates are s-sparse PTFs, for s almost quadratic in the input size of the circuit; here a PTF is called s-sparse if its underlying polynomial has at most s monomials. More specifically, we show that, for any large enough constant c, given a depth-d circuit with (n^{2-1/c})-sparse PTF gates that has at most n^{1+epsilon_d} wires, where epsilon_d depends only on c and d, the number of satisfying assignments of the circuit can be computed in randomized time 2^{n-n^{epsilon_d}} with zero error. This generalizes the result by Chen, Santhanam and Srinivasan (CCC, 2016) who gave a SAT algorithm for constant-depth circuits of super-linear wire complexity with linear threshold function (LTF) gates only.
- Quantified derandomization. The quantified derandomization problem, introduced by Goldreich and Wigderson (STOC, 2014), asks to compute the majority value of a given Boolean circuit, under the promise that the minority-value inputs to the circuit are very few. We give a quantified derandomization algorithm for constant-depth PTF circuits with a super-linear number of wires that runs in quasi-polynomial time. More specifically, we show that for any sufficiently large constant c, there is an algorithm that, given a degree-Delta PTF circuit C of depth d with n^{1+1/c^d} wires such that C has at most 2^{n^{1-1/c}} minority-value inputs, runs in quasi-polynomial time exp ((log n)^{O (Delta^2)}) and determines the majority value of C. (We obtain a similar quantified derandomization result for PTF circuits with n^{Delta}-sparse PTF gates.) This extends the recent result of Tell (STOC, 2018) for constant-depth LTF circuits of super-linear wire complexity.
- Pseudorandom generators. We show how the classical Nisan-Wigderson (NW) generator (JCSS, 1994) yields a nontrivial pseudorandom generator for PTF circuits (of unrestricted depth) with sub-linearly many gates. As a corollary, we get a PRG for degree-Delta PTFs with the seed length exp (sqrt{Delta * log n})* log^2(1/epsilon).
constant-depth circuits
polynomial threshold functions
circuit analysis algorithms
SAT
derandomization
quantified derandomization
pseudorandom generators.
Theory of computation~Circuit complexity
46:1-46:19
Regular Paper
https://eccc.weizmann.ac.il/report/2018/115
Valentine
Kabanets
Valentine Kabanets
School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
Zhenjian
Lu
Zhenjian Lu
School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
10.4230/LIPIcs.APPROX-RANDOM.2018.46
Josh Alman, Timothy M. Chan, and R. Ryan Williams. Polynomial representations of threshold functions and algorithmic applications. In FOCS, pages 467-476, 2016.
Martin Anthony. Discrete Mathematics of Neural Networks: Selected Topics. SIAM monographs on discrete mathematics and applications. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001. URL: http://dx.doi.org/10.1137/1.9780898718539.
http://dx.doi.org/10.1137/1.9780898718539
Timothy M. Chan and Ryan Williams. Deterministic APSP, orthogonal vectors, and more: Quickly derandomizing razborov-smolensky. In SODA, pages 1246-1255, 2016.
Ruiwen Chen and Rahul Santhanam. Improved algorithms for sparse MAX-SAT and MAX-k-CSP. In SAT, pages 33-45, 2015.
Ruiwen Chen, Rahul Santhanam, and Srikanth Srinivasan. Average-case lower bounds and satisfiability algorithms for small threshold circuits. In CCC, pages 1:1-1:35, 2016.
Shiteng Chen and Periklis A. Papakonstantinou. Depth-reduction for composites. In FOCS, pages 99-108, 2016.
Mikael Goldmann, Johan Håstad, and Alexander A. Razborov. Majority gates vs. general weighted threshold gates. Computational Complexity, 2:277-300, 1992.
Oded Goldreich and Avi Wigderson. On derandomizing algorithms that err extremely rarely. In STOC, pages 109-118, 2014.
Johan Håstad. Almost optimal lower bounds for small depth circuits. In S. Micali, editor, Randomness and Computation, pages 143-170, Greenwich, Connecticut, 1989. Advances in Computing Research, vol. 5, JAI Press.
Russell Impagliazzo, Ramamohan Paturi, and Stefan Schneider. A satisfiability algorithm for sparse depth two threshold circuits. In FOCS, pages 479-488, 2013.
Valentine Kabanets, Daniel M. Kane, and Zhenjian Lu. A polynomial restriction lemma with applications. In STOC, pages 615-628, 2017.
Daniel Kane and Sankeerth Rao. A PRG for Boolean PTF of degree 2 with seed length subpolynomial in ε and logarithmic in n. In CCC, 2018.
Daniel M. Kane. A structure theorem for poorly anticoncentrated gaussian chaoses and applications to the study of polynomial threshold functions. In FOCS, pages 91-100, 2012.
Shachar Lovett and Srikanth Srinivasan. Correlation bounds for poly-size AC⁰ circuits with n^1 - o(1) symmetric gates. In APPROX/RANDOM, pages 640-651, 2011.
Warren S. McCulloch and Walter Pitts. A logical calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics, 5(4):115-133, 1943.
Raghu Meka and David Zuckerman. Pseudorandom generators for polynomial threshold functions. SIAM J. Comput., 42(3):1275-1301, 2013.
Saburo Muroga, Iwao Toda, and Satoru Takasu. Theory of majority decision elements. Journal of the Franklin Institute, 271:376-418, 1961.
Noam Nisan. The communication complexity of threshold gates. In Proceedings of Combinatorics, Paul Erdős is Eighty, pages 301-315, 1994.
Noam Nisan and Avi Wigderson. Hardness vs randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994.
Takayuki Sakai, Kazuhisa Seto, Suguru Tamaki, and Junichi Teruyama. Bounded depth circuits with weighted symmetric gates: Satisfiability, lower bounds and compression. In MFCS, pages 82:1-82:16, 2016.
Suguru Tamaki. A satisfiability algorithm for depth two circuits with a sub-quadratic number of symmetric and threshold gates. Electronic Colloquium on Computational Complexity (ECCC), 23:100, 2016.
Roei Tell. Improved bounds for quantified derandomization of constant-depth circuits and polynomials. In CCC, pages 13:1-13:48, 2017.
Roei Tell. A note on the limitations of two black-box techniques in quantified derandomization. Electronic Colloquium on Computational Complexity (ECCC), 24:187, 2017.
Roei Tell. Quantified derandomization of linear threshold circuits. In STOC, 2018.
Salil P. Vadhan. Pseudorandomness. Foundations and Trends in Theoretical Computer Science, 7(1-3):1-336, 2012.
Ryan Williams. Improving exhaustive search implies superpolynomial lower bounds. In STOC, pages 231-240, 2010.
Ryan Williams. Non-uniform ACC circuit lower bounds. In CCC, pages 115-125, 2011.
Ryan Williams. New algorithms and lower bounds for circuits with linear threshold gates. In STOC, pages 194-202, 2014.
