Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015), APPROX/RANDOM 2015, August 24-26, 2015, Princeton, USA
APPROX/RANDOM 2015
August 24-26, 2015
Princeton, 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
Naveen
Garg
Naveen Garg
Klaus
Jansen
Klaus Jansen
Anup
Rao
Anup Rao
José D. P.
Rolim
José D. P. Rolim
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
40
2015
978-3-939897-89-7
https://www.dagstuhl.de/dagpub/978-3-939897-89-7
Frontmatter, Table of Contents, Preface, Program Commitees, External Reviewers, List of Authors
Frontmatter, Table of Contents, Preface, Program Commitees, External Reviewers, List of Authors
Frontmatter
Table of Contents
Preface
Program Commitees
External Reviewers
List of Authors
i-xviii
Front Matter
Naveen
Garg
Naveen Garg
Klaus
Jansen
Klaus Jansen
Anup
Rao
Anup Rao
José D. P.
Rolim
José D. P. Rolim
10.4230/LIPIcs.APPROX-RANDOM.2015.i
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On Guillotine Cutting Sequences
Imagine a wooden plate with a set of non-overlapping geometric objects painted on it. How many of them can a carpenter cut out using a panel saw making guillotine cuts, i.e., only moving forward through the material along a straight line until it is split into two pieces? Already fifteen years ago, Pach and Tardos investigated whether one can always cut out a constant fraction if all objects are axis-parallel rectangles. However, even for the case of axis-parallel squares this question is still open. In this paper, we answer the latter affirmatively. Our result is constructive and holds even in a more general setting where the squares have weights and the goal is to save as much weight as possible. We further show that when solving the more general question for rectangles affirmatively with only axis-parallel cuts, this would yield a combinatorial O(1)-approximation algorithm for the Maximum Independent Set of Rectangles problem, and would thus solve a long-standing open problem. In practical applications, like the mentioned carpentry and many other settings, we can usually place the items freely that we want to cut out, which gives rise to the two-dimensional guillotine knapsack problem: Given a collection of axis-parallel rectangles without presumed coordinates, our goal is to place as many of them as possible in a square-shaped knapsack respecting the constraint that the placed objects can be separated by a sequence of guillotine cuts. Our main result for this problem is a quasi-PTAS, assuming the input data to be quasi-polynomially bounded integers. This factor matches the best known (quasi-polynomial time) result for (non-guillotine) two-dimensional knapsack.
Guillotine cuts
Rectangles
Squares
Independent Sets
Packing
1-19
Regular Paper
Fidaa
Abed
Fidaa Abed
Parinya
Chalermsook
Parinya Chalermsook
José
Correa
José Correa
Andreas
Karrenbauer
Andreas Karrenbauer
Pablo
Pérez-Lantero
Pablo Pérez-Lantero
José A.
Soto
José A. Soto
Andreas
Wiese
Andreas Wiese
10.4230/LIPIcs.APPROX-RANDOM.2015.1
Anna Adamaszek and Andreas Wiese. Approximation schemes for maximum weight independent set of rectangles. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013), pages 400-409. IEEE, 2013.
Anna Adamaszek and Andreas Wiese. A quasi-PTAS for the two-dimensional geometric knapsack problem. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015), pages 1491-1505, 2015.
Nikhil Bansal, Alberto Caprara, Klaus Jansen, Lars Prädel, and Maxim Sviridenko. A structural lemma in 2-dimensional packing, and its implications on approximability. In Algorithms and Computation (ISAAC 2009), volume 5878 of LNCS, pages 77-86. Springer, 2009.
Parinya Chalermsook and Julia Chuzhoy. Maximum independent set of rectangles. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), pages 892-901. SIAM, 2009.
Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48:373-392, 2012.
Edward G Coffman, Jr, Michael R Garey, David S Johnson, and Robert Endre Tarjan. Performance bounds for level-oriented two-dimensional packing algorithms. SIAM J. Comput., 9(4):808-826, 1980.
Thomas Erlebach, Klaus Jansen, and Eike Seidel. Polynomial-time approximation schemes for geometric intersection graphs. SIAM Journal on Computing, 34(6):1302-1323, 2005.
Wenceslas Fernandez de la Vega and George S. Lueker. Bin packing can be solved within 1 + ε in linear time. Combinatorica, 1:349-355, 1981.
Aleksei V. Fishkin, Olga Gerber, Klaus Jansen, and Roberto Solis-Oba. Packing weighted rectangles into a square. In Mathematical Foundations of Computer Science (MFCS 2005), volume 2337 of LNCS, pages 352-363. Springer, 2005.
Rolf Harren. Approximating the orthogonal knapsack problem for hypercubes. In Automata, Languages and Programming (ICALP 2006), volume 4051 of LNCS, pages 238-249. Springer, 2006.
Klaus Jansen and Roberto Solis-Oba. New approximability results for 2-dimensional packing problems. In Mathematical Foundations of Computer Science (MFCS 2007), volume 4708 of LNCS, pages 103-114. Springer, 2007.
Klaus Jansen and Roberto Solis-Oba. A polynomial time approximation scheme for the square packing problem. In Proceedings of the 13th Integer Programming and Combinatorial Optimization Conference (IPCO 2008), pages 184-198. Springer, 2008.
Klaus Jansen and Guochuan Zhang. Maximizing the number of packed rectangles. In Proceedings of the 9th Scandinavian Workshop on Algorithm Theory (SWAT 2004), pages 362-371. Springer, 2004.
Klaus Jansen and Guochuan Zhang. Maximizing the total profit of rectangles packed into a rectangle. Algorithmica, 47(3):323-342, 2007.
János Pach and Gábor Tardos. Cutting glass. In Proceedings of the 16th Annual Symposium on Computational Geometry (SOCG 2000), pages 360-369. ACM, 2000.
Jorge Urrutia. Problem presented at ACCOTA'96: Combinatorial and Computational Aspects of Optimization, Topology, and Algebra, Taxco, Mexico, 1996.
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Approximate Nearest Neighbor Search in Metrics of Planar Graphs
We investigate the problem of approximate Nearest-Neighbor Search (NNS) in graphical metrics: The task is to preprocess an edge-weighted graph G=(V,E) on m vertices and a small "dataset" D \subset V of size n << m, so that given a query point q \in V, one can quickly approximate dist(q,D) (the distance from q to its closest vertex in D) and find a vertex a \in D within this approximated distance. We assume the query algorithm has access to a distance oracle, that quickly evaluates the exact distance between any pair of vertices.
For planar graphs G with maximum degree Delta, we show how to efficiently construct a compact data structure -- of size ~O(n(Delta+1/epsilon)) -- that answers (1+epsilon)-NNS queries in time ~O(Delta+1/epsilon). Thus, as far as NNS applications are concerned, metrics derived from bounded-degree planar graphs behave as low-dimensional metrics, even though planar metrics do not necessarily have a low doubling dimension, nor can they be embedded with low distortion into l_2. We complement our algorithmic result by lower bounds showing that the access to an exact distance oracle (rather than an approximate one) and the dependency on Delta (in query time) are both essential.
Data Structures
Nearest Neighbor Search
Planar Graphs
Planar Metrics
Planar Separator
20-42
Regular Paper
Ittai
Abraham
Ittai Abraham
Shiri
Chechik
Shiri Chechik
Robert
Krauthgamer
Robert Krauthgamer
Udi
Wieder
Udi Wieder
10.4230/LIPIcs.APPROX-RANDOM.2015.20
Ittai Abraham, Shiri Chechik, and Cyril Gavoille. Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels. In Proceedings of the 44th symposium on Theory of Computing, pages 1199-1218. ACM, 2012.
Ittai Abraham, Daniel Delling, Andrew V. Goldberg, and Renato F. Werneck. Hierarchical hub labelings for shortest paths. In Proceedings of the 20th Annual European conference on Algorithms, ESA'12, pages 24-35. Springer-Verlag, 2012.
Ittai Abraham and Cyril Gavoille. Object location using path separators. In Proceedings of the Twenty-fifth Annual ACM Symposium on Principles of Distributed Computing, PODC'06, pages 188-197. ACM, 2006.
Stephen Alstrup, Gerth Stølting Brodal, and Theis Rauhe. New data structures for orthogonal range searching. In 41st Annual Symposium on Foundations of Computer Science, pages 198-207, 2000.
Eyal Amir. Approximation algorithms for treewidth. Algorithmica, 56(4):448-479, January 2010.
Holger Bast, Stefan Funke, Peter Sanders, and Dominik Schultes. Fast routing in road networks with transit nodes. Science, 316(5824):566, 2007.
Jon Louis Bentley. Multidimensional binary search trees used for associative searching. Commun. ACM, 18(9):509-517, September 1975.
A. Beygelzimer, S. Kakade, and J. Langford. Cover trees for nearest neighbor. In 23rd international conference on Machine learning, pages 97-104. ACM, 2006.
Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, December 1996.
Hans L. Bodlaender, Pal Gronas Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michal Pilipczuk. An O(c^k n) 5-approximation algorithm for treewidth. In 54th Annual Symposium on Foundations of Computer Science, pages 499-508, 2013.
Hans L. Bodlaender, John R. Gilbert, Hjálmtýr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238-255, March 1995.
R. Cole and L.-A. Gottlieb. Searching dynamic point sets in spaces with bounded doubling dimension. In 38th Annual ACM Symposium on Theory of Computing, pages 574-583. ACM, 2006.
David Eisenstat, Philip N. Klein, and Claire Mathieu. Approximating k-center in planar graphs. In 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 617-627. SIAM, 2014.
Robert Geisberger, Peter Sanders, Dominik Schultes, and Daniel Delling. Contraction hierarchies: Faster and simpler hierarchical routing in road networks. In Proceedings of the 7th International Conference on Experimental Algorithms, WEA'08, pages 319-333. Springer-Verlag, 2008.
S. Har-Peled and M. Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM Journal on Computing, 35(5):1148-1184, 2006.
Goos Kant and Hans L. Bodlaender. Triangulating planar graphs while minimizing the maximum degree. Inf. Comput., 135(1):1-14, May 1997.
D. Karger and M. Ruhl. Finding nearest neighbors in growth-restricted metrics. In 34th Annual ACM Symposium on the Theory of Computing, pages 63-66, 2002.
R. Krauthgamer and J. R. Lee. Navigating nets: Simple algorithms for proximity search. In 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 791-801, January 2004.
R. Krauthgamer and J. R. Lee. The black-box complexity of nearest-neighbor search. Theoret. Comput. Sci., 348(2-3):262-276, 2005.
R. Krauthgamer, H. Nguyen, and T. Zondiner. Preserving terminal distances using minors. SIAM Journal on Discrete Mathematics, 28(1):127-141, 2014.
R. J. Lipton and R. E. Tarjan. A separator theorem for planar graphs. SIAM J. Appl. Math., 36(2):177-189, 1979.
C. G. Plaxton, R. Rajaraman, and A. W. Richa. Accessing nearby copies of replicated objects in a distributed environment. Theory Comput. Syst., 32(3):241-280, 1999.
Mikkel Thorup. Compact oracles for reachability and approximate distances in planar digraphs. J. ACM, 51(6):993-1024, November 2004.
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How to Tame Rectangles: Solving Independent Set and Coloring of Rectangles via Shrinking
In the Maximum Weight Independent Set of Rectangles (MWISR) problem, we are given a collection of weighted axis-parallel rectangles in the plane. Our goal is to compute a maximum weight subset of pairwise non-overlapping rectangles. Due to its various applications, as well as connections to many other problems in computer science, MWISR has received a lot of attention from the computational geometry and the approximation algorithms community. However, despite being extensively studied, MWISR remains not very well understood in terms of polynomial time approximation algorithms, as there is a large gap between the upper and lower bounds, i.e., O(log n\ loglog n) v.s. NP-hardness. Another important, poorly understood question is whether one can color rectangles with at most O(omega(R)) colors where omega(R) is the size of a maximum clique in the intersection graph of a set of input rectangles R. Asplund and Grünbaum obtained an upper bound of O(omega(R)^2) about 50 years ago, and the result has remained asymptotically best. This question is strongly related to the integrality gap of the canonical LP for MWISR.
In this paper, we settle above three open problems in a relaxed model where we are allowed to shrink the rectangles by a tiny bit (rescaling them by a factor of 1-delta for an arbitrarily small constant delta > 0. Namely, in this model, we show (i) a PTAS for MWISR and (ii) a coloring with O(omega(R)) colors which implies a constant upper bound on the integrality gap of the canonical LP.
For some applications of MWISR the possibility to shrink the rectangles has a natural, well-motivated meaning. Our results can be seen as an evidence that the shrinking model is a promising way to relax a geometric problem for the purpose of better algorithmic results.
Approximation algorithms
independent set
resource augmentation
rectangle intersection graphs
PTAS
43-60
Regular Paper
Anna
Adamaszek
Anna Adamaszek
Parinya
Chalermsook
Parinya Chalermsook
Andreas
Wiese
Andreas Wiese
10.4230/LIPIcs.APPROX-RANDOM.2015.43
Anna Adamaszek and Andreas Wiese. Approximation schemes for maximum weight independent set of rectangles. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013), pages 400-409. IEEE, 2013.
Anna Adamaszek and Andreas Wiese. A QPTAS for maximum weight independent set of polygons with polylogarithmically many vertices. In Proceedings of the 25th ACM-SIAM Symposium on Discrete Algorithms (SODA 2014), pages 645-656. SIAM, 2014.
Pankaj K. Agarwal and Nabil H. Mustafa. Independent set of intersection graphs of convex objects in 2D. Computational Geometry: Theory and Applications, 34(2):83-95, 2006.
Pankaj K. Agarwal, Marc J. van Kreveld, and Subhash Suri. Label placement by maximum independent set in rectangles. Computational Geometry: Theory and Applications, 11(3-4):209-218, 1998.
Sanjeev Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM (JACM), 45:753-782, 1998.
E. Asplund and Branko Grünbaum. On a coloring problem. Mathematica Scandinavica, 8:181-188, 1960.
Piotr Berman, Bhaskar DasGupta, S. Muthukrishnan, and Suneeta Ramaswami. Improved approximation algorithms for rectangle tiling and packing. In Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), pages 427-436. SIAM, 2001.
A. Bielecki. Problem 56. Colloquium Mathematicum, 1:333, 1948.
Parinya Chalermsook. Coloring and maximum independent set of rectangles. In Proceedings of the 14th International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX 2011), pages 123-134. Springer, 2011.
Parinya Chalermsook and Julia Chuzhoy. Maximum independent set of rectangles. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), pages 892-901. SIAM, 2009.
Timothy M. Chan. Polynomial-time approximation schemes for packing and piercing fat objects. Journal of Algorithms, 46(2):178-189, 2003.
Timothy M. Chan. A note on maximum independent sets in rectangle intersection graphs. Information Processing Letters, 89(1):19-23, 2004.
Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. In Proceedings of the 25th Annual Symposium on Computational Geometry (SoCG 2009), pages 333-340. ACM, 2009.
George Christodoulou, Khaled M. Elbassioni, and Mahmoud Fouz. Truthful mechanisms for exhibitions. In Proceedings of the 6th International Workshop on Internet and Network Economics (WINE 2010), pages 170-181. Springer, 2010.
José R. Correa, Laurent Feuilloley, and José A. Soto. Independent and hitting sets of rectangles intersecting a diagonal line. In Proceedings of the 11th Latin American Symposium on Theoretical Informatics (LATIN 2014), pages 35-46. Springer, 2014.
Thomas Erlebach and Klaus Jansen. The complexity of path coloring and call scheduling. Theoretical Computer Science, 255(1-2):33-50, 2001.
Robert J Fowler, Michael S Paterson, and Steven L Tanimoto. Optimal packing and covering in the plane are np-complete. Information processing letters, 12(3):133-137, 1981.
Jacob Fox and János Pach. Computing the independence number of intersection graphs. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2011), pages 1161-1165. SIAM, 2011.
Takeshi Fukuda, Yasuhiko Morimoto, Shinichi Morishita, and Takeshi Tokuyama. Data mining with optimized two-dimensional association rules. ACM Transactions on Database Systems (TODS), 26(2):179-213, 2001.
Sariel Har-Peled. Quasi-polynomial time approximation scheme for sparse subsets of polygons. In Proceedings of the 30th Annual Symposium on Computational Geometry (SoCG 2014), page 120. ACM, 2014.
Sariel Har-Peled and Mira Lee. Weighted geometric set cover problems revisited. JoCG, 3(1):65-85, 2012.
Johan Håstad. Clique is hard to approximate within n^1-ε. Electronic Colloquium on Computational Complexity, 4(38), 1997.
C. Hendler. Schranken für Färbungs-und Cliquenüberdeckungszahl geometrisch repräsentierbarer Graphen. Master’s thesis, FU Berlin, Germany, 1998.
Dorit S. Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing problems in image processing and VLSI. Journal of the ACM, 32(1):130-136, 1985.
Alexandr V. Kostochka. Coloring intersection graphs of geometric figures with a given clique number. Contemporary Mathematics, 342, 2004.
Liane Lewin-Eytan, Joseph Naor, and Ariel Orda. Routing and admission control in networks with advance reservations. In Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX 2002), pages 215-228. Springer, 2002.
Frank Nielsen. Fast stabbing of boxes in high dimensions. Theoretical Computer Science, 246:53-72, 2000.