Valentine Kabanets and Zhenjian Lu
Creative Commons Attribution 3.0 Unported license
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High Order Random Walks: Beyond Spectral Gap
We study high order random walks on high dimensional expanders on simplicial complexes (i.e., hypergraphs). These walks walk from a k-face (i.e., a k-hyperedge) to a k-face if they are both contained in a k+1-face (i.e, a k+1 hyperedge). This naturally generalizes the random walks on graphs that walk from a vertex (0-face) to a vertex if they are both contained in an edge (1-face).
Recent works have studied the spectrum of high order walks operators and deduced fast mixing. However, the spectral gap of high order walks operators is inherently small, due to natural obstructions (called coboundaries) that do not happen for walks on expander graphs.
In this work we go beyond spectral gap, and relate the expansion of a function on k-faces (called k-cochain, for k=0, this is a function on vertices) to its structure.
We show a Decomposition Theorem: For every k-cochain defined on high dimensional expander, there exists a decomposition of the cochain into i-cochains such that the square norm of the k-cochain is a sum of the square norms of the i-chains and such that the more weight the k-cochain has on higher levels of the decomposition the better is its expansion, or equivalently, the better is its shrinkage by the high order random walk operator.
The following corollaries are implied by the Decomposition Theorem:
- We characterize highly expanding k-cochains as those whose mass is concentrated on the highest levels of the decomposition that we construct. For example, a function on edges (i.e. a 1-cochain) which is locally thin (i.e. it contains few edges through every vertex) is highly expanding, while a function on edges that contains all edges through a single vertex is not highly expanding.
- We get optimal mixing for high order random walks on Ramanujan complexes. Ramanujan complexes are recently discovered bounded degree high dimensional expanders. The optimality in their mixing that we prove here, enable us to get from them more efficient Two-Layer-Samplers than those presented by the previous work of Dinur and Kaufman.
High Dimensional Expanders
Simplicial Complexes
Random Walk
Theory of computation~Random walks and Markov chains
Mathematics of computing~Spectra of graphs
Mathematics of computing~Hypergraphs
47:1-47:17
Regular Paper
https://arxiv.org/abs/1707.02799
Tali
Kaufman
Tali Kaufman
Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
This work was partially funded by ERC grant no. 336283 and BSF grant no. 2012256
Izhar
Oppenheim
Izhar Oppenheim
Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Be'er-Sheva, Israel
10.4230/LIPIcs.APPROX-RANDOM.2018.47
W. Ballmann and J. Świątkowski. On L²-cohomology and property (T) for automorphism groups of polyhedral cell complexes. Geom. Funct. Anal., 7(4):615-645, 1997. URL: http://dx.doi.org/10.1007/s000390050022.
http://dx.doi.org/10.1007/s000390050022
Boaz Barak, Parikshit Gopalan, Johan Håstad, Raghu Meka, Prasad Raghavendra, and David Steurer. Making the long code shorter. SIAM J. Comput., 44(5):1287-1324, 2015.
Irit Dinur and Tali Kaufman. High dimensional expanders imply agreement expanders. Electronic Colloquium on Computational Complexity (ECCC), 2017.
Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Towards a proof of the 2-to-1 games conjecture? Electronic Colloquium on Computational Complexity (ECCC), 23:198, 2016. URL: http://eccc.hpi-web.de/report/2016/198.
http://eccc.hpi-web.de/report/2016/198
Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. On non-optimally expanding sets in grassmann graphs. Electronic Colloquium on Computational Complexity (ECCC), 24:94, 2017. URL: https://eccc.weizmann.ac.il/report/2017/094.
https://eccc.weizmann.ac.il/report/2017/094
Howard Garland. p-adic curvature and the cohomology of discrete subgroups of p-adic groups. Ann. of Math. (2), 97:375-423, 1973.
Anna Gundert and Uli Wagner. On Laplacians of random complexes. In Computational geometry (SCG'12), pages 151-160. ACM, New York, 2012. URL: http://dx.doi.org/10.1145/2261250.2261272.
http://dx.doi.org/10.1145/2261250.2261272
Tali Kaufman and David Mass. High dimensional combinatorial random walks and colorful expansion. In ITCS, 2017.
Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Explicit constructions of ramanujan complexes of type. Eur. J. Comb., 26(6):965-993, 2005. URL: http://dx.doi.org/10.1016/j.ejc.2004.06.007.
http://dx.doi.org/10.1016/j.ejc.2004.06.007
Izhar Oppenheim. Vanishing of cohomology and property (T) for groups acting on weighted simplicial complexes. Groups Geom. Dyn., 9(1):67-101, 2015. URL: http://dx.doi.org/10.4171/GGD/306.
http://dx.doi.org/10.4171/GGD/306
Izhar Oppenheim. Local spectral expansion approach to high dimensional expanders Part I: Descent of spectral gaps. Discrete Comput. Geom., 59(2):293-330, 2018. URL: http://dx.doi.org/10.1007/s00454-017-9948-x.
http://dx.doi.org/10.1007/s00454-017-9948-x
Tali Kaufman and Izhar Oppenheim
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Improved Composition Theorems for Functions and Relations
One of the central problems in complexity theory is to prove super-logarithmic depth bounds for circuits computing a problem in P, i.e., to prove that P is not contained in NC^1. As an approach for this question, Karchmer, Raz and Wigderson [Mauricio Karchmer et al., 1995] proposed a conjecture called the KRW conjecture, which if true, would imply that P is not cotained in NC^{1}.
Since proving this conjecture is currently considered an extremely difficult problem, previous works by Edmonds, Impagliazzo, Rudich and Sgall [Edmonds et al., 2001], Håstad and Wigderson [Johan Håstad and Avi Wigderson, 1990] and Gavinsky, Meir, Weinstein and Wigderson [Dmitry Gavinsky et al., 2014] considered weaker variants of the conjecture. In this work we significantly improve the parameters in these variants, achieving almost tight lower bounds.
circuit complexity
communication complexity
KRW conjecture
composition
Theory of computation~Communication complexity
Theory of computation~Circuit complexity
Theory of computation~Complexity classes
48:1-48:18
Regular Paper
Sajin
Koroth
Sajin Koroth
Department of Computer Science, University of Haifa, Haifa 3498838, Israel
https://orcid.org/0000-0002-7989-1963
Supported by the Israel Science Foundation (grant No. 1445/16)
Or
Meir
Or Meir
Department of Computer Science, University of Haifa, Haifa 3498838,Israel
https://orcid.org/0000-0001-5031-0750
Partially supported by the Israel Science Foundation (grant No. 1445/16)
10.4230/LIPIcs.APPROX-RANDOM.2018.48
J. Edmonds, R. Impagliazzo, S. Rudich, and J. Sgall. Communication complexity towards lower bounds on circuit depth. computational complexity, 10(3):210-246, Dec 2001. URL: http://dx.doi.org/10.1007/s00037-001-8195-x.
http://dx.doi.org/10.1007/s00037-001-8195-x
Dmitry Gavinsky, Or Meir, Omri Weinstein, and Avi Wigderson. Toward better formula lower bounds: an information complexity approach to the KRW composition conjecture. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 213-222, 2014.
Johan Håstad and Avi Wigderson. Composition of the universal relation. In Advances In Computational Complexity Theory, volume 13 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 119-134. DIMACS/AMS, 1990.
Mauricio Karchmer, Eyal Kushilevitz, and Noam Nisan. Fractional covers and communication complexity. SIAM J. Discrete Math., 8(1):76-92, 1995.
Mauricio Karchmer, Ran Raz, and Avi Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Computational Complexity, 5(3/4):191-204, 1995.
Mauricio Karchmer and Avi Wigderson. Monotone circuits for connectivity require super-logarithmic depth. SIAM J. Discrete Math., 3(2):255-265, 1990.
Ran Raz and Pierre McKenzie. Separation of the monotone NC hierarchy. In 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, pages 234-243, 1997.
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http://dx.doi.org/10.1109/CCC.1997.612320
Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (preliminary report). In STOC, pages 209-213, 1979.
Sajin Koroth and Or Meir
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Round Complexity Versus Randomness Complexity in Interactive Proofs
Consider an interactive proof system for some set S that has randomness complexity r(n) for instances of length n, and arbitrary round complexity. We show a public-coin interactive proof system for S of round complexity O(r(n)/log n). Furthermore, the randomness complexity is preserved up to a constant factor, and the resulting interactive proof system has perfect completeness.
Interactive Proofs
Theory of computation~Interactive proof systems
49:1-49:16
Regular Paper
https://eccc.weizmann.ac.il/report/2017/055/
Maya
Leshkowitz
Maya Leshkowitz
Weizmann Institute of Science, Rehovot, Israel
10.4230/LIPIcs.APPROX-RANDOM.2018.49
Laszlo Babai. Trading group theory for randomness. In ACM Symposium on the Theory of Computing, pages 421-429, 1985.
Laszlo Babai and Shlomo Moran. Arthur-merlin games: A randomized proof system and a hierarchy of complexity classes. Journal of Computer and System Science, 36:254-276, 1988.
Mihir Bellare, Oded Goldreich, and Shafi Goldwasser. Randomness in interactive proofs. Computational Complexity, 3:319-354, 1993.
Martin Fürer, Oded Goldreich, Yishay Mansour, Michael Sipser, and Stathis Zachos. On completeness and soundness in interactive proof systems. Advances in Computing Research, 5:429-442, 1989.
Oded Goldreich and Maya Leshkowitz. On emulating interactive proofs with public coins. Electronic Colloquium on Computational Complexity (ECCC), 23:66, 2016.
Oded Goldreich, Silvio Micali, and Avi Wigderson. Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. Journal of the ACM, 38(3):691-729, 1991. Preliminary version in 27th FOCS, 1986.
Oded Goldreich, Salil P. Vadhan, and Avi Wigderson. On interactive proofs with a laconic prover. Computational Complexity, 11(1-2):1-53, 2002.
Shafi Goldwasser, Silvio Micali, and Charles Rackoff. The knowledge complexity of interactive proof systems. SIAM Journal on Computing, 18(1):186-208, 1989. Preliminary version in 17th STOC, 1985. Earlier versions date to 1982.
Shafi Goldwasser and Michael Sipser. Private coins versus public coins in interactive proof systems. Advances in Computing Research, 5:73-90, 1989. Extended abstract in 18th STOC, 1986.
Adam R. Klivans and Dieter van Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing, 31(5):1501-1526, 2002.
Maya Leshkowitz. Round complexity versus randomness complexity in interactive proofs. Electronic Colloquium on Computational Complexity (ECCC), 24:55, 2017. Full version of the paper.
Carsten Lund, Lance Fortnow, Howard J. Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. Journal of the ACM, 39(4):859-868, 1992. Extended abstract in 31st FOCS, 1990.
Ronen Shaltiel and Christopher Umans. Low-end uniform hardness versus randomness tradeoffs for AM. SIAM Journal on Computing, 39(3):1006-1037, 2009.
Adi Shamir. IP = PSPACE. Journal of the ACM, 39(4):869-877, 1992. Preliminary version in 31st FOCS, 1990.
Maya Leshkowitz
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Improved List-Decodability of Random Linear Binary Codes
There has been a great deal of work establishing that random linear codes are as list-decodable as uniformly random codes, in the sense that a random linear binary code of rate 1 - H(p) - epsilon is (p,O(1/epsilon))-list-decodable with high probability. In this work, we show that such codes are (p, H(p)/epsilon + 2)-list-decodable with high probability, for any p in (0, 1/2) and epsilon > 0. In addition to improving the constant in known list-size bounds, our argument - which is quite simple - works simultaneously for all values of p, while previous works obtaining L = O(1/epsilon) patched together different arguments to cover different parameter regimes.
Our approach is to strengthen an existential argument of (Guruswami, Håstad, Sudan and Zuckerman, IEEE Trans. IT, 2002) to hold with high probability. To complement our upper bound for random linear binary codes, we also improve an argument of (Guruswami, Narayanan, IEEE Trans. IT, 2014) to obtain a tight lower bound of 1/epsilon on the list size of uniformly random binary codes; this implies that random linear binary codes are in fact more list-decodable than uniformly random binary codes, in the sense that the list sizes are strictly smaller.
To demonstrate the applicability of these techniques, we use them to (a) obtain more information about the distribution of list sizes of random linear binary codes and (b) to prove a similar result for random linear rank-metric codes.
List-decoding
Random linear codes
Rank-metric codes
Theory of computation~Error-correcting codes
50:1-50:19
Regular Paper
https://arxiv.org/abs/1801.07839
Ray
Li
Ray Li
Department of Computer Science, Stanford University, USA
Research supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE - 1656518.
Mary
Wootters
Mary Wootters
Departments of Computer Science and Electrical Engineering, Stanford University, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.50
Volodia M. Blinovsky. Bounds for codes in the case of list decoding of finite volume. Problems of Information Transmission, 22(1):7–19, 1986.
Mahdi Cheraghchi, Venkatesan Guruswami, and Ameya Velingker. Restricted isometry of Fourier matrices and list decodability of random linear codes. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 432-442. ACM-SIAM, 2013. URL: http://dx.doi.org/10.1137/1.9781611973105.31.
http://dx.doi.org/10.1137/1.9781611973105.31
Philippe Delsarte. Bilinear forms over a finite field, with applications to coding theory. Journal of Combinatorial Theory, Series A, 25(3):226-241, 1978.
Yang Ding. On list-decodability of random rank metric codes and subspace codes. IEEE Trans. Information Theory, 61(1):51-59, 2015. URL: http://dx.doi.org/10.1109/TIT.2014.2371915.
http://dx.doi.org/10.1109/TIT.2014.2371915
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Peter Elias. List decoding for noisy channels. Wescon Convention Record, Part 2, pages 94-104, 1957.