David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing, 3:103-128, 2007.
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Non-Uniform Robust Network Design in Planar Graphs
Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every resource is equally vulnerable, and that the set of scenarios is implicitly given by a single budget constraint. This paper studies a robustness model of a different kind. We focus on Bulk-Robustness, a model recently introduced (Adjiashvili, Stiller, Zenklusen 2015) for addressing the need to model non-uniform failure patterns in systems.
We significantly extend the techniques used by Adjiashvili et al. to design approximation algorithm for bulk-robust network design problems in planar graphs. Our techniques use an augmentation framework, combined with linear programming (LP) rounding that depends on a planar embedding of the input graph. A connection to cut covering problems and the dominating set problem in circle graphs is established. Our methods use few of the specifics of bulk-robust optimization, hence it is conceivable that they can be adapted to solve other robust network design problems.
Robust optimization
Network design
Planar graph
Approximation algorithm
LP rounding
61-77
Regular Paper
David
Adjiashvili
David Adjiashvili
10.4230/LIPIcs.APPROX-RANDOM.2015.61
D. Adjiashvili. Structural Robustness in Combinatorial Optimization. PhD thesis, ETH Zürich, Zürich, Switzerland, 2012.
D. Adjiashvili, G. Oriolo, and M. Senatore. The online replacement path problem. In Proceedings of 18th Annual European Symposium on Algorithms (ESA), pages 1-12. Springer, 2013.
D. Adjiashvili, S. Stiller, and R. Zenklusen. Bulk-robust combinatorial optimization. Mathematical Programming, 149(1-2):361-390, 2014.
D. Adjiashvili and R. Zenklusen. An s-t connection problem with adaptability. Discrete Applied Mathematics, 159:695-705, 2011.
H. Aissi, C. Bazgan, and D. Vanderpooten. Min–max and min–max regret versions of combinatorial optimization problems: A survey. European Journal of Operational Research, 197(2):427-438, 2009.
N. Bansal and K. Pruhs. The geometry of scheduling. In 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 407-414, 2010.
D. Bertsimas, D. B. Brown, and C. Caramanis. Theory and applications of robust optimization. SIAM review, 53:464-501, 2011.
C. Chekuri, J. Vondrák, and R. Zenklusen. Multi-budgeted matchings and matroid intersection via dependent rounding. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1080-1097, 2011.
J. Cheriyan and R. Thurimella. Approximating Minimum-Size k-Connected Spanning Subgraphs via Matching. SIAM J. Comput, 30:292-301, 2000.
M. Damian-Iordache and S. V. Pemmaraju. A (2+ ε)-approximation scheme for minimum domination on circle graphs. Journal of Algorithms, 42(2):255-276, 2002.
K. Dhamdhere, V. Goyal, R. Ravi, and M. Singh. How to pay, come what may: approximation algorithms for demand-robust covering problems. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 367-376, October 2005.
U. Feige, K. Jain, M. Mahdian, and V. Mirrokni. Robust combinatorial optimization with exponential scenarios. In Matteo Fischetti and David Williamson, editors, IPCO 2007, volume 4513 of Lecture Notes in Computer Science, pages 439-453. Springer Berlin / Heidelberg, 2007.
H. N. Gabow, M. X. Goemans, É. Tardos, and D. P. Williamson. Approximating the smallest k-edge connected spanning subgraph by lp-rounding. Networks, 53(4):345-357, 2009.
H. N. Gabow, M.X. Goemans, and D.P. Williamson. An efficient approximation algorithm for the survivable network design problem. Mathematical Programming, Series B, 82(1-2):13-40, 1998.
D. Golovin, V. Goyal, and R. Ravi. Pay today for a rainy day: Improved approximation algorithms for demand-robust min-cut and shortest path problems. In Bruno Durand and Wolfgang Thomas, editors, STACS 2006, volume 3884 of Lecture Notes in Computer Science, pages 206-217. Springer Berlin / Heidelberg, 2006.
F. Grandoni, R. Ravi, M. Singh, and R. Zenklusen. New approaches to multi-objective optimization. Mathematical Programming, 146(1-2):525-554, 2014.
V. Guruswami, S. Sachdeva, and R. Saket. Inapproximability of minimum vertex cover on k-uniform k-partite hypergraphs. SIAM Journal on Discrete Mathematics, 29(1):36-58, 2015.
E. Israeli and R. K. Wood. Shortest-path network interdiction. Networks, 40:97-111, 2002.
K. Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica, 21:39-60, 2001.
H. Kerivin and A.R. Mahjoub. Design of survivable networks: A survey. Networks, 46(1):1-21, 2005.
R. Khandekar, G. Kortsarz, V. Mirrokni, and M. Salavatipour. Two-stage robust network design with exponential scenarios. In Dan Halperin and Kurt Mehlhorn, editors, ESA 2008, volume 5193 of Lecture Notes in Computer Science, pages 589-600. Springer Berlin / Heidelberg, 2008.
P. Kouvelis and G. Yu. Robust Discrete Optimization and Its Applications. Kluwer Academic Publishers, Boston., 1997.
N.-K. Olver. Robust network design. PhD thesis, McGill University, Montreal, Quebec, Canada, 2010.
C. H. Papadimitriou and M. Yannakakis. On the approximability of trade-offs and optimal access of web sources. In 41st Annual Symposium on Foundations of Computer Science (FOCS), pages 86-92, 2000.
R. Ravi, M. V. Marathe, Ravi S. S., Rosenkrantz D. J., and Hunt H. B. Many birds with one stone: Multi-objective approximation algorithms. In 25th annual ACM Symposium on the Theory of Computing (STOC), pages 438-447, 1993.
D.P. Williamson, M.X. Goemans, M. Mihail, and V.V. Vazirani. A primal-dual approximation algorithm for generalized steiner network problems. Combinatorica, 15(3):435-454, 1995.
G. Yu and J. Yang. On the robust shortest path problem. Computers & Operations Research, 25(6):457-468, 1998.
R. Zenklusen. Matching interdiction. Discrete Applied Mathematics, 158(15):1676-1690, 2010.
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Large Supports are Required for Well-Supported Nash Equilibria
We prove that for any constant k and any epsilon < 1, there exist bimatrix win-lose games for which every epsilon-WSNE requires supports of cardinality greater than k. To do this, we provide a graph-theoretic characterization of win-lose games that possess epsilon-WSNE with constant cardinality supports. We then apply a result in additive number theory of Haight to construct win-lose games that do not satisfy the requirements of the characterization. These constructions disprove graph theoretic conjectures of Daskalakis, Mehta and Papadimitriou and Myers.
bimatrix games
well-supported Nash equilibria
78-84
Regular Paper
Yogesh
Anbalagan
Yogesh Anbalagan
Hao
Huang
Hao Huang
Shachar
Lovett
Shachar Lovett
Sergey
Norin
Sergey Norin
Adrian
Vetta
Adrian Vetta
Hehui
Wu
Hehui Wu
10.4230/LIPIcs.APPROX-RANDOM.2015.78
I. Althöfer. On sparse approximations to randomized strategies and convex combinations. Linear Algebra and its Applications, 199:339-355, 1994.
Y. Anbalagan, S. Norin, R. Savani, and A. Vetta. Polylogarithmic supports are required for approximate well-supported Nash equilibria below 2/3. In Proceedings of Ninth Conference on Web and Internet Economics (WINE), pages 15-23, 2013.
L. Caccetta and R. Häggkvist. On minimal digraphs with given girth. Linear Algebra and its Applications, 21:181-187, 1978.
P. Charbit. Circuits in Graphs and Digraphs via Embeddings. Doctoral Thesis, University of Lyon, 2005.
X. Chen, X. Deng, and S. Teng. Settling the complexity of computing two-player Nash equilibria. Journal of the ACM, 56(3):1-57, 2009.
C. Daskalakis, P. Goldberg, and C. Papadimitriou. The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195-259, 2009.
C. Daskalakis, A. Mehta, and C. Papadimitriou. A note on approximate Nash equilibria. Theoretical Computer Science, 410(17):1581-1588, 2009.
J. Haight. Difference covers which have small k-sums for any k. Mathematika, 20:109-118, 1973.
R. Lipton, E. Markakis, and Mehta A. Playing large games using simple startegies. In Proceedings of Fourth Conference on Electronic Commerce (EC), pages 36-41, 2003.
J. Myers. Extremal Theory of Graph Minors and Directed Graphs. Doctoral Thesis, University of Cambridge, 2003.
J. Nash. Non-cooperative games. Annals of Mathematics, 54:289-295, 1951.
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Minimizing Maximum Flow-time on Related Machines
We consider the online problem of minimizing the maximum flow-time on related machines. This is a natural generalization of the extensively studied makespan minimization problem to the setting where jobs arrive over time. Interestingly, natural algorithms such as Greedy or Slow-fit that work for the simpler identical machines case or for makespan minimization on related machines, are not O(1)-competitive. Our main result is a new O(1)-competitive algorithm for the problem. Previously, O(1)-competitive algorithms were known only with resource augmentation, and in fact no O(1) approximation was known even in the offline case.
Related machines scheduling
Maximum flow-time minimization
On-line algorithm
Approximation algorithm
85-95
Regular Paper
Nikhil
Bansal
Nikhil Bansal
Bouke
Cloostermans
Bouke Cloostermans
10.4230/LIPIcs.APPROX-RANDOM.2015.85
Susanne Albers. Introduction to scheduling, chapter Online scheduling, pages 51-73. Chapman and Hall/CRC, 2010.
Christoph Ambühl and Monaldo Mastrolilli. On-line scheduling to minimize max flow time: an optimal preemptive algorithm. Oper. Res. Lett., 33(6):597-602, 2005.
S. Anand, Karl Bringmann, Tobias Friedrich, Naveen Garg, and Amit Kumar. Minimizing maximum (weighted) flow-time on related and unrelated machines. In ICALP (1), pages 13-24, 2013.
S. Anand, Naveen Garg, and Amit Kumar. Resource augmentation for weighted flow-time explained by dual fitting. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1228-1241, 2012.
Yossi Azar. On-line load balancing. In Amos Fiat and Gerhard J. Woeginger, editors, Online Algorithms, volume 1442 of Lecture Notes in Computer Science, pages 178-195. Springer, 1998.
Yossi Azar, Bala Kalyanasundaram, Serge A. Plotkin, Kirk Pruhs, and Orli Waarts. On-line load balancing of temporary tasks. J. Algorithms, 22(1):93-110, 1997.
Nikhil Bansal and Janardhan Kulkarni. Minimizing flow-time on unrelated machines. In Symposium on Theory of Computing, STOC, 2015, to appear.
Michael A. Bender, Soumen Chakrabarti, and S. Muthukrishnan. Flow and stretch metrics for scheduling continuous job streams. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 270-279, 1998.
Niv Buchbinder and Joseph Naor. The design of competitive online algorithms via a primal-dual approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009.
Chandra Chekuri and Benjamin Moseley. Online scheduling to minimize the maximum delay factor. In Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1116-1125. Society for Industrial and Applied Mathematics, 2009.
Yookun Cho and Sartaj Sahni. Bounds for list schedules on uniform processors. SIAM Journal on Computing, 9(1):91-103, 1980.
Naveen Garg. Minimizing average flow-time. In Efficient Algorithms, Essays Dedicated to Kurt Mehlhorn on the Occasion of His 60th Birthday, pages 187-198, 2009.
Sungjin Im, Benjamin Moseley, and Kirk Pruhs. A tutorial on amortized local competitiveness in online scheduling. SIGACT News, 42(2):83-97, 2011.
Bala Kalyanasundaram and Kirk Pruhs. Speed is as powerful as clairvoyance. J. ACM, 47(4):617-643, 2000.
Kirk Pruhs, Jiri Sgall, and Eric Torng. Handbook of Scheduling: Algorithms, Models, and Performance Analysis, chapter Online Scheduling. CRC Press, 2004.
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A 2-Competitive Algorithm For Online Convex Optimization With Switching Costs
We consider a natural online optimization problem set on the real line. The state of the online algorithm at each integer time is a location on the real line. At each integer time, a convex function arrives online. In response, the online algorithm picks a new location. The cost paid by the online algorithm for this response is the distance moved plus the value of the function at the final destination. The objective is then to minimize the aggregate cost over all time. The motivating application is rightsizing power-proportional data centers. We give a 2-competitive algorithm for this problem. We also give a 3-competitive memoryless algorithm, and show that this is the best competitive ratio achievable by a deterministic memoryless algorithm. Finally we show that this online problem is strictly harder than the standard ski rental problem.
Stochastic
Scheduling
96-109
Regular Paper
Nikhil
Bansal
Nikhil Bansal
Anupam
Gupta
Anupam Gupta
Ravishankar
Krishnaswamy
Ravishankar Krishnaswamy
Kirk
Pruhs
Kirk Pruhs
Kevin
Schewior
Kevin Schewior
Cliff
Stein
Cliff Stein
10.4230/LIPIcs.APPROX-RANDOM.2015.96
Lachlan L. H. Andrew, Siddharth Barman, Katrina Ligett, Minghong Lin, Adam Meyerson, Alan Roytman, and Adam Wierman. A tale of two metrics: Simultaneous bounds on competitiveness and regret. In COLT 2013 - The 26th Annual Conference on Learning Theory, June 12-14, 2013, Princeton University, NJ, USA, pages 741-763, 2013.
Nikhil Bansal, Niv Buchbinder, and Joseph Naor. Towards the randomized k-server conjecture: A primal-dual approach. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 40-55, 2010.
Yair Bartal, Béla Bollobás, and Manor Mendel. Ramsey-type theorems for metric spaces with applications to online problems. J. Comput. Syst. Sci., 72(5):890-921, 2006.
Yair Bartal, Nathan Linial, Manor Mendel, and Assaf Naor. On metric ramsey-type phenomena. In Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing, STOC'03, pages 463-472, New York, NY, USA, 2003. ACM.
Allan Borodin, Nathan Linial, and Michael E. Saks. An optimal on-line algorithm for metrical task system. J. ACM, 39(4):745-763, 1992.
M. Chrobak, H. Karloff, T. Payne, and S. Vishwanathan. New results on server problems. In Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'90, pages 291-300, Philadelphia, PA, USA, 1990. Society for Industrial and Applied Mathematics.
Aaron Coté, Adam Meyerson, and Laura Poplawski. Randomized k-server on hierarchical binary trees. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC'08, pages 227-234, New York, NY, USA, 2008. ACM.
Amos Fiat and Manor Mendel. Better algorithms for unfair metrical task systems and applications. SIAM J. Comput., 32(6):1403-1422, 2003.
Anna R. Karlin, Mark S. Manasse, Lyle A. McGeoch, and Susan S. Owicki. Competitive randomized algorithms for nonuniform problems. Algorithmica, 11(6):542-571, 1994.
Minghong Lin, Zhenhua Liu, Adam Wierman, and Lachlan L. H. Andrew. Online algorithms for geographical load balancing. In 2012 International Green Computing Conference, IGCC 2012, San Jose, CA, USA, June 4-8, 2012, pages 1-10, 2012.
Minghong Lin, Adam Wierman, Lachlan L. H. Andrew, and Eno Thereska. Online dynamic capacity provisioning in data centers. In 2011 49th Annual Allerton Conference on Communication, Control, and Computing, Allerton Park & Retreat Center, Monticello, IL, USA, 28-30 September, 2011, pages 1159-1163, 2011.
Minghong Lin, Adam Wierman, Lachlan L. H. Andrew, and Eno Thereska. Dynamic right-sizing for power-proportional data centers. IEEE/ACM Trans. Netw., 21(5):1378-1391, 2013.
Minghong Lin, Adam Wierman, Alan Roytman, Adam Meyerson, and Lachlan L. H. Andrew. Online optimization with switching cost. SIGMETRICS Performance Evaluation Review, 40(3):98-100, 2012.
Zhenhua Liu, Minghong Lin, Adam Wierman, Steven H. Low, and Lachlan L. H. Andrew. Greening geographical load balancing. In SIGMETRICS 2011, Proceedings of the 2011 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, San Jose, CA, USA, 07-11 June 2011 (Co-located with FCRC 2011), pages 233-244, 2011.
Kai Wang, Minghong Lin, Florin Ciucu, Adam Wierman, and Chuang Lin. Characterizing the impact of the workload on the value of dynamic resizing in data centers. In Proceedings of the IEEE INFOCOM 2013, Turin, Italy, April 14-19, 2013, pages 515-519, 2013.
Adam Wierman. Personal communication, 2015.
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Beating the Random Assignment on Constraint Satisfaction Problems of Bounded Degree
We show that for any odd k and any instance I of the max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a 1/2 + Omega(1/sqrt(D)) fraction of I's constraints, where D is a bound on the number of constraints that each variable occurs in.
This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a quantum algorithm to find an assignment satisfying a 1/2 Omega(D^{-3/4}) fraction of the equations.