Peter Elias. Error-correcting codes for list decoding. IEEE Trans. Information Theory, 37(1):5-12, 1991.
Ernst M. Gabidulin, A. V. Paramonov, and O. V. Tretjakov. Ideals over a non-commutative ring and their applications in cryptology. In Proceedings of Advances in Cryptology - EUROCRYPT '91, Workshop on the Theory and Application of of Cryptographic Techniques, pages 482-489, 1991. URL: http://dx.doi.org/10.1007/3-540-46416-6_41.
http://dx.doi.org/10.1007/3-540-46416-6_41
Ernst Mukhamedovich Gabidulin. Theory of codes with maximum rank distance. Problemy Peredachi Informatsii, 21(1):3-16, 1985.
Maximilien Gadouleau and Zhiyuan Yan. On the decoder error probability of bounded rank-distance decoders for maximum rank-distance codes. IEEE Trans. Information Theory, 54(7):3202-3206, 2008. URL: http://dx.doi.org/10.1109/TIT.2008.924697.
http://dx.doi.org/10.1109/TIT.2008.924697
Venkatesan Guruswami. List decoding of binary codes-a brief survey of some recent results. In Proceedings of Coding and Cryptology, Second International Workshop (IWCC), pages 97-106, 2009. URL: http://dx.doi.org/10.1007/978-3-642-01877-0_10.
http://dx.doi.org/10.1007/978-3-642-01877-0_10
Venkatesan Guruswami, Johan Håstad, and Swastik Kopparty. On the list-decodability of random linear codes. IEEE Trans. Information Theory, 57(2):718-725, 2011. URL: http://dx.doi.org/10.1109/TIT.2010.2095170.
http://dx.doi.org/10.1109/TIT.2010.2095170
Venkatesan Guruswami, Johan Håstad, Madhu Sudan, and David Zuckerman. Combinatorial bounds for list decoding. IEEE Trans. Information Theory, 48(5):1021-1034, 2002. URL: http://dx.doi.org/10.1109/18.995539.
http://dx.doi.org/10.1109/18.995539
Venkatesan Guruswami and Piotr Indyk. Efficiently decodable codes meeting gilbert-varshamov bound for low rates. In Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, pages 756-757. Society for Industrial and Applied Mathematics, 2004.
Venkatesan Guruswami and Srivatsan Narayanan. Combinatorial limitations of average-radius list-decoding. IEEE Trans. Information Theory, 60(10):5827-5842, 2014. URL: http://dx.doi.org/10.1109/TIT.2014.2343224.
http://dx.doi.org/10.1109/TIT.2014.2343224
Venkatesan Guruswami and Nicolas Resch. On the list-decodability of random linear rank-metric codes. arXiv preprint arXiv:1710.11516, 2017.
Venkatesan Guruswami and Atri Rudra. Concatenated codes can achieve list-decoding capacity. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20-22, 2008, pages 258-267, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347111.
http://dl.acm.org/citation.cfm?id=1347082.1347111
Venkatesan Guruswami and Atri Rudra. Explicit codes achieving list decoding capacity: Error-correction with optimal redundancy. IEEE Trans. Information Theory, 54(1):135-150, 2008.
Venkatesan Guruswami and Salil Vadhan. A lower bound on list size for list decoding. In Chandra Chekuri, Klaus Jansen, José D. P. Rolim, and Luca Trevisan, editors, Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 318-329, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg.
Venkatesan Guruswami, Carol Wang, and Chaoping Xing. Explicit list-decodable rank-metric and subspace codes via subspace designs. IEEE Trans. Information Theory, 62(5):2707-2718, 2016.
Venkatesan Guruswami and Chaoping Xing. Folded codes from function field towers and improved optimal rate list decoding. In Proceedings of the Forty-Fourth annual ACM Symposium on Theory of Computing (STOC), pages 339-350. ACM, 2012.
Venkatesan Guruswami and Chaoping Xing. List decoding Reed-Solomon, Algebraic-Geometric, and Gabidulin subcodes up to the Singleton bound. In Proceedings of the Forty-Fifth annual ACM Symposium on Theory of Computing (STOC), pages 843-852. ACM, 2013.
Brett Hemenway, Noga Ron-Zewi, and Mary Wootters. Local list recovery of high-rate tensor codes &applications. In 58th Annual IEEE Symposium on Foundations of Computer Science, 2017.
Brett Hemenway and Mary Wootters. Linear-time list recovery of high-rate expander codes. In International Colloquium on Automata, Languages, and Programming, pages 701-712. Springer, 2015.
Ralf Koetter and Frank R Kschischang. Coding for errors and erasures in random network coding. IEEE Trans. Information Theory, 54(8):3579-3591, 2008.
Ray Li and Mary Wootters. Improve list-decodability of random linear binary code. CoRR, abs/1801.07839, 2018. URL: http://arxiv.org/abs/1801.07839.
http://arxiv.org/abs/1801.07839
Pierre Loidreau. Designing a rank metric based mceliece cryptosystem. In Proceedings of the Post-Quantum Cryptography, Third International Workshop on Post-Quantum Cryptography, (PQCrypto), pages 142-152, 2010. URL: http://dx.doi.org/10.1007/978-3-642-12929-2_11.
http://dx.doi.org/10.1007/978-3-642-12929-2_11
Pierre Loidreau. A new rank metric codes based encryption scheme. In Proceedings of the Post-Quantum Cryptography, 8th International Workshop on Post-Quantum Cryptography, (PQCrypto), pages 3-17, 2017. URL: http://dx.doi.org/10.1007/978-3-319-59879-6_1.
http://dx.doi.org/10.1007/978-3-319-59879-6_1
Hsiao-feng Lu and P. Vijay Kumar. A unified construction of space-time codes with optimal rate-diversity tradeoff. IEEE Trans. Information Theory, 51(5):1709-1730, 2005. URL: http://dx.doi.org/10.1109/TIT.2005.846403.
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P. Lusina, Ernst M. Gabidulin, and Martin Bossert. Maximum rank distance codes as space-time codes. IEEE Trans. Information Theory, 49(10):2757-2760, 2003. URL: http://dx.doi.org/10.1109/TIT.2003.818023.
http://dx.doi.org/10.1109/TIT.2003.818023
Ron M. Roth. Maximum-rank array codes and their application to crisscross error correction. IEEE Trans. Information Theory, 37(2):328-336, 1991. URL: http://dx.doi.org/10.1109/18.75248.
http://dx.doi.org/10.1109/18.75248
Atri Rudra. Limits to list decoding of random codes. IEEE Trans. Information Theory, 57(3):1398-1408, 2011. URL: http://dx.doi.org/10.1109/TIT.2010.2054750.
http://dx.doi.org/10.1109/TIT.2010.2054750
Atri Rudra and Steve Uurtamo. Two theorems on list decoding. In Maria Serna, Ronen Shaltiel, Klaus Jansen, and José Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 696-709, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg.
Atri Rudra and Mary Wootters. Every list-decodable code for high noise has abundant near-optimal rate puncturings. In Proceedings of the Forty-Sixth annual ACM Symposium on Theory of Computing (STOC), pages 764-773. ACM, 2014.
Atri Rudra and Mary Wootters. It'll probably work out: Improved list-decoding through random operations. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science (ITCS), pages 287-296. ACM, 2015. URL: http://dx.doi.org/10.1145/2688073.2688092.
http://dx.doi.org/10.1145/2688073.2688092
Atri Rudra and Mary Wootters. Average-radius list-recovery of random linear codes. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). ACM-SIAM, 2018.
Natalia Silberstein, Ankit Singh Rawat, O Ozan Koyluoglu, and Sriram Vishwanath. Optimal locally repairable codes via rank-metric codes. In Proceedings of the 2013 IEEE International Symposium on Information Theory (ISIT), pages 1819-1823. IEEE, 2013.
Natalia Silberstein, Ankit Singh Rawat, and Sriram Vishwanath. Error resilience in distributed storage via rank-metric codes. In Proceedings of the 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 1150-1157. IEEE, 2012.
Danilo Silva, Frank R Kschischang, and Ralf Koetter. A rank-metric approach to error control in random network coding. IEEE Trans. Information Theory, 54(9):3951-3967, 2008.
Madhu Sudan. List decoding: algorithms and applications. SIGACT News, 31(1):16-27, 2000. URL: http://dx.doi.org/10.1145/346048.346049.
http://dx.doi.org/10.1145/346048.346049
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http://dx.doi.org/10.1561/0400000010
Antonia Wachter-Zeh. Bounds on list decoding of rank-metric codes. IEEE Trans. Information Theory, 59(11):7268-7277, 2013.
Mary Wootters. On the list decodability of random linear codes with large error rates. In Proceedings of the Forty-Fifth Symposium on Theory of Computing Conference (STOC), pages 853-860, 2013. URL: http://dx.doi.org/10.1145/2488608.2488716.
http://dx.doi.org/10.1145/2488608.2488716
Jack Wozencraft. List decoding. Quarter Progress Report, 48:90-95, 1958.
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Ray Li and Mary Wootters
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Sunflowers and Quasi-Sunflowers from Randomness Extractors
The Erdös-Rado sunflower theorem (Journal of Lond. Math. Soc. 1960) is a fundamental result in combinatorics, and the corresponding sunflower conjecture is a central open problem. Motivated by applications in complexity theory, Rossman (FOCS 2010) extended the result to quasi-sunflowers, where similar conjectures emerge about the optimal parameters for which it holds.
In this work, we exhibit a surprising connection between the existence of sunflowers and quasi-sunflowers in large enough set systems, and the problem of constructing (or existing) certain randomness extractors. This allows us to re-derive the known results in a systematic manner, and to reduce the relevant conjectures to the problem of obtaining improved constructions of the randomness extractors.
Sunflower conjecture
Quasi-sunflowers
Randomness Extractors
Theory of computation~Randomness, geometry and discrete structures
51:1-51:13
Regular Paper
Xin
Li
Xin Li
Johns Hopkins University, Baltimore, USA
Research supported by NSF award CCF-1617713
Shachar
Lovett
Shachar Lovett
University of California San Diego, La Jolla, USA
Research supported by NSF CCF-1614023
Jiapeng
Zhang
Jiapeng Zhang
University of California San Diego, La Jolla, USA
Research supported by NSF CCF-1614023
10.4230/LIPIcs.APPROX-RANDOM.2018.51
Noga Alon, Amir Shpilka, and Christopher Umans. On sunflowers and matrix multiplication. Computational Complexity, pages 214-223, 2012.
Boaz Barak, Guy Kindler, Ronen Shaltiel, Benny Sudakov, and Avi Wigderson. Simulating independence: New constructions of condensers, ramsey graphs, dispersers, and extractors. In Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pages 1-10. ACM, 2005.
Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma. Explicit two-source extractors for near-logarithmic min-entropy. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing, 2017.
Eshan Chattopadhyay and David Zuckerman. Explicit two-source extractors and resilient functions. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing, 2016.
Benny Chor and Oded Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM Journal on Computing, 17(2):230-261, 1988.
Gil Cohen. Two-source extractors for quasi-logarithmic min-entropy and improved privacy amplification protocols. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing, 2017.
Ernie Croot, Vsevolod F. Lev, and Péter Pál Pach. Progression-free sets in z₄ⁿ are exponentially small. Annals of Mathematics, 185(1):331-337, 2017.
Paul Erdős. Some remarks on the theory of graphs. Bull. Amer. Math. Soc., 53(4):292-294, 1947.
Paul Erdős and R Rado. Intersection theorems for systems of sets. Journal of the London Mathematical Society, 35(1):85-90, 1960.
Paul Erdős and George Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463-470, 1935.
M. Goos, S. Lovett, R. Meka, T. Watson, and D. Zuckerman. Rectangles are nonnegative juntas. SIAM Journal on Computing, 45(5):1835-1869, 2016.
Parikshit Gopalan, Raghu Meka, and Omer Reingold. Dnf sparsification and a faster deterministic counting algorithm. computational complexity, 22(2):275-310, 2013.
Ben Green and Terence Tao. The primes contain arbitrarily long arithmetic progressions. Annals of Mathematics, 167(2):481-547, 2008.
Pravesh K. Kothari, Raghu Meka, and Prasad Raghavendra. Approximating rectangles by juntas and weakly-exponential lower bounds for lp relaxations of csps. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, 2017.
Xin Li. Three source extractors for polylogarithmic min-entropy. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science, 2015.
Xin Li. Improved non-malleable extractors, non-malleable codes and independent source extractors. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing, 2017.
Alexander Razborov. Some lower bounds for the monotone complexity of some boolean functions. Soviet Math. Dokl., 31:354-357, 1985.
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Jordan S. Ellenberg and Dion Gijswijt. On large subsets of f_qⁿ with no three-term arithmetic progression. Annals of Mathematics, 185(1):339-343, 2017.
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Xin Li, Shachar Lovett, and Jiapeng Zhang
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Torpid Mixing of Markov Chains for the Six-vertex Model on Z^2
In this paper, we study the mixing time of two widely used Markov chain algorithms for the six-vertex model, Glauber dynamics and the directed-loop algorithm, on the square lattice Z^2. We prove, for the first time that, on finite regions of the square lattice these Markov chains are torpidly mixing under parameter settings in the ferroelectric phase and the anti-ferroelectric phase.
the six-vertex model
Eulerian orientations
square lattice
torpid mixing
Theory of computation~Design and analysis of algorithms
Theory of computation~Random walks and Markov chains
52:1-52:15
Regular Paper
This work was supported by NSF CCF-1714275.