For arbitrary constraint satisfaction problems, we give a similar result for "triangle-free" instances; i.e., an efficient algorithm that finds an assignment satisfying at least a mu + Omega(1/sqrt(degree)) fraction of constraints, where mu is the fraction that would be satisfied by a uniformly random assignment.
constraint satisfaction problems
bounded degree
advantage over random
110-123
Regular Paper
Boaz
Barak
Boaz Barak
Ankur
Moitra
Ankur Moitra
Ryan
O’Donnell
Ryan O’Donnell
Prasad
Raghavendra
Prasad Raghavendra
Oded
Regev
Oded Regev
David
Steurer
David Steurer
Luca
Trevisan
Luca Trevisan
Aravindan
Vijayaraghavan
Aravindan Vijayaraghavan
David
Witmer
David Witmer
John
Wright
John Wright
10.4230/LIPIcs.APPROX-RANDOM.2015.110
Noga Alon. On the edge-expansion of graphs. Combin. Probab. Comput., 6(2):145-152, 1997.
Bonnie Berger and Peter W. Shor. Approximation alogorithms for the maximum acyclic subgraph problem. In Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'90, pages 236-243, Philadelphia, PA, USA, 1990. Society for Industrial and Applied Mathematics.
Irit Dinur, Ehud Friedgut, Guy Kindler, and Ryan O'Donnell. On the Fourier tails of bounded functions over the discrete cube. Israel J. Math., 160:389-412, 2007.
Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm applied to a bounded occurrence constraint problem, 2014. arXiv:1412.6062.
Venkatesan Guruswami and Yuan Zhou. Approximating bounded occurrence ordering csps. In Anupam Gupta, Klaus Jansen, José Rolim, and Rocco Servedio, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 7408 of Lecture Notes in Computer Science, pages 158-169. Springer Berlin Heidelberg, 2012.
Johan Håstad. On bounded occurrence constraint satisfaction. Inform. Process. Lett., 74(1-2):1-6, 2000.
Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001.
Johan Håstad and S. Venkatesh. On the advantage over a random assignment. Random Structures Algorithms, 25(2):117-149, 2004.
Subhash Khot and Assaf Naor. Linear equations modulo 2 and the L \sb 1 diameter of convex bodies. SIAM J. Comput., 38(4):1448-1463, 2008.
Konstantin Makarychev. Local search is better than random assignment for bounded occurrence ordering k-csps. In 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, February 27 - March 2, 2013, Kiel, Germany, pages 139-147, 2013.
Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014.
James B. Shearer. A note on bipartite subgraphs of triangle-free graphs. Random Structures Algorithms, 3(2):223-226, 1992.
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Improved Bounds in Stochastic Matching and Optimization
We consider two fundamental problems in stochastic optimization: approximation algorithms for stochastic matching, and sampling bounds in the black-box model. For the former, we improve the current-best bound of 3.709 due to Adamczyk et al. (2015), to 3.224; we also present improvements on Bansal et al. (2012) for hypergraph matching and for relaxed versions of the problem. In the context of stochastic optimization, we improve upon the sampling bounds of Charikar et al. (2005).
stochastic matching
approximation algorithms
sampling complexity
124-134
Regular Paper
Alok
Baveja
Alok Baveja
Amit
Chavan
Amit Chavan
Andrei
Nikiforov
Andrei Nikiforov
Aravind
Srinivasan
Aravind Srinivasan
Pan
Xu
Pan Xu
10.4230/LIPIcs.APPROX-RANDOM.2015.124
Marek Adamczyk, Fabrizio Grandoni, and Joydeep Mukherjee. Improved approximation algorithms for stochastic matching. CoRR, abs/1505.01439, 2015.
Nikhil Bansal, Anupam Gupta, Jian Li, Julián Mestre, Viswanath Nagarajan, and Atri Rudra. When LP is the cure for your matching woes: Improved bounds for stochastic matchings. Algorithmica, 63(4):733-762, 2012.
Moses Charikar, Chandra Chekuri, and Martin Pál. Sampling bounds for stochastic optimization. In Proceedings of the 8th International Workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th International Conference on Randamization and Computation: Algorithms and Techniques, APPROX'05/RANDOM'05, pages 257-269. Springer-Verlag, 2005.
Ning Chen, Nicole Immorlica, Anna R Karlin, Mohammad Mahdian, and Atri Rudra. Approximating matches made in heaven. Automata, Languages and Programming, pages 266-278, 2009.
Brian C Dean, Michel X Goemans, and Jan Vondrák. Approximating the stochastic knapsack problem: The benefit of adaptivity. Mathematics of Operations Research, 33(4):945-964, 2008.
Zoltán Füredi, Jeff Kahn, and Paul D. Seymour. On the fractional matching polytope of a hypergraph. Combinatorica, 13(2):167-180, 1993.
Rajiv Gandhi, Samir Khuller, Srinivasan Parthasarathy, and Aravind Srinivasan. Dependent rounding and its applications to approximation algorithms. Journal of the ACM (JACM), 53(3):324-360, 2006.
Naveen Garg, Anupam Gupta, Stefano Leonardi, and Piotr Sankowski. Stochastic analyses for online combinatorial optimization problems. In SODA, pages 942-951, 2008.
Anupam Gupta and Amit Kumar. A constant-factor approximation for stochastic steiner forest. In STOC, pages 659-668, 2009.
Anupam Gupta, R Ravi, and Amitabh Sinha. An edge in time saves nine: Lp rounding approximation algorithms for stochastic network design. In Foundations of Computer Science, 2004. Proceedings. 45th Annual IEEE Symposium on, pages 218-227. IEEE, 2004.
Nicole Immorlica, David Karger, Maria Minkoff, and Vahab S Mirrokni. On the costs and benefits of procrastination: approximation algorithms for stochastic combinatorial optimization problems. In Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, pages 691-700. Society for Industrial and Applied Mathematics, 2004.
Anton J Kleywegt, Alexander Shapiro, and Tito Homem-de Mello. The sample average approximation method for stochastic discrete optimization. SIAM Journal on Optimization, 12(2):479-502, 2002.
Jeff Linderoth, Alexander Shapiro, and Stephen Wright. The empirical behavior of sampling methods for stochastic programming. Annals of Operations Research, 142(1):215-241, 2006.
Andrzej Ruszczynski and Alexander Shapiro. Stochastic programming. Handbooks in operations research and management science, 2003.
Alexander Shapiro. Monte Carlo sampling methods. Handbooks in operations research and management science, 10:353-425, 2003.
David B Shmoys and Chaitanya Swamy. An approximation scheme for stochastic linear programming and its application to stochastic integer programs. Journal of the ACM (JACM), 53(6):978-1012, 2006.
A. Srinivasan. Approximation algorithms for stochastic and risk-averse optimization. In Proc. ACM-SIAM Symposium on Discrete Algorithms, pages 1305-1313, 2007.
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Fully Dynamic Bin Packing Revisited
We consider the fully dynamic bin packing problem, where items arrive and depart in an online fashion and repacking of previously packed items is allowed. The goal is, of course, to minimize both the number of bins used as well as the amount of repacking. A recently introduced way of measuring the repacking costs at each timestep is the migration factor, defined as the total size of repacked items divided by the size of an arriving or departing item. Concerning the trade-off between number of bins and migration factor, if we wish to achieve an asymptotic competitive ratio of 1 + epsilon for the number of bins, a relatively simple argument proves a lower bound of Omega(1/epsilon) of the migration factor. We establish a fairly close upper bound of O(1/epsilon^4 log(1/epsilon)) using a new dynamic rounding technique and new ideas to handle small items in a dynamic setting such that no amortization is needed. The running time of our algorithm is polynomial in the number of items n and in 1/epsilon. The previous best trade-off was for an asymptotic competitive ratio of 5/4 for the bins (rather than 1+epsilon) and needed an amortized number of O(log n) repackings (while in our scheme the number of repackings is independent of n and non-amortized).
online
bin packing
migration factor
robust
AFPTAS
135-151
Regular Paper
Sebastian
Berndt
Sebastian Berndt
Klaus
Jansen
Klaus Jansen
Kim-Manuel
Klein
Kim-Manuel Klein
10.4230/LIPIcs.APPROX-RANDOM.2015.135
J. Balogh, J. Békési, and G. Galambos. New lower bounds for certain classes of bin packing algorithms. In Workshop on Approximation and Online Algorithms(WAOA), volume 6534 of LNCS, pages 25-36, 2010.
J. Balogh, J. Békési, G. Galambos, and G. Reinelt. Lower bound for the online bin packing problem with restricted repacking. SIAM Journal on Computing, 38(1):398-410, 2008.
A. Beloglazov and R. Buyya. Energy efficient allocation of virtual machines in cloud data centers. In 10th IEEE/ACM International Conference on Cluster, Cloud and Grid Computing, CCGrid 2010, pages 577-578, 2010.
N. Bobroff, A. Kochut, and K.A. Beaty. Dynamic placement of virtual machines for managing SLA violations. In Integrated Network Management, IM 2007. 10th IFIP/IEEE International Symposium on Integrated Network Management, pages 119-128, 2007.
D.J. Brown. A lower bound for on-line one-dimensional bin packing algorithms. Technical Report R-864, Coordinated Sci Lab Univ of Illinois Urbana, 1979.
J.W. Chan, T. Lam, and P.W.H. Wong. Dynamic bin packing of unit fractions items. Theoretical Computer Science, 409(3):521-529, 2008.
J.W. Chan, P.W.H. Wong, and F.C.C. Yung. On dynamic bin packing: An improved lower bound and resource augmentation analysis. Algorithmica, 53(2):172-206, 2009.
E.G. Coffman, M.R. Garey, and D.S. Johnson. Dynamic bin packing. SIAM Journal on Computing, 12(2):227-258, 1983.
Khuzaima D., Shahin K., and Alejandro L. On the online fault-tolerant server consolidation problem. In 26th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA'14, pages 12-21, 2014.
K. Eisemann. The Trim Problem. Management Science, 3(3):279-284, 1957.
L. Epstein and A. Levin. A robust APTAS for the classical bin packing problem. Mathematical Programming, 119(1):33-49, 2009.
L. Epstein and A. Levin. Robust approximation schemes for cube packing. SIAM Journal on Optimization, 23(2):1310-1343, 2013.
W. Fernandez de la Vega and G.S. Lueker. Bin packing can be solved within 1+ ε in linear time. Combinatorica, 1(4):349-355, 1981.
G. Gambosi, A. Postiglione, and M. Talamo. Algorithms for the relaxed online bin-packing model. SIAM Journal on Computing, 30(5):1532-1551, 2000.
Z. Ivković and E.L. Lloyd. Partially dynamic bin packing can be solved within 1 + ε in (amortized) polylogarithmic time. Information Processing Letter, 63(1):45-50, 1997.
Z. Ivković and E.L. Lloyd. Fully dynamic algorithms for bin packing: Being (mostly) myopic helps. SIAM Journal on Computing, 28(2):574-611, 1998.
Z. Ivković and E.L. Lloyd. Fully dynamic bin packing. In Fundamental Problems in Computing, pages 407-434. Springer, 2009.
K. Jansen and K. Klein. A robust AFPTAS for online bin packing with polynomial migration. In International Colloquium on Automata, Languages, and Programming(ICALP), pages 589-600, 2013.
D.S. Johnson. Fast algorithms for bin packing. Journal of Computer and System Sciences, 8(3):272-314, 1974.
D.S. Johnson, A. Demers, J.D. Ullman, M.R. Garey, and R.L. Graham. Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM Journal on Computing, 3(4):299-325, 1974.
D.S. Johnson, A.J. Demers, J.D. Ullman, M.R. Garey, and R.L. Graham. Worst-case performance bounds for simple one-dimensional packing algorithms. SIAM Journal on Computing, 3(4):299-325, 1974.
G. Jung, K.R. Joshi, M.A. Hiltunen, R.D. Schlichting, and C. Pu. Generating adaptation policies for multi-tier applications in consolidated server environments. In 2008 International Conference on Autonomic Computing, ICAC 2008, June 2-6, 2008, Chicago, Illinois, USA, pages 23-32, 2008.
G. Jung, K.R. Joshi, M.A. Hiltunen, R.D. Schlichting, and C. Pu. A cost-sensitive adaptation engine for server consolidation of multitier applications. In Middleware 2009, ACM/IFIP/USENIX, 10th International Middleware Conference, Proceedings, pages 163-183, 2009.
N. Karmarkar and R.M. Karp. An efficient approximation scheme for the one-dimensional bin-packing problem. In 23rd Annual Symposium on Foundations of Computer Science (FOCS), pages 312-320. IEEE Computer Society, 1982.
C.C. Lee and D. Lee. A simple on-line bin-packing algorithm. Journal of the ACM (JACM), 32(3):562-572, 1985.
Y. Li, X. Tang, and W. Cai. On dynamic bin packing for resource allocation in the cloud. In 26th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA'14, pages 2-11, 2014.
F.M. Liang. A lower bound for on-line bin packing. Information processing letters, 10(2):76-79, 1980.
J.M. Park, Uday R. Savagaonkar, E.K.P. Chong, H.J. Siegel, and S.D. Jones. Efficient resource allocation for qos channels in mf-tdma satellite systems. In MILCOM 2000. 21st Century Military Communications Conference Proceedings, volume 2, pages 645-649. IEEE, 2000.
P. Sanders, N. Sivadasan, and M. Skutella. Online scheduling with bounded migration. Mathematics of Operations Research, 34(2):481-498, 2009.
S.S. Seiden. On the online bin packing problem. Journal of the ACM, 49(5):640-671, 2002.
M. Skutella and J. Verschae. A robust PTAS for machine covering and packing. In European Symposium on Algorithms(ESA), volume 6346 of LNCS, pages 36-47, 2010.
S. Srikantaiah, A. Kansal, and F. Zhao. Energy aware consolidation for cloud computing. In Proceedings of the 2008 Conference on Power Aware Computing and Systems, HotPower'08, pages 10-10, 2008.
A.L. Stolyar. An infinite server system with general packing constraints. Operations Research, 61(5):1200-1217, 2013.
A.L. Stolyar and Y. Zhong. A large-scale service system with packing constraints: Minimizing the number of occupied servers. In Proceedings of the ACM SIGMETRICS/international conference on Measurement and modeling of computer systems, pages 41-52. ACM, 2013.
J.D. Ullman. The Performance of a Memory Allocation Algorithm. Technical report. Princeton University, 1971.
A. Verma, P. Ahuja, and A. Neogi. pmapper: Power and migration cost aware application placement in virtualized systems. In Middleware 2008, ACM/IFIP/USENIX 9th International Middleware Conference, Proceedings, pages 243-264, 2008.
A. Vliet. An improved lower bound for on-line bin packing algorithms. Information Processing Letters, 43(5):277-284, 1992.
Prudence WH Wong, Fencol CC Yung, and Mihai Burcea. An 8/3 lower bound for online dynamic bin packing. In Algorithms and Computation, pages 44-53. Springer, 2012.
A.C. Yao. New algorithms for bin packing. Journal of the ACM (JACM), 27(2):207-227, 1980.
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Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be X-colorable if its vertices can be colored with X colors so that no hyperedge is monochromatic. 2-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a 2-colorable k-uniform hypergraph, it is NP-hard to find a 2-coloring miscoloring fewer than a fraction 2^(-k+1) of hyperedges (which is trivially achieved by a random 2-coloring), and the best algorithms to color the hypergraph properly require about n^(1-1/k) colors, approaching the trivial bound of n as k increases.
In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a 2-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than 2-colorability:
(A) Low-discrepancy: If the hypergraph has a 2-coloring of discrepancy l << sqrt(k), we give an algorithm to color the hypergraph with about n^(O(l^2/k)) colors. However, for the maximization version, we prove NP-hardness of finding a 2-coloring miscoloring a smaller than 2^(-O(k)) (resp. k^(-O(k))) fraction of the hyperedges when l = O(log k) (resp. l=2). Assuming the Unique Games conjecture, we improve the latter hardness factor to 2^(-O(k)) for almost discrepancy-1 hypergraphs.
(B) Rainbow colorability: If the hypergraph has a (k-l)-coloring such that each hyperedge is polychromatic with all these colors (this is stronger than a (l+1)-discrepancy 2-coloring), we give a 2-coloring algorithm that miscolors at most k^(-Omega(k)) of the hyperedges when l << sqrt(k), and complement this with a matching Unique Games hardness result showing that when l = sqrt(k), it is hard to even beat the 2^(-k+1) bound achieved by a random coloring.
(C) Strong Colorability: We obtain similar (stronger) Min- and Max-2-Coloring algorithmic results in the case of (k+l)-strong colorability.
Hypergraph Coloring
Discrepancy
Rainbow Coloring
Stong Coloring
Algorithms
Semidefinite Programming
Hardness of Approximation
152-174
Regular Paper
Vijay V. S. P.
Bhattiprolu
Vijay V. S. P. Bhattiprolu
Venkatesan
Guruswami
Venkatesan Guruswami
Euiwoong
Lee
Euiwoong Lee
10.4230/LIPIcs.APPROX-RANDOM.2015.152
Geir Agnarsson and Magnús M Halldórsson. Strong colorings of hypergraphs. In Approximation and Online Algorithms, pages 253-266. Springer, 2005.
Noga Alon. Personal communication, 2014.
Noga Alon, Pierre Kelsen, Sanjeev Mahajan, and Ramesh Hariharan. Approximate hypergraph coloring. Nordic Journal of Computing, 3(4):425-439, 1996.
Kazuhiko Aomoto et al. Analytic structure of schläfli function. Nagoya Math. J, 68:1-16, 1977.