Tianyu
Liu
Tianyu Liu
University of Wisconsin-Madison, Madison, WI, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.52
David Allison and Nicolai Reshetikhin. Numerical study of the 6-vertex model with domain wall boundary conditions. Annales de l'Institut Fourier, 55(6):1847-1869, 2005. URL: http://eudml.org/doc/116236.
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http://dx.doi.org/10.1007/978-3-642-40328-6_27
Christian Borgs, Jennifer T. Chayes, Jeong Han Kim, Alan Frieze, Prasad Tetali, Eric Vigoda, and Van Ha Vu. Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS), pages 218-229, 1999. URL: http://dl.acm.org/citation.cfm?id=795665.796518.
http://dl.acm.org/citation.cfm?id=795665.796518
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http://arxiv.org/abs/1712.05880
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http://dx.doi.org/10.1103/PhysRevLett.19.108
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F. Martinelli and E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region. I. the attractive case. Communications in Mathematical Physics, 161(3):447-486, Apr 1994. URL: http://dx.doi.org/10.1007/BF02101929.
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http://dx.doi.org/10.1016/0301-0104(79)85201-5
Tianyu Liu
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On the Testability of Graph Partition Properties
In this work we study the testability of a family of graph partition properties that generalizes a family previously studied by Goldreich, Goldwasser, and Ron (Journal of the ACM, 1998 ). While the family studied by Goldreich, Goldwasser, and Ron includes a variety of natural properties, such as k-colorability and containing a large cut, it does not include other properties of interest, such as split graphs, and more generally (p,q)-colorable graphs. The generalization we consider allows us to impose constraints on the edge-densities within and between parts (relative to the sizes of the parts). We denote the family studied in this work by GPP.
We first show that all properties in GPP have a testing algorithm whose query complexity is polynomial in 1/epsilon, where epsilon is the given proximity parameter (and there is no dependence on the size of the graph). As the testing algorithm has two-sided error, we next address the question of which properties in GPP can be tested with one-sided error and query complexity polynomial in 1/epsilon. We answer this question by establishing a characterization result. Namely, we define a subfamily GPP_{0,1} of GPP and show that a property P in GPP is testable by a one-sided error algorithm that has query complexity poly(1/epsilon) if and only if P in GPP_{0,1}.
Graph Partition Properties
Theory of computation~Graph algorithms analysis
53:1-53:13
Regular Paper
Yonatan
Nakar
Yonatan Nakar
Tel Aviv University, Tel Aviv, Israel
This research was partially supported by the Israel Science Foundation grant No. 671/13.
Dana
Ron
Dana Ron
Tel Aviv University, Tel Aviv, Israel
10.4230/LIPIcs.APPROX-RANDOM.2018.53
Noga Alon. Testing subgraphs in large graphs. Random Structures &Algorithms, 21(3-4):359-370, 2002.
Noga Alon, Eldar Fischer, Michael Krivelevich, and Mario Szegedy. Efficient testing of large graphs. Combinatorica, 20(4):451-476, 2000.
Noga Alon, Eldar Fischer, Ilan Newman, and Asaf Shapira. A combinatorial characterization of the testable graph properties: it’s all about regularity. SIAM Journal on Computing, 39(1):143-167, 2009.
Noga Alon and Jacob Fox. Easily testable graph properties. Combinatorics, Probability and Computing, 24(4):646-657, 2015.
Noga Alon and Michael Krivelevich. Testing k-colorability. SIAM Journal on Discrete Mathematics, 15(2):211-227, 2002.
Noga Alon and Asaf Shapira. Testing subgraphs in directed graphs. Journal of Computer and System Sciences, 69(3):354-382, 2004.
Noga Alon and Asaf Shapira. A characterization of easily testable induced subgraphs. Combinatorics, Probability and Computing, 15(6):791-805, 2006.
Noga Alon and Asaf Shapira. A characterization of the (natural) graph properties testable with one-sided error. SIAM Journal on Computing, 37(6):1703-1727, 2008.
Christian Borgs, Jennifer Chayes, László Lovász, Vera T Sós, Balázs Szegedy, and Katalin Vesztergombi. Graph limits and parameter testing. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 261-270. ACM, 2006.
David Conlon and Jacob Fox. Graph removal lemmas. Surveys in combinatorics, 1(2):3, 2013.
Lior Gishboliner and Asaf Shapira. Removal lemmas with polynomial bounds. arXiv preprint arXiv:1611.10315, 2016.
Oded Goldreich, Shari Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. Journal of the ACM (JACM), 45(4):653-750, 1998.
Oded Goldreich and Luca Trevisan. Three theorems regarding testing graph properties. Random Structures &Algorithms, 23(1):23-57, 2003.
Yonatan Nakar and Dana Ron
Creative Commons Attribution 3.0 Unported license
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On Closeness to k-Wise Uniformity
A probability distribution over {-1, 1}^n is (epsilon, k)-wise uniform if, roughly, it is epsilon-close to the uniform distribution when restricted to any k coordinates. We consider the problem of how far an (epsilon, k)-wise uniform distribution can be from any globally k-wise uniform distribution. We show that every (epsilon, k)-wise uniform distribution is O(n^{k/2}epsilon)-close to a k-wise uniform distribution in total variation distance. In addition, we show that this bound is optimal for all even k: we find an (epsilon, k)-wise uniform distribution that is Omega(n^{k/2}epsilon)-far from any k-wise uniform distribution in total variation distance. For k=1, we get a better upper bound of O(epsilon), which is also optimal.
One application of our closeness result is to the sample complexity of testing whether a distribution is k-wise uniform or delta-far from k-wise uniform. We give an upper bound of O(n^{k}/delta^2) (or O(log n/delta^2) when k = 1) on the required samples. We show an improved upper bound of O~(n^{k/2}/delta^2) for the special case of testing fully uniform vs. delta-far from k-wise uniform. Finally, we complement this with a matching lower bound of Omega(n/delta^2) when k = 2.
Our results improve upon the best known bounds from [Alon et al., 2007], and have simpler proofs.
k-wise independence
property testing
Fourier analysis
Boolean function
Theory of computation~Design and analysis of algorithms
54:1-54:19
Regular Paper
Supported by NSF grants CCF-1618679, CCF-1717606. This material is based upon work supported by the National Science Foundation under grant numbers listed above. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF).
https://arxiv.org/abs/1806.03569
Ryan
O'Donnell
Ryan O'Donnell
Carnegie Mellon University, Pittsburgh, PA, USA
Yu
Zhao
Yu Zhao
Carnegie Mellon University, Pittsburgh, PA, USA
10.4230/LIPIcs.APPROX-RANDOM.2018.54
Jayadev Acharya, Constantinos Daskalakis, and Gautam Kamath. Optimal testing for properties of distributions. In Advances in Neural Information Processing Systems, pages 3591-3599, 2015.