Per Austrin, Venkatesan Guruswami, and Johan Håstad. (2 + ε)-SAT is NP-hard. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on. IEEE, 2014.
Per Austrin and Johan Håstad. On the usefulness of predicates. ACM Trans. Comput. Theory, 5(1):1:1-1:24, May 2013.
Per Austrin and Elchanan Mossel. Approximation resistant predicates from pairwise independence. computational complexity, 18(2):249-271, 2009.
Nikhil Bansal. Constructive algorithms for discrepancy minimization. In Foundations of Computer Science (FOCS), 2014 IEEE 51st Annual Symposium on, FOCS'10, pages 3-10. IEEE, 2010.
Nikhil Bansal and Subhash Khot. Inapproximability of hypergraph vertex cover and applications to scheduling problems. In Proceedings of the 37th International Colloquium on Automata, Languages and Programming, ICALP'10, pages 250-261, 2010.
V. Bhattiprolu, V. Guruswami, and E. Lee. Approximate Hypergraph Coloring under Low-discrepancy and Related Promises. Arxiv, 2015. arXiv:1506.06444.
Béla Bollobás, David Pritchard, Thomas Rothvoß, and Alex Scott. Cover-decomposition and polychromatic numbers. SIAM Journal on Discrete Mathematics, 27(1):240-256, 2013.
Károly Böröczky Jr and Martin Henk. Random projections of regular polytopes. Archiv der Mathematik, 73(6):465-473, 1999.
Hui Chen and Alan M Frieze. Coloring bipartite hypergraphs. In Proceedings of the 5th international conference on Integer Programming and Combinatorial Optimization, IPCO'96, pages 345-358, 1996.
Irit Dinur and Venkatesan Guruswami. PCPs via low-degree long code and hardness for constrained hypergraph coloring. In Foundations of Computer Science (FOCS), 2014 IEEE 54th Annual Symposium on, FOCS 13, pages 340-349, 2013.
Irit Dinur, Oded Regev, and Clifford D. Smyth. The hardness of 3-Uniform hypergraph coloring. Combinatorica, 25(1):519-535, 2005.
Paul Erdos and László Lovász. Problems and results on 3-chromatic hypergraphs and some related questions. Infinite and finite sets, 10(2):609-627, 1975.
Venkatesan Guruswami, Johan Håstad, Prahladh Harsha, Srikanth Srinivasan, and Girish Varma. Super-polylogarithmic hypergraph coloring hardness via low-degree long codes. In Proceedings of the 46th annual ACM Symposium on Theory of Computing, STOC'14, 2014.
Venkatesan Guruswami, Johan Håstad, and Madhu Sudan. Hardness of approximate hypergraph coloring. SIAM Journal on Computing, 31(6):1663-1686, 2002.
Venkatesan Guruswami and Euiwoong Lee. Strong inapproximability results on balanced rainbow-colorable hypergraphs. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'15, pages 822-836, 2015.
Venkatesan Guruswami and Euiwoong Lee. Towards a characterization of approximation resistance for symmetric CSPs. To Appear, The 18th. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2015.
Johan Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, July 2001.
Sangxia Huang. 2^(log N)^1/4 - o(1) hardness for hypergraph coloring. Electronic Colloquium on Computational Complexity (ECCC), 15-062, 2015.
David Karger, Rajeev Motwani, and Madhu Sudan. Approximate graph coloring by semidefinite programming. Journal of the ACM (JACM), 45(2):246-265, 1998.
Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for Max-Cut and other 2-variable CSPs? SIAM Journal on Computing, 37(1):319-357, 2007.
Subhash Khot and Rishi Saket. Hardness of coloring 2-colorable 12-uniform hypergraphs with exp(log^Ω (1) n) colors. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 206-215. IEEE, 2014.
Hellmuth Kneser. Der simplexinhalt in der nichteuklidischen geometrie. Deutsche Math, 1:337-340, 1936.
László Lovász. On the shannon capacity of a graph. Information Theory, IEEE Transactions on, 25(1):1-7, 1979.
Shachar Lovett and Raghu Meka. Constructive discrepancy minimization by walking on the edges. In Foundations of Computer Science (FOCS), 2014 IEEE 53rd Annual Symposium on, FOCS 12, pages 61-67, 2012.
Colin McDiarmid. A random recolouring method for graphs and hypergraphs. Combinatorics, Probability and Computing, 2(03):363-365, 1993.
Jun Murakami and Masakazu Yano. On the volume of a hyperbolic and spherical tetrahedron. Communications in analysis and geometry, 13(2):379, 2005.
CA Rogers. An asymptotic expansion for certain schläfli functions. Journal of the London Mathematical Society, 1(1):78-80, 1961.
Claude Ambrose Rogers. Packing and covering. Number 54 in Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge University Press, 1964.
Thomas Rothvoß. Constructive discrepancy minimization for convex sets. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 140-145. IEEE, 2014.
S. Sachdeva and R. Saket. Optimal inapproximability for scheduling problems via structural hardness for hypergraph vertex cover. In Proceedings of the 28th annual IEEE Conference on Computational Complexity, CCC'13, pages 219-229, 2013.
Ludwig Schläfli. On the multiple integral ∫ … ∫ dx dy... dz, whose limits are p1 = a1x+ b1y+… + h1z > 0, p2 > 0,..., pn > 0, and x2+ y2+… + z2 < 1. Quart. J. Math, 2(1858):269-300, 1858.
Joel Spencer. Six standard deviations suffice. Transactions of the American Mathematical Society, 289(2):679-706, 1985.
Cenny Wenner. Parity is positively useless. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: The 17th. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pages 433-448. Schloss Dagstuhl, 2014.
Avi Wigderson. Improving the performance guarantee for approximate graph coloring. Journal of the ACM (JACM), 30(4):729-735, 1983.
Uri Zwick. Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, volume 98 of SODA'98, pages 201-210, 1998.
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Stochastic and Robust Scheduling in the Cloud
Users of cloud computing services are offered rapid access to computing resources via the Internet. Cloud providers use different pricing options such as (i) time slot reservation in advance at a fixed price and (ii) on-demand service at a (hourly) pay-as-used basis. Choosing the best combination of pricing options is a challenging task for users, in particular, when the instantiation of computing jobs underlies uncertainty.
We propose a natural model for two-stage scheduling under uncertainty that captures such resource provisioning and scheduling problem in the cloud. Reserving a time unit for processing jobs incurs some cost, which depends on when the reservation is made: a priori decisions, based only on distributional information, are much cheaper than on-demand decisions when the actual scenario is known. We consider both stochastic and robust versions of scheduling unrelated machines with objectives of minimizing the sum of weighted completion times and the makespan. Our main contribution is an (8+eps)-approximation algorithm for the min-sum objective for the stochastic polynomial-scenario model. The same technique gives a (7.11+eps)-approximation for minimizing the makespan. The key ingredient is an LP-based separation of jobs and time slots to be considered in either the first or the second stage only, and then approximately solving the separated problems. At the expense of another epsilon our results hold for any arbitrary scenario distribution given by means of a black-box. Our techniques also yield approximation algorithms for robust two-stage scheduling.
Approximation Algorithms
Robust Optimization
Stochastic Optimization
Unrelated Machine Scheduling
Cloud Computing
175-186
Regular Paper
Lin
Chen
Lin Chen
Nicole
Megow
Nicole Megow
Roman
Rischke
Roman Rischke
Leen
Stougie
Leen Stougie
10.4230/LIPIcs.APPROX-RANDOM.2015.175
Amazon EC2 Pricing Options: https://aws.amazon.com/ec2/pricing/.
Talal Al-Khamis and Rym M'Hallah. A two-stage stochastic programming model for the parallel machine scheduling problem with machine capacity. Computers & OR, 38(12):1747-1759, 2011.
Gerard M. Campbell. A two-stage stochastic program for scheduling and allocating cross-trained workers. JORS, 62(6):1038-1047, 2011.
Sivadon Chaisiri, Bu-Sung Lee, and Dusit Niyato. Optimization of resource provisioning cost in cloud computing. IEEE Trans. Serv. Comput., 5(2):164-177, 2012.
Moses Charikar, Chandra Chekuri, and Martin Pál. Sampling bounds for stochastic optimization. In Proc. of APPROX and RANDOM 2005, pages 257-269, 2005.
Lin Chen, Nicole Megow, Roman Rischke, Leen Stougie, and José Verschae. Optimal algorithms and a PTAS for cost-aware scheduling. To appear in Proc. of MFCS 2015, 2015.
Kedar Dhamdhere, Vineet Goyal, R. Ravi, and Mohit Singh. How to pay, come what may: Approximation algorithms for demand-robust covering problems. In Proc. of FOCS 2005, pages 367-376, 2005.
Shane Dye, Leen Stougie, and Asgeir Tomasgard. The stochastic single resource service-provision problem. Naval Res. Logist., 50(8):869-887, 2003.
Uriel Feige, Kamal Jain, Mohammad Mahdian, and Vahab S. Mirrokni. Robust combinatorial optimization with exponential scenarios. In Proc. of IPCO 2007, pages 439-453, 2007.
Michel X. Goemans. Improved approximation algorthims for scheduling with release dates. In Proc. of SODA 1997, pages 591-598, 1997.
Anupam Gupta, Martin Pál, R. Ravi, and Amitabh Sinha. Sampling and cost-sharing: Approximation algorithms for stochastic optimization problems. SIAM J. Comput., 40(5):1361-1401, 2011.
Rohit Khandekar, Guy Kortsarz, Vahab S. Mirrokni, and Mohammad R. Salavatipour. Two-stage robust network design with exponential scenarios. Algorithmica, 65(2):391-408, 2013.
Anton J. Kleywegt, Alexander Shapiro, and Tito Homem-de Mello. The sample average approximation method for stochastic discrete optimization. SIAM J. Optim., 12(2):479-502, 2001.
Janardhan Kulkarni and Kamesh Munagala. Algorithms for cost aware scheduling. In Proc. of WAOA 2012, pages 201-214, 2013.
Eugene L. Lawler and Jacques Labetoulle. On preemptive scheduling of unrelated parallel processors by linear programming. J. ACM, 25(4):612-619, 1978.
Stefano Leonardi, Nicole Megow, Roman Rischke, Leen Stougie, Chaitanya Swamy, and José Verschae. Scheduling with time-varying cost: Deterministic and stochastic models. Presentation at the 11th Workshop on Models and Algorithms for Planning and Scheduling Problems (MAPSP 2013), 2013.
Michael L. Pinedo. Scheduling: Theory, Algorithms, and Systems. Springer, 2008.
Maurice Queyranne and Maxim Sviridenko. A (2+ε)-approximation algorithm for the generalized preemptive open shop problem with minsum objective. J. Algorithms, 45(2):202-212, 2002.
Andreas S. Schulz and Martin Skutella. Random-based scheduling: New approximations and LP lower bounds. In Proc. of APPROX and RANDOM 1997, pages 119-133, 1997.
Andreas S. Schulz and Martin Skutella. Scheduling unrelated machines by randomized rounding. SIAM J. Discrete Math., 15(4):450-469, 2002.
David B. Shmoys and Mauro Sozio. Approximation algorithms for 2-stage stochastic scheduling problems. In Proc. of IPCO 2007, pages 145-157. Springer, 2007.
David B. Shmoys and Chaitanya Swamy. An approximation scheme for stochastic linear programming and its application to stochastic integer programs. J. ACM, 53(6):978-1012, 2006.
René A. Sitters. Approximability of average completion time scheduling on unrelated machines. In Proc. of ESA 2008, pages 768-779, 2008.
Martin Skutella. List scheduling in order of α-points on a single machine. In Evripidis Bampis, Klaus Jansen, and Claire Kenyon, editors, Efficient Approximation and Online Algorithms, volume 3484 of LNCS, pages 250-291. Springer, 2006.
Chaitanya Swamy and David B. Shmoys. Approximation algorithms for 2-stage stochastic optimization problems. SIGACT News, 37(1):33-46, 2006.
Chaitanya Swamy and David B. Shmoys. Sampling-based approximation algorithms for multistage stochastic optimization. SIAM J. Comput., 41(4):975-1004, 2012.
G. Wan and X. Qi. Scheduling with variable time slot costs. Naval Research Logistics, 57:159-171, 2010.
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On Approximating Node-Disjoint Paths in Grids
In the Node-Disjoint Paths (NDP) problem, the input is an undirected n-vertex graph G, and a collection {(s_1,t_1),...,(s_k,t_k)} of pairs of vertices called demand pairs. The goal is to route the largest possible number of the demand pairs (s_i,t_i), by selecting a path connecting each such pair, so that the resulting paths are node-disjoint. NDP is one of the most basic and extensively studied routing problems. Unfortunately, its approximability is far from being well-understood: the best current upper bound of O(sqrt(n)) is achieved via a simple greedy algorithm, while the best current lower bound on its approximability is Omega(log^{1/2-\delta}(n)) for any constant delta. Even for seemingly simpler special cases, such as planar graphs, and even grid graphs, no better approximation algorithms are currently known. A major reason for this impasse is that the standard technique for designing approximation algorithms for routing problems is LP-rounding of the standard multicommodity flow relaxation of the problem, whose integrality gap for NDP is Omega(sqrt(n)) even on grid graphs.
Our main result is an O(n^{1/4} * log(n))-approximation algorithm for NDP on grids. We distinguish between demand pairs with both vertices close to the grid boundary, and pairs where at least one of the two vertices is far from the grid boundary. Our algorithm shows that when all demand pairs are of the latter type, the integrality gap of the multicommodity flow LP-relaxation is at most O(n^{1/4} * log(n)), and we deal with demand pairs of the former type by other methods. We complement our upper bounds by proving that NDP is APX-hard on grid graphs.
Node-disjoint paths
approximation algorithms
routing and layout
187-211
Regular Paper
Julia
Chuzhoy
Julia Chuzhoy
David H. K.
Kim
David H. K. Kim
10.4230/LIPIcs.APPROX-RANDOM.2015.187
Alok Aggarwal, Jon Kleinberg, and David P. Williamson. Node-disjoint paths on the mesh and a new trade-off in VLSI layout. SIAM J. Comput., 29(4):1321-1333, February 2000.
M. Ajtai, V. Chvátal, M.M. Newborn, and E. Szemerédi. Crossing-free subgraphs. In Gert Sabidussi Peter L. Hammer, Alexander Rosa and Jean Turgeon, editors, Theory and Practice of Combinatorics A collection of articles honoring Anton Kotzig on the occasion of his sixtieth birthday, volume 60 of North-Holland Mathematics Studies, pages 9-12. North-Holland, 1982.
Matthew Andrews. Approximation algorithms for the edge-disjoint paths problem via Raecke decompositions. In Proceedings of IEEE FOCS, pages 277-286, 2010.
Matthew Andrews, Julia Chuzhoy, Venkatesan Guruswami, Sanjeev Khanna, Kunal Talwar, and Lisa Zhang. Inapproximability of edge-disjoint paths and low congestion routing on undirected graphs. Combinatorica, 30(5):485-520, 2010.
Matthew Andrews and Lisa Zhang. Hardness of the undirected edge-disjoint paths problem. In STOC, pages 276-283. ACM, 2005.
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45:501-555, May 1998.
Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: a new characterization of np. J. ACM, 45:70-122, January 1998.
Yonatan Aumann and Yuval Rabani. Improved bounds for all optical routing. In Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms, SODA'95, pages 567-576, Philadelphia, PA, USA, 1995. Society for Industrial and Applied Mathematics.
Chandra Chekuri and Julia Chuzhoy. Half-integral all-or-nothing flow. Unpublished Manuscript.
Chandra Chekuri and Alina Ene. Poly-logarithmic approximation for maximum node disjoint paths with constant congestion. In Proc. of ACM-SIAM SODA, 2013.
Chandra Chekuri, Sanjeev Khanna, and F Bruce Shepherd. Edge-disjoint paths in planar graphs. In Foundations of Computer Science, 2004. Proceedings. 45th Annual IEEE Symposium on, pages 71-80. IEEE, 2004.
Chandra Chekuri, Sanjeev Khanna, and F. Bruce Shepherd. Multicommodity flow, well-linked terminals, and routing problems. In Proc. of ACM STOC, pages 183-192, 2005.
Chandra Chekuri, Sanjeev Khanna, and F. Bruce Shepherd. An O(√ n) approximation and integrality gap for disjoint paths and unsplittable flow. Theory of Computing, 2(1):137-146, 2006.
Julia Chuzhoy. Routing in undirected graphs with constant congestion. In Proc. of ACM STOC, pages 855-874, 2012.
Julia Chuzhoy and Shi Li. A polylogarithimic approximation algorithm for edge-disjoint paths with congestion 2. In Proc. of IEEE FOCS, 2012.
M. Cutler and Y. Shiloach. Permutation layout. Networks, 8:253-278, 1978.
R. Karp. On the complexity of combinatorial problems. Networks, 5:45-68, 1975.
Ken-Ichi Kawarabayashi and Yusuke Kobayashi. An o(log n)-approximation algorithm for the edge-disjoint paths problem in eulerian planar graphs. ACM Trans. Algorithms, 9(2):16:1-16:13, March 2013.
Jon M. Kleinberg. An approximation algorithm for the disjoint paths problem in even-degree planar graphs. In FOCS'05, pages 627-636, 2005.
Jon M. Kleinberg and Éva Tardos. Disjoint paths in densely embedded graphs. In Proceedings of the 36th Annual Symposium on Foundations of Computer Science, pages 52-61, 1995.
Jon M. Kleinberg and Éva Tardos. Approximations for the disjoint paths problem in high-diameter planar networks. J. Comput. Syst. Sci., 57(1):61-73, 1998.
Stavros G. Kolliopoulos and Clifford Stein. Approximating disjoint-path problems using packing integer programs. Mathematical Programming, 99:63-87, 2004.
Frank Thomson Leighton. Complexity Issues in VLSI: Optimal Layouts for the Shuffle-exchange Graph and Other Networks. MIT Press, Cambridge, MA, USA, 1983.
Harald Räcke. Minimizing congestion in general networks. In Proc. of IEEE FOCS, pages 43-52, 2002.
Prabhakar Raghavan and Clark D. Tompson. Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica, 7:365-374, December 1987.
Satish Rao and Shuheng Zhou. Edge disjoint paths in moderately connected graphs. SIAM J. Comput., 39(5):1856-1887, 2010.
N. Robertson and P. D. Seymour. Outline of a disjoint paths algorithm. In Paths, Flows and VLSI-Layout. Springer-Verlag, 1990.
Neil Robertson and Paul D. Seymour. Graph minors. VII. disjoint paths on a surface. J. Comb. Theory, Ser. B, 45(2):212-254, 1988.
Neil Robertson and Paul D Seymour. Graph minors. XIII. the disjoint paths problem. Journal of Combinatorial Theory, Series B, 63(1):65-110, 1995.
Loïc Seguin-Charbonneau and F. Bruce Shepherd. Maximum edge-disjoint paths in planar graphs with congestion 2. In Proceedings of the 2011 IEEE 52Nd Annual Symposium on Foundations of Computer Science, FOCS'11, pages 200-209, Washington, DC, USA, 2011. IEEE Computer Society.
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Approximating Upper Degree-Constrained Partial Orientations
In the Upper Degree-Constrained Partial Orientation (UDPO) problem we are given an undirected graph G=(V,E), together with two degree constraint functions d^-,d^+:V -> N. The goal is to orient as many edges as possible, in such a way that for each vertex v in V the number of arcs entering v is at most d^-(v), whereas the number of arcs leaving v is at most d^+(v). This problem was introduced by Gabow [SODA'06], who proved it to be MAXSNP-hard (and thus APX-hard). In the same paper Gabow presented an LP-based iterative rounding 4/3-approximation algorithm.
As already observed by Gabow, the problem in question is a special case of the classic 3-Dimensional Matching, which in turn is a special case of the k-Set Packing problem. Back in 2006 the best known polynomial time approximation algorithm for 3-Dimensional Matching was a simple local search by Hurkens and Schrijver [SIDMA'89], the approximation ratio of which is (3+epsilon)/2; hence the algorithm of Gabow was an improvement over the approach brought from the more general problems.
In this paper we show that the UDPO problem when cast as 3-Dimensional Matching admits a special structure, which is obliviously exploited by the known approximation algorithms for k-Set Packing. In fact, we show that already the local-search routine of Hurkens and Schrijver gives (4+epsilon)/3-approximation when used for the instances coming from UDPO. Moreover, the recent approximation algorithm for 3-Set Packing [Cygan, FOCS'13] turns out to be a (5+epsilon)/4-approximation for UDPO. This improves over 4/3 as the best ratio known up to date for UDPO.
graph orientations
degree-constrained orientations
approximation algorithm
local search
212-224
Regular Paper
Marek
Cygan
Marek Cygan
Tomasz
Kociumaka
Tomasz Kociumaka
10.4230/LIPIcs.APPROX-RANDOM.2015.212
Esther M. Arkin and Refael Hassin. On local search for weighted k-set packing. In Rainer Burkard and Gerhard Woeginger, editors, Algorithms - ESA 1997, volume 1284 of LNCS, pages 13-22. Springer Berlin Heidelberg, 1997.
Jørgen Bang-Jensen and Gregory Z. Gutin. Digraphs: theory, algorithms and applications. Springer Monographs in Mathematics. Springer, second edition, 2009.
Piotr Berman. A d/2 approximation for maximum weight independent set in d-claw free graphs. In Magnús M. Halldórsson, editor, Algorithm Theory - SWAT 2000, volume 1851 of LNCS, pages 214-219. Springer Berlin Heidelberg, 2000.
Piotr Berman and Marek Karpinski. Improved approximation lower bounds on small occurrence optimization. Electronic Colloquium on Computational Complexity (ECCC), 10(008), 2003.
Yuk Hei Chan and Lap Chi Lau. On linear and semidefinite programming relaxations for hypergraph matching. Mathematical Programming, 135(1-2):123-148, 2012.
Barun Chandra and Magnús M. Halldórsson. Greedy local improvement and weighted set packing approximation. Journal of Algorithms, 39(2):223-240, 2001.
Marek Cygan. Improved approximation for 3-dimensional matching via bounded pathwidth local search. In 54th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 509-518. IEEE Computer Society, 2013.
Marek Cygan, Fabrizio Grandoni, and Monaldo Mastrolilli. How to sell hyperedges: The hypermatching assignment problem. In 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 342-351. SIAM, 2013.
Harold N. Gabow. Upper degree-constrained partial orientations. In 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 554-563. SIAM, 2006.
Magnús M. Halldórsson. Approximating discrete collections via local improvements. In 6th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 160-169. SIAM, 1995.
Elad Hazan, Shmuel Safra, and Oded Schwartz. On the complexity of approximating k-dimensional matching. In Sanjeev Arora, Klaus Jansen, José D. P. Rolim, and Amit Sahai, editors, Approximation, Randomization, and Combinatorial Optimization - APPROX-RANDOM 2003, volume 2764 of LNCS, pages 83-97. Springer Berlin Heidelberg, 2003.
Cor A. J. Hurkens and Alexander Schrijver. On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM Journal on Discrete Mathematics, 2(1):68-72, 1989.
Alexander Schrijver. Combinatorial Optimization - Polyhedra and Efficiency, volume 24 of Algorithms and Combinatorics. Springer, 2003.
Maxim Sviridenko and Justin Ward. Large neighborhood local search for the maximum set packing problem. In Fedor V. Fomin, Rūsiņš Freivalds, Marta Kwiatkowska, and David Peleg, editors, Automata, Languages, and Programming - ICALP 2013, volume 7965 of LNCS, pages 792-803. Springer Berlin Heidelberg, 2013.
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Approximating Hit Rate Curves using Streaming Algorithms
A hit rate curve is a function that maps cache size to the proportion of requests that can be served from the cache. (The caching policy and sequence of requests are assumed to be fixed.) Hit rate curves have been studied for decades in the operating system, database and computer architecture communities. They are useful tools for designing appropriate cache sizes, dynamically allocating memory between competing caches, and for summarizing locality properties of the request sequence. In this paper we focus on the widely-used LRU caching policy.
Computing hit rate curves is very efficient from a runtime standpoint, but existing algorithms are not efficient in their space usage. For a stream of m requests for n cacheable objects, all existing algorithms that provably compute the hit rate curve use space linear in n. In the context of modern storage systems, n can easily be in the billions or trillions, so the space usage of these algorithms makes them impractical.
We present the first algorithm for provably approximating hit rate curves for the LRU policy with sublinear space. Our algorithm uses O( p^2 * log(n) * log^2(m) / epsilon^2 ) bits of space and approximates the hit rate curve at p uniformly-spaced points to within additive error epsilon. This is not far from optimal. Any single-pass algorithm with the same guarantees must use Omega(p^2 + epsilon^{-2} + log(n)) bits of space. Furthermore, our use of additive error is necessary. Any single-pass algorithm achieving multiplicative error requires Omega(n) bits of space.
Cache analysis
hit rate curves
miss rate curves
streaming algorithms
225-241
Regular Paper
Zachary
Drudi
Zachary Drudi
Nicholas J. A.
Harvey
Nicholas J. A. Harvey
Stephen
Ingram
Stephen Ingram
Andrew
Warfield
Andrew Warfield
Jake
Wires
Jake Wires
10.4230/LIPIcs.APPROX-RANDOM.2015.225
George S. Almási, Călin Caşcaval, and David A. Padua. Calculating stack distances efficiently. In Proceedings of the 2002 workshop on memory system performance (MSP'02), pages 37-43, 2002.
Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 20-29. ACM, 1996.
Ziv Bar-Yossef, TS Jayram, Ravi Kumar, D Sivakumar, and Luca Trevisan. Counting distinct elements in a data stream. In Randomization and Approximation Techniques in Computer Science, pages 1-10. Springer, 2002.
L. A. Belady. A study of replacement algorithms for a virtual-storage computer. IBM Systems Journal, 5(2):78-101, 1966.
Brian T Bennett and Vincent J. Kruskal. LRU stack processing. IBM Journal of Research and Development, 19(4):353-357, 1975.
Hjortur Bjornsson, Gregory Chockler, Trausti Saemundsson, and Ymir Vigfusson. Dynamic performance profiling of cloud caches. In Proceedings of the 4th annual Symposium on Cloud Computing (SoCC). ACM, 2013.
Vladimir Braverman and Rafail Ostrovsky. Smooth histograms for sliding windows. In Foundations of Computer Science, 2007. Proceedings. 48th Annual IEEE Symposium on, pages 283-293. IEEE, 2007.
Amit Chakrabarti and Oded Regev. An optimal lower bound on the communication complexity of gap-hamming-distance. SIAM Journal on Computing, 41(5):1299-1317, 2012.
Mayur Datar, Aristides Gionis, Piotr Indyk, and Rajeev Motwani. Maintaining stream statistics over sliding windows. SIAM J. Comput., 31(6):1794-1813, 2002.
Chen Ding and Yutao Zhong. Predicting whole-program locality through reuse distance analysis. In PLDI, pages 245-257. ACM, 2003.
Zachary Drudi. A streaming algorithms approach to approximating hit rate curves. Master’s thesis, University of British Columbia, 2014.
Marianne Durand and Philippe Flajolet. Loglog counting of large cardinalities. In Algorithms-ESA 2003, pages 605-617. Springer, 2003.
David Eklov and Erik Hagersten. StatStack: Efficient modeling of LRU caches. In Performance Analysis of Systems & Software (ISPASS), 2010 IEEE International Symposium on, pages 55-65. IEEE, 2010.
Philippe Flajolet, Éric Fusy, Olivier Gandouet, and Frédéric Meunier. HyperLogLog: the analysis of a near-optimal cardinality estimation algorithm. DMTCS Proceedings, 0(1), 2008.
Sumit Ganguly, Minos Garofalakis, and Rajeev Rastogi. Tracking set-expression cardinalities over continuous update streams. The VLDB Journal, 13(4):354-369, 2004.
Stephen T. Jones, Andrea C. Arpaci-Dusseau, and Remzi H. Arpaci-Dusseau. Geiger: Monitoring the buffer cache in a virtual machine environment. In ASPLOS, pages 14-24. ACM, 2006.
Daniel M Kane, Jelani Nelson, and David P Woodruff. An optimal algorithm for the distinct elements problem. In Proceedings of the twenty-ninth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, pages 41-52. ACM, 2010.
E Kushilevitz and N Nisan. Communication complexity, 1997.
Richard L. Mattson, Jan Gecsei, Donald R. Slutz, and Irving L. Traiger. Evaluation techniques for storage hierarchies. IBM Systems Journal, 9(2):78-117, 1970.
Nimrod Megiddo and Dharmendra S Modha. ARC: A self-tuning, low overhead replacement cache. In FAST, volume 3, pages 115-130, 2003.
Qingpeng Niu, James Dinan, Qingda Lu, and P Sadayappan. Parda: A fast parallel reuse distance analysis algorithm. In Parallel & Distributed Processing Symposium (IPDPS), 2012 IEEE 26th International, pages 1284-1294. IEEE, 2012.
Frank Olken. Efficient methods for calculating the success function of fixed space replacement policies. Master’s thesis, University of California, Berkeley, 1981.
Alexander A. Razborov. On the distributional complexity of disjointness. Theoretical Computer Science, 106(2):385-390, 1992.
Xipeng Shen, Yutao Zhong, and Chen Ding. Locality phase prediction. In ASPLOS, pages 165-176. ACM, 2004.
Alan Jay Smith. Two methods for the efficient analysis of memory address trace data. Software Engineering, IEEE Transactions on, 3(1):94-101, 1977.
Gokul Soundararajan, Daniel Lupei, Saeed Ghanbari, Adrian Daniel Popescu, Jin Chen, and Cristiana Amza. Dynamic resource allocation for database servers running on virtual storage. In FAST. USENIX, 2009.
Harold S Stone, John Turek, and Joel L. Wolf. Optimal partitioning of cache memory. Computers, IEEE Transactions on, 41(9):1054-1068, 1992.
Carl A. Waldspurger, Nohhyun Park, Alexander Garthwaite, and Irfan Ahmad. Efficient MRC construction with SHARDS. In Proceedings of the 13th USENIX Conference on File and Storage Technologies (FAST'15), pages 95-110. USENIX, 2015.
Jake Wires, Stephen Ingram, Zachary Drudi, Nicholas J. A. Harvey, and Andrew Warfield. Characterizing storage workloads with counter stacks. In OSDI, 2014.
Ting Yang, Emery D. Berger, Scott F. Kaplan, and J. Eliot B. Moss. CRAMM: Virtual memory support for garbage-collected applications. In OSDI, pages 103-116. ACM, 2006.
Yutao Zhong, Maksim Orlovich, Xipeng Shen, and Chen Ding. Array regrouping and structure splitting using whole-program reference affinity. In PLDI, pages 255-266. ACM, 2004.
Pin Zhou, Vivek Pandey, Jagadeesan Sundaresan, Anand Raghuraman, Yuanyuan Zhou, and Sanjeev Kumar. Dynamic tracking of page miss ratio curve for memory management. In ASPLOS, pages 177-188. ACM, 2004.
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Terminal Embeddings
In this paper we study terminal embeddings, in which one is given a finite metric (X,d_X) (or a graph G=(V,E)) and a subset K of X of its points are designated as terminals. The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve approx |K| * |X| pairs, the distortion depends only on |K|, rather than on |X|.
We also strengthen this notion, and consider embeddings that approximately preserve the distances between all pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to X \times X and with respect to K * X.
Moreover, our embeddings have implications to the areas of Approximation and Online Algorithms. In particular, Arora et. al. devised an ~O(sqrt(log(r))-approximation algorithm for sparsest-cut instances with r demands. Building on their framework, we provide an ~O(sqrt(log |K|)-approximation for sparsest-cut instances in which each demand is incident on one of the vertices of K (aka, terminals). Since |K| <= r, our bound generalizes that of Arora et al.
embedding
distortion
terminals
242-264
Regular Paper
Michael
Elkin
Michael Elkin
Arnold
Filtser
Arnold Filtser
Ofer
Neiman
Ofer Neiman
10.4230/LIPIcs.APPROX-RANDOM.2015.242
Ittai Abraham, Yair Bartal, and Ofer Neiman. Embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion. In SODA, pages 502-511, 2007.
Ittai Abraham, Yair Bartal, and Ofer Neiman. Advances in metric embedding theory. Advances in Mathematics, 228(6):3026-3126, 2011.
Ittai Abraham and Ofer Neiman. Using petal-decompositions to build a low stretch spanning tree. In STOC, pages 395-406, 2012.
Noga Alon, Richard M. Karp, David Peleg, and Douglas West. A graph-theoretic game and its application to the k-server problem. SIAM J. Comput., 24(1):78-100, 1995.
I. Altḧofer, G. Das, D. P. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81-100, 1993.
Reid Andersen and Uriel Feige. Interchanging distance and capacity in probabilistic mappings, 2009.
S. Arora, J. R. Lee, and A. Naor. Euclidean distortion and the sparsest cut. Journal of the American Mathematical Society 21, 1:1-21, 2008.
Sanjeev Arora, James R. Lee, and Assaf Naor. Fréchet embeddings of negative type metrics. Discrete & Computational Geometry, 38(4):726-739, 2007.
Sanjeev Arora, Satish Rao, and Umesh V. Vazirani. Expander flows, geometric embeddings and graph partitioning. J. ACM, 56(2), 2009.
Yonatan Aumann and Yuval Rabani. An O(log k) approximate min-cut max-flow theorem and approximation algorithm. SIAM Journal on Computing, 27(1):291-301, 1998.
B. Awerbuch, A. Baratz, and D. Peleg. Efficient broadcast and light-weight spanners. Technical Report CS92-22, The Weizmann Institute of Science, Rehovot, Israel., 1992.
Yair Bartal. Probabilistic approximations of metric spaces and its algorithmic applications. In FOCS, pages 184-193, 1996.
Yair Bartal, Amos Fiat, and Yuval Rabani. Competitive algorithms for distributed data management. J. Comput. Syst. Sci., 51(3):341-358, 1995.
Yair Bartal, Nathan Linial, Manor Mendel, and Assaf Naor. Some low distortion metric ramsey problems. Discrete & Computational Geometry, 33(1):27-41, 2005.
J. Bourgain. On lipschitz embedding of finite metric spaces in hilbert space. Israel Journal of Mathematics, 52(1-2):46-52, 1985.