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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, pages 496-505, 2007.
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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 Symposium on Foundations of Computer Science, pages 442-451, 2001. URL: http://dx.doi.org/10.1109/SFCS.2001.959920.
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http://dx.doi.org/10.1109/SFCS.2000.892113
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Mark Braverman. Polylogarithmic independence fools AC^0 circuits. Journal of the ACM, 57(5):28:1-28:10, 2010. URL: http://dx.doi.org/10.1145/1754399.1754401.
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Ning Xie. Testing k-wise independent distributions. PhD thesis, Massachusetts Institute of Technology, 2012.
Ryan O'Donnell and Yu Zhao
Creative Commons Attribution 3.0 Unported license
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Pseudo-Derandomizing Learning and Approximation
We continue the study of pseudo-deterministic algorithms initiated by Gat and Goldwasser [Eran Gat and Shafi Goldwasser, 2011]. A pseudo-deterministic algorithm is a probabilistic algorithm which produces a fixed output with high probability. We explore pseudo-determinism in the settings of learning and approximation. Our goal is to simulate known randomized algorithms in these settings by pseudo-deterministic algorithms in a generic fashion - a goal we succinctly term pseudo-derandomization. Learning. In the setting of learning with membership queries, we first show that randomized learning algorithms can be derandomized (resp. pseudo-derandomized) under the standard hardness assumption that E (resp. BPE) requires large Boolean circuits. Thus, despite the fact that learning is an algorithmic task that requires interaction with an oracle, standard hardness assumptions suffice to (pseudo-)derandomize it. We also unconditionally pseudo-derandomize any {quasi-polynomial} time learning algorithm for polynomial size circuits on infinitely many input lengths in sub-exponential time.
Next, we establish a generic connection between learning and derandomization in the reverse direction, by showing that deterministic (resp. pseudo-deterministic) learning algorithms for a concept class C imply hitting sets against C that are computable deterministically (resp. pseudo-deterministically). In particular, this suggests a new approach to constructing hitting set generators against AC^0[p] circuits by giving a deterministic learning algorithm for AC^0[p]. Approximation. Turning to approximation, we unconditionally pseudo-derandomize any poly-time randomized approximation scheme for integer-valued functions infinitely often in subexponential time over any samplable distribution on inputs. As a corollary, we get that the (0,1)-Permanent has a fully pseudo-deterministic approximation scheme running in sub-exponential time infinitely often over any samplable distribution on inputs.
Finally, we {investigate} the notion of approximate canonization of Boolean circuits. We use a connection between pseudodeterministic learning and approximate canonization to show that if BPE does not have sub-exponential size circuits infinitely often, then there is a pseudo-deterministic approximate canonizer for AC^0[p] computable in quasi-polynomial time.
derandomization
learning
approximation
boolean circuits
Theory of computation~Pseudorandomness and derandomization
55:1-55:19
Regular Paper
This work was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant No. 615075.
https://eccc.weizmann.ac.il/report/2018/122/
Igor
Carboni Oliveira
Igor Carboni Oliveira
Department of Computer Science, University of Oxford, United Kingdom.
Rahul
Santhanam
Rahul Santhanam
Department of Computer Science, University of Oxford, United Kingdom.
10.4230/LIPIcs.APPROX-RANDOM.2018.55
Boaz Barak, Oded Goldreich, Russell Impagliazzo, Steven Rudich, Amit Sahai, Salil P. Vadhan, and Ke Yang. On the (im)possibility of obfuscating programs. Journal of the ACM, 59(2):6:1-6:48, 2012.
Dan Boneh and Richard J. Lipton. Amplification of weak learning under the uniform distribution. In Conference on Computational Learning Theory (COLT), pages 347-351, 1993.
Zvika Brakerski, Christina Brzuska, and Nils Fleischhacker. On statistically secure obfuscation with approximate correctness. In International Cryptology Conference (CRYPTO), pages 551-578, 2016.
Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Learning algorithms from natural proofs. In Conference on Computational Complexity (CCC), pages 10:1-10:24, 2016.
Bill Fefferman, Ronen Shaltiel, Christopher Umans, and Emanuele Viola. On beating the hybrid argument. Theory of Computing, 9:809-843, 2013.
Lance Fortnow and Adam R. Klivans. Efficient learning algorithms yield circuit lower bounds. Journal of Computer and System Sciences, 75(1):27-36, 2009. URL: http://dx.doi.org/10.1016/j.jcss.2008.07.006.
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Eran Gat and Shafi Goldwasser. Probabilistic search algorithms with unique answers and their cryptographic applications. Electronic Colloquium on Computational Complexity (ECCC), 18:136, 2011.
Oded Goldreich, Noam Nisan, and Avi Wigderson. On Yao’s XOR lemma. In Studies in Complexity and Cryptography, pages 273-301. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22670-0_23.
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Shafi Goldwasser and Ofer Grossman. Bipartite perfect matching in pseudo-deterministic NC. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 87:1-87:13, 2017.
Shafi Goldwasser, Ofer Grossman, and Dhiraj Holden. Pseudo-deterministic proofs. Electronic Colloquium on Computational Complexity (ECCC), 24:105, 2017.
Shafi Goldwasser, Dan Gutfreund, Alexander Healy, Tali Kaufman, and Guy N. Rothblum. Verifying and decoding in constant depth. In Symposium on Theory of Computing (STOC), pages 440-449, 2007.
Parikshit Gopalan and Rocco A. Servedio. Learning and lower bounds for AC⁰ with threshold gates. In International Workshop on Approximation, Randomization, and Combinatorial Optimization (RANDOM-APPROX), pages 588-601, 2010.
Ofer Grossman. Finding primitive roots pseudo-deterministically. Electronic Colloquium on Computational Complexity (ECCC), 22:207, 2015.
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Ryan C. Harkins and John M. Hitchcock. Exact learning algorithms, betting games, and circuit lower bounds. Transactions on Computation Theory (TOCT), 5(4):18:1-18:11, 2013. URL: http://dx.doi.org/10.1145/2539126.2539130.
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Dhiraj Holden. A note on unconditional subexponential-time pseudo-deterministic algorithms for BPP search problems. arXiv, 2017. URL: http://arxiv.org/abs/1707.05808.
http://arxiv.org/abs/1707.05808
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Moni Naor and Omer Reingold. Number-theoretic constructions of efficient pseudo-random functions. Journal of the ACM, 51(2):231-262, 2004.
Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of Computer and System Sciences, 49(2):149-167, 1994.
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Igor C. Oliveira and Rahul Santhanam. Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness. In Computational Complexity Conference (CCC), pages 18:1-18:49, 2017.