T.-H. Chan, Donglin Xia, Goran Konjevod, and Andrea Richa. A tight lower bound for the steiner point removal problem on trees. In Proceedings of the 9th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th International Conference on Randomization and Computation, APPROX'06/RANDOM'06, pages 70-81, Berlin, Heidelberg, 2006. Springer-Verlag.
Moses Charikar, Tom Leighton, Shi Li, and Ankur Moitra. Vertex sparsifiers and abstract rounding algorithms. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 265-274, 2010.
Don Coppersmith and Michael Elkin. Sparse source-wise and pair-wise distance preservers. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'05, pages 660-669, Philadelphia, PA, USA, 2005. Society for Industrial and Applied Mathematics.
Michael Elkin, Yuval Emek, Daniel A. Spielman, and Shang-Hua Teng. Lower-stretch spanning trees. SIAM Journal on Computing, 38(2):608-628, 2008.
Michael Elkin, Arnold Filtser, and Ofer Neiman. Prioritized metric structures and embedding. In Proceedings of the 47th ACM Symposium on Theory of Computing, STOC'15, 2015.
Michael Elkin and Shay Solomon. Steiner shallow-light trees are exponentially lighter than spanning ones. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 373-382, 2011.
Matthias Englert, Anupam Gupta, Robert Krauthgamer, Harald Räcke, Inbal Talgam-Cohen, and Kunal Talwar. Vertex sparsifiers: New results from old techniques. SIAM J. Comput., 43(4):1239-1262, 2014.
Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci., 69(3):485-497, 2004.
T. Figiel, J. Lindenstrauss, and V.D. Milman. The dimension of almost spherical sections of convex bodies. Acta Mathematica, 139(1):53-94, 1977.
E. N. Gilbert and H. O. Pollak. Steiner minimal trees. SIAM Journal on Applied Mathematics, 16(1):1-29, Jan 1968.
Anupam Gupta. Steiner points in tree metrics don't (really) help. In Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'01, pages 220-227, Philadelphia, PA, USA, 2001. Society for Industrial and Applied Mathematics.
Anupam Gupta, Viswanath Nagarajan, and R. Ravi. An improved approximation algorithm for requirement cut. Oper. Res. Lett., 38(4):322-325, 2010.
Sariel Har-Peled, Piotr Indyk, and Anastasios Sidiropoulos. Euclidean spanners in high dimensions. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'13, pages 804-809. SIAM, 2013.
William Johnson and Joram Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. Contemporary Mathematics, 26:189–206, 1984.
Lior Kamma, Robert Krauthgamer, and Huy L. Nguyen. Cutting corners cheaply, or how to remove steiner points. In SODA, pages 1029-1040, 2014.
Telikepalli Kavitha and Nithin M. Varma. Small stretch pairwise spanners. In Proceedings of the 40th International Conference on Automata, Languages, and Programming - Volume Part I, ICALP'13, pages 601-612, Berlin, Heidelberg, 2013. Springer-Verlag.
Samir Khuller, Balaji Raghavachari, and Neal E. Young. Balancing minimum spanning trees and shortest-path trees. Algorithmica, 14(4):305-321, 1995.
Robert Krauthgamer, James R. Lee, Manor Mendel, and Assaf Naor. Measured descent: a new embedding method for finite metrics. Geometric and Functional Analysis, 15(4):839-858, 2005.
Frank Thomson Leighton and Satish Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM, 46(6):787-832, 1999.
N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995.
Manor Mendel and Assaf Naor. Ramsey partitions and proximity data structures. In FOCS, pages 109-118, 2006.
Ankur Moitra. Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size. In FOCS, pages 3-12, 2009.
Yuri Rabinovich and Ran Raz. Lower bounds on the distortion of embedding finite metric spaces in graphs. Discrete & Computational Geometry, 19(1):79-94, 1998.
Harald Räcke. Minimizing congestion in general networks. In Proceedings of the 43rd Symposium on Foundations of Computer Science, FOCS'02, pages 43-52, Washington, DC, USA, 2002. IEEE Computer Society.
Harald Räcke. Optimal hierarchical decompositions for congestion minimization in networks. In STOC, pages 255-264, 2008.
Liam Roditty, Mikkel Thorup, and Uri Zwick. Deterministic constructions of approximate distance oracles and spanners. In Proceedings of the 32Nd International Conference on Automata, Languages and Programming, ICALP'05, pages 261-272, Berlin, Heidelberg, 2005. Springer-Verlag.
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On Linear Programming Relaxations for Unsplittable Flow in Trees
We study some linear programming relaxations for the Unsplittable Flow problem on trees (UFP-Tree). Inspired by results obtained by Chekuri, Ene, and Korula for Unsplittable Flow on paths (UFP-Path), we present a relaxation with polynomially many constraints that has an integrality gap bound of O(log n * min(log m, log n)) where n denotes the number of tasks and m denotes the number of edges in the tree. This matches the approximation guarantee of their combinatorial algorithm and is the first demonstration of an efficiently-solvable relaxation for UFP-Tree with a sub-linear integrality gap.
The new constraints in our LP relaxation are just a few of the (exponentially many) rank constraints that can be added to strengthen the natural relaxation. A side effect of how we prove our upper bound is an efficient O(1)-approximation for solving the rank LP. We also show that our techniques can be used to prove integrality gap bounds for similar LP relaxations for packing demand-weighted subtrees of an edge-capacitated tree.
On the other hand, we show that the inclusion of all rank constraints does not reduce the integrality gap for UFP-Tree to a constant. Specifically, we show the integrality gap is Omega(sqrt(log n)) even in cases where all tasks share a common endpoint. In contrast, intersecting instances of UFP-Path are known to have an integrality gap of O(1) even if just a few of the rank 1 constraints are included.
We also observe that applying two rounds of the Lovász-Schrijver SDP procedure to the natural LP for UFP-Tree derives an SDP whose integrality gap is also O(log n * min(log m, log n)).
Unsplittable flow
Linear programming relaxation
Approximation algorithm
265-283
Regular Paper
Zachary
Friggstad
Zachary Friggstad
Zhihan
Gao
Zhihan Gao
10.4230/LIPIcs.APPROX-RANDOM.2015.265
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Inapproximability of H-Transversal/Packing
Given an undirected graph G=(V,E) and a fixed pattern graph H with k vertices, we consider the H-Transversal and H-Packing problems. The former asks to find the smallest subset S of vertices such that the subgraph induced by V - S does not have H as a subgraph, and the latter asks to find the maximum number of pairwise disjoint k-subsets S1, ..., Sm such that the subgraph induced by each Si has H as a subgraph.
We prove that if H is 2-connected, H-Transversal and H-Packing are almost as hard to approximate as general k-Hypergraph Vertex Cover and k-Set Packing, so it is NP-hard to approximate them within a factor of Omega(k) and Omega(k / polylog(k)) respectively. We also show that there is a 1-connected H where H-Transversal admits an O(log k)-approximation algorithm, so that the connectivity requirement cannot be relaxed from 2 to 1. For a special case of H-Transversal where H is a (family of) cycles, we mention the implication of our result to the related Feedback Vertex Set problem, and give a different hardness proof for directed graphs.
Constraint Satisfaction Problems
Approximation resistance
284-304
Regular Paper
Venkatesan
Guruswami
Venkatesan Guruswami
Euiwoong
Lee
Euiwoong Lee
10.4230/LIPIcs.APPROX-RANDOM.2015.284
Matthew Andrews, Julia Chuzhoy, Venkatesan Guruswami, Sanjeev Khanna, Kunal Talwar, and Lisa Zhang. Inapproximability of edge-disjoint paths and low congestion routing on undirected graphs. Combinatorica, 30(5):485-520, 2010.
Matthew Andrews and Lisa Zhang. Logarithmic hardness of the undirected edge-disjoint paths problem. Journal of the ACM, 53(5):745-761, 2006.
P. Austrin, S. Khot, and M. Safra. Inapproximability of vertex cover and independent set in bounded degree graphs. In Proceedings of the 24th annual IEEE Conference on Computational Complexity, CCC'09, pages 74-80, 2009.
V. Bafna, P. Berman, and T. Fujito. Constant ratio approximations of the weighted feedback vertex set problem for undirected graphs. In Proceedings of the 6th International Symposium on Algorithms and Computation, ISAAC'95, pages 142-151, 1995.
N. Bansal and S. Khot. Inapproximability of hypergraph vertex cover and applications to scheduling problems. In Proceedings of the 37th International Colloquium on Automata, Languages and Programming, ICALP'10, pages 250-261, 2010.
R. Bar-Yehuda, D. Geiger, J. Naor, and R. 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.
A. Becker and D. Geiger. Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artificial Intelligence, 83(1):167-188, 1996.
H. L. Bodlaender. On disjoint cycles. International Journal of Foundations of Computer Science, 5(01):59-68, 1994.
A. Brandstädt, V. Chepoi, and F. Dragan. Clique r-domination and clique r-packing problems on dually chordal graphs. SIAM J. on Discrete Mathematics, 10(1):109-127, 1997.
Parinya Chalermsook, Bundit Laekhanukit, and Danupon Nanongkai. Pre-reduction graph products: Hardnesses of properly learning DFAs and approximating EDP on DAGs. In Proc. of the 55rd Annual Symp. on Foundations of Computer Science (FOCS'14). IEEE, 2014.
S. Chan. Approximation resistance from pairwise independent subgroups. In Proc. of the 45th Annual ACM Symp. on Symposium on Theory of Computing, STOC'13, pages 447-456, 2013.
Maw-Shang Chang, Ton Kloks, and Chuan-Min Lee, editors. Graph-Theoretic Concepts in Computer Science, volume 2204 of Lecture Notes in Computer Science. Springer, 2001.
F. Chataigner, G. Manić, Y. Wakabayashi, and R. Yuster. Approximation algorithms and hardness results for the clique packing problem. Discrete Applied Mathematics, 157(7):1396-1406, 2009.
J. Chen, Y. Liu, S. Lu, B. O’sullivan, and I. Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. Journal of the ACM, 55(5):21:1-21:19, 2008.
R. Chitnis, M. Cygan, M. Hajiaghayi, and D. Marx. Directed subset feedback vertex set is fixed-parameter tractable. In Proceedings of the 39th International Colloquium on Automata, Languages, and Programming, ICALP'12, pages 230-241, 2012.
Miroslav Chlebík and Janka Chlebíková. Approximation hardness of dominating set problems in bounded degree graphs. Information and Computation, 206(11):1264-1275, 2008.
F. A. Chudak, M. X. Goemans, D. S. Hochbaum, and D. P. Williamson. A primal-dual interpretation of two 2-approximation algorithms for the feedback vertex set problem in undirected graphs. Operations Research Letters, 22(4–5):111-118, 1998.
M. Cygan, M. Pilipczuk, M. Pilipczuk, and J. O. Wojtaszczyk. Subset feedback vertex set is fixed-parameter tractable. In Proceedings of the 38th International Colloquim on Automata, Languages and Programming, ICALP'11, pages 449-461, 2011.
Marek Cygan. Improved approximation for 3-dimensional matching via bounded pathwidth local search. In Proceedings of the 54th annual IEEE symposium on Foundations of Computer Science, FOCS'13, pages 509-518, 2013.
I. Dinur and S. Safra. On the hardness of approximating minimum vertex cover. Annals of Mathematics, pages 439-485, 2005.
Irit Dinur, Venkatesan Guruswami, Subhash Khot, and Oded Regev. A new multilayered PCP and the hardness of hypergraph vertex cover. SIAM Journal on Computing, 34(5):1129-1146, 2005.
D. Dor and M. Tarsi. Graph decomposition is NP-complete: A complete proof of Holyer’s conjecture. SIAM Journal on Computing, 26(4):1166-1187, 1997.
Guillermo Durán, Min Chih Lin, Sergio Mera, and Jayme L Szwarcfiter. Algorithms for finding clique-transversals of graphs. Annals of Operations Research, 157(1):37-45, 2008.
Paul Erdős, Tibor Gallai, and Zsolt Tuza. Covering the cliques of a graph with vertices. Discrete Mathematics, 108(1):279-289, 1992.
G. Even, J. Naor, S. Rao, and B. Schieber. Divide-and-conquer approximation algorithms via spreading metrics. Journal of the ACM, 47(4):585-616, 2000.
G. Even, J. Naor, B. Schieber, and M. Sudan. Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica, 20:151-174, 1998.
Tomas Feder and Carlos Subi. Packing edge-disjoint triangles in given graphs. Electronic Colloquium on Computational Complexity (ECCC), 2012. TR12-013.
Z. Friggstad and M. R. Salavatipour. Approximability of packing disjoint cycles. In Proc. of the 18th Int'l Conf. on Algorithms and Computation, ISAAC'07, pages 304-315, 2007.
V. Guruswami, R. Manokaran, and P. Raghavendra. Beating the random ordering is hard: Inapproximability of maximum acyclic subgraph. In Proceedings of the 49th annual IEEE symposium on Foundations of Computer Science, FOCS'08, pages 573-582, 2008.
Venkatesan Guruswami and Euiwoong Lee. Inapproximability of feedback vertex set for bounded length cycles. Electronic Colloquium on Computational Complexity (ECCC), TR 14-006, 2014.
Venkatesan Guruswami and Euiwoong Lee. Inapproximability of H-transversal/matching. arXiv preprint arXiv:1506.06302, 2015.
Venkatesan Guruswami and C. Pandu Rangan. Algorithmic aspects of clique-transversal and clique-independent sets. Discrete Applied Mathematics, 100(3):183-202, 2000.
Venkatesan Guruswami, C. Pandu Rangan, M. S. Chang, G. J. Chang, and C. K. Wong. The K_r-packing problem. Computing, 66(1):79-89, 2001.
Elad Hazan, Shmuel Safra, and Oded Schwartz. On the complexity of approximating k-set packing. Computational Complexity, 15(1):20-39, 2006.
P. Hell, S. Klein, L.T. Nogueira, and F. Protti. Packing r-cliques in weighted chordal graphs. Annals of Operations Research, 138(1):179-187, 2005.
Bart M. P. Jansen and Dániel Marx. Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and turing kernels. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 616-629, 2015.
Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the 34th annual ACM Symposium on Theory of Computing, STOC'02, pages 767-775, 2002.
Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-ε. Journal of Computer and System Sciences, 74(3):335-349, 2008.
D. Kirkpatrick and P. Hell. On the complexity of general graph factor problems. SIAM Journal on Computing, 12(3):601-609, 1983.
Ton Kloks. Packing interval graphs with vertex-disjoint triangles. arXiv preprint arXiv:1202.1041, 2012.
Guy Kortsarz, Michael Langberg, and Zeev Nutov. Approximating maximum subgraphs without short cycles. SIAM Journal on Discrete Mathematics, 24(1):255-269, 2010.
M. Krivelevich, Z. Nutov, M. R. Salavatipour, J. V. Yuster, and R. Yuster. Approximation algorithms and hardness results for cycle packing problems. ACM Transactions on Algorithms, 3(4), November 2007.
Chuan-Min Lee. Weighted maximum-clique transversal sets of graphs. ISRN Discrete Mathematics, 2011, 2012.
CM Lee, MS Chang, and SC Sheu. The clique transversal and clique independence of distance hereditary graphs. In Proceedings of the 19th Workshop on Combinatorial Mathematics and Computation Theory, Taiwan, pages 64-69, 2002.
Carsten Lund and Mihalis Yannakakis. The approximation of maximum subgraph problems. In Automata, Languages and Programming, volume 700 of Lecture Notes in Computer Science, pages 40-51. Springer, 1993.
Sridhar Rajagopalan and Vijay V. Vazirani. Primal-dual rnc approximation algorithms for set cover and covering integer programs. SIAM J. on Computing, 28(2):525-540, 1998.
Dieter Rautenbach and Friedrich Regen. On packing shortest cycles in graphs. Information Processing Letters, 109(14):816-821, 2009.
P. D. Seymour. Packing directed circuits fractionally. Combinatorica, 15(2):281-288, 1995.
Erfang Shan, Zuosong Liang, and Liying Kang. Clique-transversal sets and clique-coloring in planar graphs. European Journal of Combinatorics, 36:367-376, 2014.
Ola Svensson. Hardness of vertex deletion and project scheduling. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 7408 of Lecture Notes in Computer Science, pages 301-312. Springer, 2012.
Zsolt Tuza. Covering all cliques of a graph. In S.T. Hedetniemi, editor, Topics on Domination, volume 48 of Annals of Discrete Mathematics, pages 117-126. Elsevier, 1991.
R. Yuster. Edge-disjoint cliques in graphs with high minimum degree. SIAM Journal on Discrete Mathematics, 28(2):893-910, 2014.
Raphael Yuster. Combinatorial and computational aspects of graph packing and graph decomposition. Computer Science Review, 1(1):12-26, 2007.
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Towards a Characterization of Approximation Resistance for Symmetric CSPs
A Boolean constraint satisfaction problem (CSP) is called approximation resistant if independently setting variables to 1 with some probability achieves the best possible approximation ratio for the fraction of constraints satisfied. We study approximation resistance of a natural subclass of CSPs that we call Symmetric Constraint Satisfaction Problems (SCSPs), where satisfaction of each constraint only depends on the number of true literals in its scope. Thus a SCSP of arity k can be described by a subset of allowed number of true literals.
For SCSPs without negation, we conjecture that a simple sufficient condition to be approximation resistant by Austrin and Hastad is indeed necessary. We show that this condition has a compact analytic representation in the case of symmetric CSPs (depending only on the gap between the largest and smallest numbers in S), and provide the rationale behind our conjecture. We prove two interesting special cases of the conjecture, (i) when S is an interval and (ii) when S is even. For SCSPs with negation, we prove that the analogous sufficient condition by Austrin and Mossel is necessary for the same two cases, though we do not pose an analogous conjecture in general.
Constraint Satisfaction Problems
Approximation resistance
305-322
Regular Paper
Venkatesan
Guruswami
Venkatesan Guruswami
Euiwoong
Lee
Euiwoong Lee
10.4230/LIPIcs.APPROX-RANDOM.2015.305
Per Austrin, Siavosh Benabbas, and Avner Magen. On quadratic threshold CSPs. Discrete Mathematics & Theoretical Computer Science, 14(2):205-228, 2012.
Per Austrin and Johan Håstad. On the usefulness of predicates. ACM Transactions on Computation Theory, 5(1):1:1-1:24, May 2013.
Per Austrin and Subhash Khot. A characterization of approximation resistance for even k-partite CSPs. In Proceedings of the 4th Conference on Innovations in Theoretical Computer Science, ITCS'13, pages 187-196, 2013.
Per Austrin and Elchanan Mossel. Approximation resistant predicates from pairwise independence. Computational Complexity, 18(2):249-271, 2009.
Boaz Barak, Siu On Chan, and Pravesh Kothari. Sum of squares lower bounds from pairwise independence. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, STOC'15, 2015. To appear.
Siavosh Benabbas, Konstantinos Georgiou, Avner Magen, and Madhur Tulsiani. SDP gaps from pairwise independence. Theory of Computing, 8(1):269-289, 2012.
Siu On Chan. Approximation resistance from pairwise independent subgroups. In Proceedings of the 45th annual ACM Symposium on Theory of Computing, STOC'13, pages 447-456, 2013.
Mahdi Cheraghchi, Johan Håstad, Marcus Isaksson, and Ola Svensson. Approximating linear threshold predicates. ACM Trans. Comput. Theory, 4(1):2:1-2:31, 2012.
Wing-Sum Cheung. Generalizations of Hölder’s inequality. International Journal of Mathematics and Mathematical Sciences, 26(1):7-10, 2001.
G Hast. Beating a random assignment. KTH, Stockholm. PhD thesis, Ph. D Thesis, 2005.
Johan Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, July 2001.
Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the 34th annual ACM Symposium on Theory of Computing, STOC'02, pages 767-775, 2002.
Subhash Khot, Madhur Tulsiani, and Pratik Worah. A characterization of strong approximation resistance. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC'14, pages 634-643, 2014.
Thomas Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th annual ACM Symposium on Theory of Computing, STOC'78, pages 216-226, 1978.
Madhur Tulsiani. CSP gaps and reductions in the Lasserre hierarchy. In Proceedings of the 41st annual ACM Symposium on Theory of Computing, STOC'09, pages 303-312, 2009.
Uri Zwick. Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'98, pages 201-210, 1998.
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Sequential Importance Sampling Algorithms for Estimating the All-Terminal Reliability Polynomial of Sparse Graphs
The all-terminal reliability polynomial of a graph counts its connected subgraphs of various sizes. Algorithms based on sequential importance sampling (SIS) have been proposed to estimate a graph's reliability polynomial. We show upper bounds on the relative error of three sequential importance sampling algorithms. We use these to create a hybrid algorithm, which selects the best SIS algorithm for a particular graph G and particular coefficient of the polynomial.
This hybrid algorithm is particularly effective when G has low degree. For graphs of average degree < 11, it is the fastest known algorithm; for graphs of average degree <= 45 it is the fastest known polynomial-space algorithm. For example, when a graph has average degree 3, this algorithm estimates to error epsilon in time O(1.26^n * epsilon^{-2}).
Although the algorithm may take exponential time, in practice it can have good performance even on medium-scale graphs. We provide experimental results that show quite practical performance on graphs with hundreds of vertices and thousands of edges. By contrast, alternative algorithms are either not rigorous or are completely impractical for such large graphs.
All-terminal reliability
sequential importance sampling
323-340
Regular Paper
David G.
Harris
David G. Harris
Francis
Sullivan
Francis Sullivan
10.4230/LIPIcs.APPROX-RANDOM.2015.323
Michael O. Ball and J. Scott Provan. Bounds on the reliablity polynomial for shellable independence systems. SIAM Journal on Algebraic and Discrete Methods, 3:166-181, 1982.
Isabel Beichl, Brian Cloteaux, and Francis Sullivan. An approximation algorithm for the coefficients of the reliability polynomial. Congressus Numerantium, 197:143-151, 2010.
Andreas Bjorklund, Thore Husfeldt, Petteri Kaski, and Mikko Koivisto. Computing the Tutte polynomial in vertex-exponential time. Foundations of Computer Science (FOCS), pages 677-686, 2008.
Adams L. Buchsbaum and Milena Mihail. Monte Carlo and Markov chain techniques for network reliability and sampling. Networks, 25:117-130, 1995.
Shiri Chechik, Yuval Emek, Boaz Patt-Shamir, and David Peleg. Sparse reliable graph backbones. Automata, Languages, and Processing, pages 261-272, 2010.
Andrew Chen. On graphs with large numbers of spanning trees. PhD dissertation for Michigan State University Department of Computer Science, 2005.
Fan RK Chung and Ronald L. Graham. On the cover polynomial of a digraph. Journal of Combinatorial Theory, Series B, 65:273-290, 1995.
Charles J. Colbourn, Bradley M. Debroni, and Wendy J. Myrold. Estimating the coefficients of the reliability polynomial. Congress Numerantium, 62:217-223, 1988.
Holger Dell, Thore Husfeldt, Dániel Marx, Nina Taslaman, and Martin Wahlén. Exponential time complexity of the permanent and Tutte polynomial. ACM Transactions on Algorithms, 10-4, 2014.
George S. Fishman. A Monte Carlo sampling plan for estimating network reliability. Operations Research, 34:581-594, 1986.
Heidi Gebauer and Yoshio Okamoto. Fast exponential-time algorithms for the forest counting in graph classes. Theory of Computing Australasian Symposium, 65:63-69, 2007.
Leslie Ann Goldberg and Mark Jerrum. Inapproximability of the Tutte polynomial. Information and Computation, 206-7:908-929, 2008.
Gary Haggard, David J. Pearce, and Gordon Royle. Computing Tutte polynomials. ACM Transactions on Mathematical Software, 37-3 Article # 24, 2010.
David G. Harris, Francis Sullivan, and Isabel Beichl. Linear algebra and sequential importance sampling for network reliability. Winter Simulation Conference, 2011.
David G. Harris, Francis Sullivan, and Isabel Beichl. Fast sequential importance sampling to estimate the graph reliability polynomial. Algorithmica, 68-4:916-939, 2014.
David Karger. A randomized fully polynomial time approximation scheme for the all terminal network reliability problem. SIAM Journal on Computing, 29:11-17, 1996.
Joseph B. Kruskal. The number of simplices in a complex. Mathematical Optimization Techniques, 1963.
Marco Laumanns and Rico Zenklusen. High-confidence estimation of small s-t reliabilities in directed acylic networks. Networks, 57-4:367-388, 2011.
Wendy Myrvold. Counting k-component forests of a graph. Networks, 22:647-652, 1992.
J. Scott Provan and Michael O. Ball. The complexity of counting cuts and of computing the probability that a graph is connected. SIAM Journal on Computing, 12:777-788, 1983.
Kyoko Sekine, Hiroshi Imai, and Seiichiro Tani. Computing the Tutte polynomial of a graph of moderate size. Lecture Notes in Computer Science, 1004:224-233, 1995.
David Bruce Wilson. Generating random spanning trees more quickly than the cover time. ACM Symposium on Theory of Computing (STOC), pages 296-303, 1996.
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Improved NP-Inapproximability for 2-Variable Linear Equations
An instance of the 2-Lin(2) problem is a system of equations of the form "x_i + x_j = b (mod 2)". Given such a system in which it's possible to satisfy all but an epsilon fraction of the equations, we show it is NP-hard to satisfy all but a C*epsilon fraction of the equations, for any C < 11/8 = 1.375 (and any 0 < epsilon <= 1/8). The previous best result, standing for over 15 years, had 5/4 in place of 11/8. Our result provides the best known NP-hardness even for the Unique Games problem, and it also holds for the special case of Max-Cut. The precise factor 11/8 is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of 3/2.
Our proof is by a modified gadget reduction from a pairwise-independent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor C greater than 2.54. Previously, no such limitation on gadget reductions was known.
approximability
unique games
linear equation
gadget
linear programming
341-360
Regular Paper
Johan
Håstad
Johan Håstad
Sangxia
Huang
Sangxia Huang
Rajsekar
Manokaran
Rajsekar Manokaran
Ryan
O’Donnell
Ryan O’Donnell
John
Wright
John Wright
10.4230/LIPIcs.APPROX-RANDOM.2015.341
Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. O(√log n) approximation algorithms for Min-Uncut, Min-2CNF-Deletion, and directed cut problems. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 573-581, 2005.
Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for Unique Games and related problems. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, pages 563-572, 2010.
Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pages 222-231, 2004.
Per Austrin and Johan Håstad. On the usefulness of predicates. ACM Trans. Comput. Theory, 5(1):1:1-1:24, 2013.
Mihir Bellare, Oded Goldreich, and Madhu Sudan. Free bits, PCPs, and non-approximability - towards tight results. SIAM Journal of Computing, 27(3):804-915, 1998.
Siu On Chan. Approximation resistance from pairwise independent subgroups. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing, pages 447-456, 2013.
Moses Charikar, Konstantin Makarychev, and Yury Makarychev. Near-optimal algorithms for Unique Games. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 205-214, 2006.
Miroslav Chlebík and Janka Chlebíková. On approximation hardness of the minimum 2SAT-DELETION problem. In Proceedings of the 29th Annual International Symposium on Mathematical Foundations of Computer Science, pages 263-273, 2004.
Irit Dinur and Samuel Safra. On the hardness of approximating minimum vertex cover. Annals of Mathematics, 162(1):439-485, 2005.
Uriel Feige and Daniel Reichman. On systems of linear equations with two variables per equation. In Proceedings of the 7th Annual International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pages 117-127, 2004.
Michel Goemans and David Williamson. A 0.878 approximation algorithm for MAX-2SAT and MAX-CUT. In Proceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 422-431, 1994.
Anupam Gupta, Kunal Talwar, and David Witmer. Sparsest Cut on bounded treewidth graphs: algorithms and hardness results. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing, pages 281-290, 2013.
Johan Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, 2001.
Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 767-775, 2002.
Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for Max-Cut and other 2-variable CSPs? SIAM Journal on Computing, 37(1):319-357, 2007.
Dana Moshkovitz and Ran Raz. Two-query PCP with subconstant error. Journal of the ACM, 57(5):29, 2010.
Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences: invariance and optimality. Annals of Mathematics, 171(1):295-341, 2010.
Ryan O'Donnell and John Wright. A new point of NP-hardness for Unique-Games. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pages 289-306, 2012.
Anup Rao. Parallel repetition in projection games and a concentration bound. SIAM Journal of Computing, 40(6):1871-1891, 2011.
Luca Trevisan, Gregory Sorkin, Madhu Sudan, and David Williamson. Gadgets, approximation, and linear programming. SIAM Journal on Computing, 29(6):2074-2097, 2000.
Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science, 348(2-3):357-365, 2005.
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A Tight Approximation Bound for the Stable Marriage Problem with Restricted Ties
The problem of finding a maximum cardinality stable matching in the presence of ties and unacceptable partners, called MAX SMTI, is a well-studied NP-hard problem. The MAX SMTI is NP-hard even for highly restricted instances where (i) ties appear only in women's preference lists and (ii) each tie appears at the end of each woman's preference list. The current best lower bounds on the approximation ratio for this variant are 1.1052 unless P=NP and 1.25 under the unique games conjecture, while the current best upper bound is 1.4616. In this paper, we improve the upper bound to 1.25, which matches the lower bound under the unique games conjecture. Note that this is the first special case of the MAX SMTI where the tight approximation bound is obtained. The improved ratio is achieved via a new analysis technique, which avoids the complicated case-by-case analysis used in earlier studies. As a by-product of our analysis, we show that the integrality gap of natural IP and LP formulations for this variant is 1.25. We also show that the unrestricted MAX SMTI cannot be approximated with less than 1.5 unless the approximation ratio of a certain special case of the minimum maximal matching problem can be improved.
stable marriage with ties and incomplete lists
approximation algorithm
integer program
linear program relaxation
integrality gap
361-380
Regular Paper
Chien-Chung
Huang
Chien-Chung Huang
Kazuo
Iwama
Kazuo Iwama
Shuichi
Miyazaki
Shuichi Miyazaki
Hiroki
Yanagisawa
Hiroki Yanagisawa
10.4230/LIPIcs.APPROX-RANDOM.2015.361
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Designing Overlapping Networks for Publish-Subscribe Systems
From the publish-subscribe systems of the early days of the Internet to the recent emergence of Web 3.0 and IoT (Internet of Things), new problems arise in the design of networks centered at producers and consumers of constantly evolving information. In a typical problem, each terminal is a source or sink of information and builds a physical network in the form of a tree or an overlay network in the form of a star rooted at itself. Every pair of pub-sub terminals that need to be coordinated (e.g. the source and sink of an important piece of control information) define an edge in a bipartite demand graph; the solution must ensure that the corresponding networks rooted at the endpoints of each demand edge overlap at some node. This simple overlap constraint, and the requirement that each network is a tree or a star, leads to a variety of new questions on the design of overlapping networks.
In this paper, for the general demand case of the problem, we show that a natural LP formulation has a non-constant integrality gap; on the positive side, we present a logarithmic approximation for the general demand case. When the demand graph is complete, however, we design approximation algorithms with small constant performance ratios, irrespective of whether the pub networks and sub networks are required to be trees or stars.
Approximation Algorithms
Steiner Trees
Publish-Subscribe Systems
Integrality Gap
VPN.
381-395
Regular Paper
Jennifer
Iglesias
Jennifer Iglesias
Rajmohan
Rajaraman
Rajmohan Rajaraman
R.
Ravi
R. Ravi
Ravi
Sundaram
Ravi Sundaram
10.4230/LIPIcs.APPROX-RANDOM.2015.381
Ajit Agrawal, Philip N. Klein, and R. Ravi. When trees collide: An approximation algorithm for the generalized steiner problem on networks. SIAM J. Comput., 24(3):440-456, 1995.
Yair Bartal. On approximating arbitrary metrices by tree metrics. In STOC, pages 161-168, 1998.
R. C. Chakinala, A. Kumarasubramanian, K. A. Laing, R. Manokaran, C. Pandu Rangan, and R. Rajaraman. Playing push vs pull: models and algorithms for disseminating dynamic data in networks. In SPAA, pages 244-253, 2006.
Yevgeniy Dodis and Sanjeev Khanna. Designing networks with bounded pairwise distance. In STOC, pages 750-759, 1999.
Friedrich Eisenbrand and Fabrizio Grandoni. An improved approximation algorithm for virtual private network design. SODA, pages 928-932, 2005.
Friedrich Eisenbrand, Fabrizio Grandoni, Thomas Rothvoss, and Guido Schafer. Connected facility location via random facility sampling and core detouring. JCSS, 76(8):709-726, 2010.
Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci., 69(3):485-497, 2004.
Naveen Garg, Goran Konjevod, and R. Ravi. A polylogarithmic approximation algorithm for the group Steiner tree problem. J. Algorithms, 37(1):66-84, 2000.
Navin Goyal, Neil Olver, and F. Bruce Shepherd. The VPN conjecture is true. J. ACM, 60(3):17, 2013.
Anupam Gupta, Jon M. Kleinberg, Amit Kumar, Rajeev Rastogi, and Bülent Yener. Provisioning a virtual private network: a network design problem for multicommodity flow. STOC, pages 389-398, 2001.
Eran Halperin, Guy Kortsarz, Robert Krauthgamer, Aravind Srinivasan, and Nan Wang. Integrality ratio for group Steiner trees and directed Steiner trees. SIAM J. Comput., 36(5):1494-1511, 2007.
Kamal Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica, 21(1):39-60, 2001.
Guy Kortsarz and David Peleg. Generating low-degree 2-spanners. SIAM Journal on Computing, 27(5):1438-1456, 1998.
Shi Li. Approximation algorithms for network routing and facility location problems. Doctoral Dissertation, Princeton University, Januray 2014.
Laura J. Poplawski and Rajmohan Rajaraman. Multicommodity facility location under group steiner access cost. SODA, pages 996-1013, 2011.
R. Ravi and F. Sibel Salman. Approximation algorithms for the traveling purchaser problem and its variants in network design. ESA, pages 29-40, 1999.
R. Ravi and Amitabh Sinha. Approximation algorithms for multicommodity facility location problems. SIAM J. Discrete Math., 24(2):538-551, 2010.
Thomas Rothvoss and Laura Sanita. On the complexity of the asymmetric VPN problem. In APPROX/RANDOM, pages 326-338, 2009.