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http://dx.doi.org/10.1137/10080703X
Igor C. Oliveira and Rahul Santhanam
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits
We study correlation bounds and pseudorandom generators for depth-two circuits that consist of a SYM-gate (computing an arbitrary symmetric function) or THR-gate (computing an arbitrary linear threshold function) that is fed by S {AND} gates. Such circuits were considered in early influential work on unconditional derandomization of Luby, Velickovi{c}, and Wigderson [Michael Luby et al., 1993], who gave the first non-trivial PRG with seed length 2^{O(sqrt{log(S/epsilon)})} that epsilon-fools these circuits.
In this work we obtain the first strict improvement of [Michael Luby et al., 1993]'s seed length: we construct a PRG that epsilon-fools size-S {SYM,THR} oAND circuits over {0,1}^n with seed length 2^{O(sqrt{log S})} + polylog(1/epsilon), an exponential (and near-optimal) improvement of the epsilon-dependence of [Michael Luby et al., 1993]. The above PRG is actually a special case of a more general PRG which we establish for constant-depth circuits containing multiple SYM or THR gates, including as a special case {SYM,THR} o AC^0 circuits. These more general results strengthen previous results of Viola [Viola, 2006] and essentially strengthen more recent results of Lovett and Srinivasan [Lovett and Srinivasan, 2011].
Our improved PRGs follow from improved correlation bounds, which are transformed into PRGs via the Nisan-Wigderson "hardness versus randomness" paradigm [Nisan and Wigderson, 1994]. The key to our improved correlation bounds is the use of a recent powerful multi-switching lemma due to Håstad [Johan Håstad, 2014].
Pseudorandom generators
correlation bounds
constant-depth circuits
Theory of computation~Pseudorandomness and derandomization
56:1-56:20
Regular Paper
Rocco A.
Servedio
Rocco A. Servedio
Department of Computer Science, Columbia University, New York, NY, USA
Supported by NSF grants CCF-1420349 and CCF-1563155.
Li-Yang
Tan
Li-Yang Tan
Department of Computer Science, Stanford University, Stanford, California, USA
Supported by NSF grant CCF-1563122. Part of this research was done during a visit to Columbia University.
10.4230/LIPIcs.APPROX-RANDOM.2018.56
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Rocco A. Servedio and Li-Yang Tan
Creative Commons Attribution 3.0 Unported license
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Randomly Coloring Graphs of Logarithmically Bounded Pathwidth
We consider the problem of sampling a proper k-coloring of a graph of maximal degree Delta uniformly at random. We describe a new Markov chain for sampling colorings, and show that it mixes rapidly on graphs of logarithmically bounded pathwidth if k >=(1+epsilon)Delta, for any epsilon>0, using a hybrid paths argument.
Random coloring
Glauber dynamics
Markov-chain Monte Carlo
Theory of computation~Random walks and Markov chains
Mathematics of computing~Markov-chain Monte Carlo methods
57:1-57:19
Regular Paper
https://hal.archives-ouvertes.fr/hal-01832102
Shai
Vardi
Shai Vardi
Krannert School of Management, Purdue University, West Lafayette, IN, 47907, USA
Supported in part by the Linde Foundation and NSF grants CNS-1254169 and CNS-1518941.
10.4230/LIPIcs.APPROX-RANDOM.2018.57
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Shai Vardi
Creative Commons Attribution 3.0 Unported license
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Explicit Strong LTCs with Inverse Poly-Log Rate and Constant Soundness
An error-correcting code C subseteq F^n is called (q,epsilon)-strong locally testable code (LTC) if there exists a tester that makes at most q queries to the input word. This tester accepts all codewords with probability 1 and rejects all non-codewords x not in C with probability at least epsilon * delta(x,C), where delta(x,C) denotes the relative Hamming distance between the word x and the code C. The parameter q is called the query complexity and the parameter epsilon is called soundness.
Goldreich and Sudan (J.ACM 2006) asked about the existence of strong LTCs with constant query complexity, constant relative distance, constant soundness and inverse polylogarithmic rate. They also asked about the explicit constructions of these codes.
Strong LTCs with the required range of parameters were obtained recently in the works of Viderman (CCC 2013, FOCS 2013) based on the papers of Meir (SICOMP 2009) and Dinur (J.ACM 2007). However, the construction of these codes was probabilistic.
In this work we show that codes presented in the works of Dinur (J.ACM 2007) and Ben-Sasson and Sudan (SICOMP 2005) provide the explicit construction of strong LTCs with the above range of parameters. Previously, such codes were proven to be weak LTCs. Using the results of Viderman (CCC 2013, FOCS 2013) we prove that such codes are, in fact, strong LTCs.
Error-Correcting Codes
Tensor Products
Locally Testable Codes
Theory of computation~Interactive proof systems
58:1-58:14
Regular Paper
The full version of this paper appeared as [Michael Viderman, 2015].
Michael
Viderman
Michael Viderman
Yahoo Research, Haifa, Israel
10.4230/LIPIcs.APPROX-RANDOM.2018.58
Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron. Testing Reed-Muller codes. IEEE Transactions on Information 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
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Eli Ben-Sasson, Elena Grigorescu, Ghid Maatouk, Amir Shpilka, and Madhu Sudan. On sums of locally testable affine invariant properties. In Proceedings of Approximation, Randomization, and Combinatorial Optimization (APPROX-RANDOM), volume 6845 of Lecture Notes in Computer Science, pages 400-411. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22935-0.
http://dx.doi.org/10.1007/978-3-642-22935-0
Eli Ben-Sasson, Noga Ron-Zewi, and Madhu Sudan. Sparse affine-invariant linear codes are locally testable. In FOCS, pages 561-570. IEEE Computer Society, 2012. URL: http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=6374356.
http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=6374356
Eli Ben-Sasson and Madhu Sudan. Robust locally testable codes and products of codes. Random Struct. Algorithms, 28(4):387-402, 2006. URL: http://dx.doi.org/10.1002/rsa.20120.
http://dx.doi.org/10.1002/rsa.20120
Eli Ben-Sasson and Madhu Sudan. Short PCPs with polylog query complexity. SIAM J. Comput, 38(2):551-607, 2008. URL: http://dx.doi.org/10.1137/050646445.
http://dx.doi.org/10.1137/050646445
Eli Ben-Sasson and Michael Viderman. Composition of Semi-LTCs by Two-Wise Tensor Products. In Proceedings of Approximation, Randomization, and Combinatorial Optimization (APPROX-RANDOM), volume 5687 of Lecture Notes in Computer Science, pages 378-391. Springer, 2009. URL: http://dx.doi.org/10.1007/978-3-642-03685-9.
http://dx.doi.org/10.1007/978-3-642-03685-9
Eli Ben-Sasson and Michael Viderman. Tensor Products of Weakly Smooth Codes are Robust. Theory of Computing, 5(1):239-255, 2009. URL: http://dx.doi.org/10.4086/toc.2009.v005a012.
http://dx.doi.org/10.4086/toc.2009.v005a012
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http://dx.doi.org/10.1007/978-3-642-15369-3
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