David B. Shmoys, Chaitanya Swamy, and Retsef Levi. Facility location with service installation costs. In SODA, pages 1088-1097, 2004.
Chaitanya Swamy and Amit Kumar. Primal-dual algorithms for connected facility location problems. Algorithmica, 40(4):245-269, 2004.
Chaitanya Swamy and David B. Shmoys. Fault-tolerant facility location. ACM Transactions on Algorithms, 4(4), 2008.
David P Williamson and David B Shmoys. The Design of Approximation Algorithms. Cambridge University Press, 2011.
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Approximating Dense Max 2-CSPs
In this paper, we present a polynomial-time algorithm that approximates sufficiently high-value Max 2-CSPs on sufficiently dense graphs to within O(N^epsilon) approximation ratio for any constant epsilon > 0. Using this algorithm, we also achieve similar results for free games, projection games on sufficiently dense random graphs, and the Densest k-Subgraph problem with sufficiently dense optimal solution. Note, however, that algorithms with similar guarantees to the last algorithm were in fact discovered prior to our work by Feige et al. and Suzuki and Tokuyama.
In addition, our idea for the above algorithms yields the following by-product: a quasi-polynomial time approximation scheme (QPTAS) for satisfiable dense Max 2-CSPs with better running time than the known algorithms.
Max 2-CSP
Dense Graphs
Densest k-Subgraph
QPTAS
Free Games
Projection Games
396-415
Regular Paper
Pasin
Manurangsi
Pasin Manurangsi
Dana
Moshkovitz
Dana Moshkovitz
10.4230/LIPIcs.APPROX-RANDOM.2015.396
S. Aaronson, R. Impagliazzo, and D. Moshkovitz. AM with multiple Merlins. In Computational Complexity (CCC), 2014 IEEE 29th Conference on, pages 44-55, June 2014.
N. Alon, W. F. de la Vega, R. Kannan, and M. Karpinski. Random sampling and approximation of max-CSPs. J. Comput. Syst. Sci., 67(2):212-243, September 2003.
S. Arora, D. Karger, and M. Karpinski. Polynomial time approximation schemes for dense instances of NP-hard problems. In Proceedings of the Twenty-seventh Annual ACM Symposium on Theory of Computing, STOC'95, pages 284-293, New York, NY, USA, 1995. ACM.
B. Barak, M. Hardt, T. Holenstein, and D. Steurer. Subsampling mathematical relaxations and average-case complexity. In Proceedings of the Twenty-second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'11, pages 512-531. SIAM, 2011.
B. Barak, A. Rao, R. Raz, R. Rosen, and R. Shaltiel. Strong parallel repetition theorem for free projection games. In Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM'09, pages 352-365, Berlin, Heidelberg, 2009. Springer-Verlag.
M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCPs, and nonapproximability - towards tight results. SIAM Journal on Computing, 27(3):804-915, 1998.
A. Bhaskara, M. Charikar, E. Chlamtac, U. Feige, and A. Vijayaraghavan. Detecting high log-densities: An O(n^1/4) approximation for densest k-subgraph. In Proceedings of the Forty-second ACM Symposium on Theory of Computing, STOC'10, pages 201-210, New York, NY, USA, 2010. ACM.
F. G.S.L. Brandao and A. W. Harrow. Quantum de finetti theorems under local measurements with applications. In Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC'13, pages 861-870, New York, NY, USA, 2013. ACM.
M. Braverman, Y. K. Ko, and O. Weinstein. Approximating the best nash equilibrium in n^o(log n)-time breaks the exponential time hypothesis. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'15, pages 970-982. SIAM, 2015.
M. Charikar, M. Hajiaghayi, and H. Karloff. Improved approximation algorithms for label cover problems. In ESA, pages 23-34, 2009.
U. Feige, D. Peleg, and G. Kortsarz. The dense k-subgraph problem. Algorithmica, 29(3):410-421, 2001.
J. Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, 2001.
S. Khot. Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS'04, pages 136-145, Washington, DC, USA, 2004. IEEE Computer Society.
P. Manurangsi and D. Moshkovitz. Improved approximation algorithms for projection games. In Algorithms – ESA 2013, volume 8125 of Lecture Notes in Computer Science, pages 683-694. Springer Berlin Heidelberg, 2013.
D. Moshkovitz. The projection games conjecture and the NP-hardness of ln n-approximating set-cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 15th International Workshop, APPROX 2012, volume 7408, pages 276-287, 2012.
R. Raz. A parallel repetition theorem. In SIAM Journal on Computing, volume 27, pages 763-803, 1998.
R. Shaltiel. Derandomized parallel repetition theorems for free games. Comput. Complex., 22(3):565-594, September 2013.
A. Suzuki and T. Tokuyama. Dense subgraph problems with output-density conditions. In Xiaotie Deng and Ding-Zhu Du, editors, Algorithms and Computation, volume 3827 of Lecture Notes in Computer Science, pages 266-276. Springer Berlin Heidelberg, 2005.
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The Container Selection Problem
We introduce and study a network resource management problem that is a special case of non-metric k-median, naturally arising in cross platform scheduling and cloud computing. In the continuous d-dimensional container selection problem, we are given a set C of input points in d-dimensional Euclidean space, for some d >= 2, and a budget k. An input point p can be assigned to a "container point" c only if c dominates p in every dimension. The assignment cost is then equal to the L1-norm of the container point. The goal is to find k container points in the d-dimensional space, such that the total assignment cost for all input points is minimized. The discrete variant of the problem has one key distinction, namely, the container points must be chosen from a given set F of points.
For the continuous version, we obtain a polynomial time approximation scheme for any fixed dimension d>= 2. On the negative side, we show that the problem is NP-hard for any d>=3. We further show that the discrete version is significantly harder, as it is NP-hard to approximate without violating the budget k in any dimension d>=3. Thus, we focus on obtaining bi-approximation algorithms. For d=2, the bi-approximation guarantee is (1+epsilon,3), i.e., for any epsilon>0, our scheme outputs a solution of size 3k and cost at most (1+epsilon) times the optimum. For fixed d>2, we present a (1+epsilon,O((1/epsilon)log k)) bi-approximation algorithm.
non-metric k-median
geometric hitting set
approximation algorithms
cloud computing
cross platform scheduling.
416-434
Regular Paper
Viswanath
Nagarajan
Viswanath Nagarajan
Kanthi K.
Sarpatwar
Kanthi K. Sarpatwar
Baruch
Schieber
Baruch Schieber
Hadas
Shachnai
Hadas Shachnai
Joel L.
Wolf
Joel L. Wolf
10.4230/LIPIcs.APPROX-RANDOM.2015.416
Marcel R. Ackermann, Johannes Blömer, and Christian Sohler. Clustering for metric and nonmetric distance measures. ACM Transactions on Algorithms (TALG), 6(4):59, 2010.
Amazon EC2. In URL: http://aws.amazon.com/ec2/.
http://aws.amazon.com/ec2/
Hyung-Chan An, Aditya Bhaskara, Chandra Chekuri, Shalmoli Gupta, Vivek Madan, and Ola Svensson. Centrality of trees for capacitated k-center. In IPCO, pages 52-63, 2014.
Sanjeev Arora, Prabhakar Raghavan, and Satish Rao. Approximation schemes for euclidean k-medians and related problems. In STOC, pages 106-113, 1998.
Hervé Brönnimann and Michael T. Goodrich. Almost optimal set covers in finite vc-dimension. Discrete & Computational Geometry, 14(4):463-479, 1995.
Jaroslaw Byrka, Thomas Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An Improved Approximation for k-median, and Positive Correlation in Budgeted Optimization. In SODA, 2015.
Tomás Feder and Daniel H. Greene. Optimal algorithms for approximate clustering. In STOC, pages 434-444, 1988.
David Haussler and Emo Welzl. ε-Nets and Simplex Range Queries. Discrete &Computational Geometry, 2:127-151, 1987.
Benjamin Hindman, Andy Konwinski, Matei Zaharia, Ali Ghodsi, Anthony Joseph, Scott Shenker, and Ion Stoica. Mesos: A Platform for Fine-Grained Resource Sharing in the Data Center. In NSDI, 2011.
A. K. Jain, M. N. Murty, and P. J. Flynn. Data clustering: A review. ACM Comput. Surv., 31(3), September 1999.
Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. In STOC, pages 901-910, 2013.
Jyh-Han Lin and Jeffrey Scott Vitter. epsilon-approximations with minimum packing constraint violation (extended abstract). In STOC, pages 771-782, 1992.
Private cloud. In URL: http://wikipedia.org/wiki/Cloud_computing#Private_cloud.
http://wikipedia.org/wiki/Cloud_computing#Private_cloud
Evangelia Pyrga and Saurabh Ray. New existence proofs epsilon-nets. In SOCG, pages 199-207, 2008.
Baruch Schieber. Computing a Minimum Weight k-Link Path in Graphs with the Concave Monge Property. Journal of Algorithms, 29(2):204-222, 1998.
V. Vavilapalli, A. Murthy, C. Douglis, A. Agarwal, M. Konar, R. Evans, T. Graves, J. Lowe, H. Shah, S. Seth, B. Saha, C. Curino, O. O'Malley, S. Radia, B. Reed, and E. Baldeschwiele. Apache Hadoop YARN: Yet Another Resource Negotiator. In SoCC, 2013.
Joel Wolf, Zubair Nabi, Viswanath Nagarajan, Robert Saccone, Rohit Wagle, Kirsten Hildrum, Edward Ping, and Kanthi Sarpatwar. The X-Flex Cross-Platform Scheduler: Who’s The Fairest Of Them All? In Middleware, 2014.
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Tight Bounds for Graph Problems in Insertion Streams
Despite the large amount of work on solving graph problems in the data stream model, there do not exist tight space bounds for almost any of them, even in a stream with only edge insertions. For example, for testing connectivity, the upper bound is O(n * log(n)) bits, while the lower bound is only Omega(n) bits. We remedy this situation by providing the first tight Omega(n * log(n)) space lower bounds for randomized algorithms which succeed with constant probability in a stream of edge insertions for a number of graph problems. Our lower bounds apply to testing bipartiteness, connectivity, cycle-freeness, whether a graph is Eulerian, planarity, H-minor freeness, finding a minimum spanning tree of a connected graph, and testing if the diameter of a sparse graph is constant. We also give the first Omega(n * k * log(n)) space lower bounds for deterministic algorithms for k-edge connectivity and k-vertex connectivity; these are optimal in light of known deterministic upper bounds (for k-vertex connectivity we also need to allow edge duplications, which known upper bounds allow). Finally, we give an Omega(n * log^2(n)) lower bound for randomized algorithms approximating the minimum cut up to a constant factor with constant probability in a graph with integer weights between 1 and n, presented as a stream of insertions and deletions to its edges. This lower bound also holds for cut sparsifiers, and gives the first separation of maintaining a sparsifier in the data stream model versus the offline model.
communication complexity
data streams
graphs
space complexity
435-448
Regular Paper
Xiaoming
Sun
Xiaoming Sun
David P.
Woodruff
David P. Woodruff
10.4230/LIPIcs.APPROX-RANDOM.2015.435
Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Graph sketches: sparsification, spanners, and subgraphs. In Proceedings of the 31st ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2012, Scottsdale, AZ, USA, May 20-24, 2012, pages 5-14, 2012.
Alexandr Andoni, Huy L. Nguyên, Yury Polyanskiy, and Yihong Wu. Tight lower bound for linear sketches of moments. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 25-32, 2013.
Khanh Do Ba, Piotr Indyk, Eric Price, and David P. Woodruff. Lower bounds for sparse recovery. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 1190-1197, 2010.
Brian Babcock, Shivnath Babu, Mayur Datar, Rajeev Motwani, and Jennifer Widom. Models and issues in data stream systems. In Proceedings of the Twenty-first ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, June 3-5, Madison, Wisconsin, USA, pages 1-16, 2002.
Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci., 68(4):702-732, 2004.
Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM J. Comput., 41(6):1704-1721, 2012.
Vladimir Braverman, Jonathan Katzman, Charles Seidell, and Gregory Vorsanger. An optimal algorithm for large frequency moments using o(n\^(1-2/k)) bits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, September 4-6, 2014, Barcelona, Spain, pages 531-544, 2014.
Amit Chakrabarti, Graham Cormode, and Andrew McGregor. A near-optimal algorithm for estimating the entropy of a stream. ACM Transactions on Algorithms, 6(3), 2010.
Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, and Andrew Chi-Chih Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14-17 October 2001, Las Vegas, Nevada, USA, pages 270-278, 2001.
Kenneth L. Clarkson and David P. Woodruff. Numerical linear algebra in the streaming model. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009, pages 205-214, 2009.
Peter Clifford and Ioana Cosma. A simple sketching algorithm for entropy estimation over streaming data. In Proceedings of the Sixteenth International Conference on Artificial Intelligence and Statistics, AISTATS 2013, Scottsdale, AZ, USA, April 29 - May 1, 2013, pages 196-206, 2013.
T. Cover and J. Thomas. Elements of Information Theory. John Wiley and Sons, Inc., 1991.
Michael S. Crouch, Andrew McGregor, and Daniel Stubbs. Dynamic graphs in the sliding-window model. In Algorithms - ESA 2013 - 21st Annual European Symposium, Sophia Antipolis, France, September 2-4, 2013. Proceedings, pages 337-348, 2013.
T.A. Dowling and R.M. Wilson. Whitney number inequalities for geometric lattices. Proc. Amer. Math. Soc., 47:504-512, 1975.
David Eppstein, Zvi Galil, Giuseppe F. Italiano, and Amnon Nissenzweig. Sparsification - a technique for speeding up dynamic graph algorithms. J. ACM, 44(5):669-696, 1997.
Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. In Automata, Languages and Programming: 31st International Colloquium, ICALP 2004, Turku, Finland, July 12-16, 2004. Proceedings, pages 531-543, 2004.
Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. Graph distances in the streaming model: the value of space. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, January 23-25, 2005, pages 745-754, 2005.
Sumit Ganguly. Polynomial estimators for high frequency moments. CoRR, abs/1104.4552, 2011.
Sudipto Guha, Andrew McGregor, and D. Tench. Vertex and hyperedge connectivity in dynamic graph streams. In PODS, 2015.
Nicholas J. A. Harvey, Jelani Nelson, and Krzysztof Onak. Sketching and streaming entropy via approximation theory. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 489-498, 2008.
T. S. Jayram and David P. Woodruff. Optimal bounds for johnson-lindenstrauss transforms and streaming problems with subconstant error. ACM Transactions on Algorithms, 9(3):26, 2013.
Daniel M. Kane, Jelani Nelson, and David P. Woodruff. On the exact space complexity of sketching and streaming small norms. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 1161-1178, 2010.
Daniel M. Kane, Jelani Nelson, and David P. Woodruff. An optimal algorithm for the distinct elements problem. In Proceedings of the Twenty-Ninth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2010, June 6-11, 2010, Indianapolis, Indiana, USA, pages 41-52, 2010.
Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 561-570, 2014.
Alexandr Kostochka. The minimum hadwiger number for graphs with a given mean degree of vertices. Metody Diskret. Analiz., 38:37-58, 1982.
Ilan Kremer, Noam Nisan, and Dana Ron. On randomized one-round communication complexity. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, 29 May-1 June 1995, Las Vegas, Nevada, USA, pages 596-605, 1995.
Yi Li and David P. Woodruff. A tight lower bound for high frequency moment estimation with small error. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013. Proceedings, pages 623-638, 2013.
Andrew McGregor. Graph stream algorithms: a survey. SIGMOD Record, 43(1):9-20, 2014.
S. Muthukrishnan. Data streams: Algorithms and applications. Foundations and Trends in Theoretical Computer Science, 1(2), 2005.
Eric Price and David P. Woodruff. (1 + eps)-approximate sparse recovery. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 295-304, 2011.
Ran Raz and Boris Spieker. On the "log rank"-conjecture in communication complexity. In 34th Annual Symposium on Foundations of Computer Science, Palo Alto, California, USA, 3-5 November 1993, pages 168-176, 1993.
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A Chasm Between Identity and Equivalence Testing with Conditional Queries
A recent model for property testing of probability distributions enables tremendous savings in the sample complexity of testing algorithms, by allowing them to condition the sampling on subsets of the domain.
In particular, Canonne, Ron, and Servedio showed that, in this setting, testing identity of an unknown distribution D (i.e., whether D = D* for an explicitly known D*) can be done with a constant number of samples, independent of the support size n - in contrast to the required sqrt(n) in the standard sampling model. However, it was unclear whether the same held for the case of testing equivalence, where both distributions are unknown. Indeed, while Canonne, Ron, and Servedio established a polylog(n)-query upper bound for equivalence testing, very recently brought down to ~O(log(log(n))) by Falahatgar et al., whether a dependence on the domain size n is necessary was still open, and explicitly posed by Fischer at the Bertinoro Workshop on Sublinear Algorithms. In this work, we answer the question in the positive, showing that any testing algorithm for equivalence must make Omega(sqrt(log(log(n)))) queries in the conditional sampling model. Interestingly, this demonstrates an intrinsic qualitative gap between identity and equivalence testing, absent in the standard sampling model (where both problems have sampling complexity n^(Theta(1))).
Turning to another question, we investigate the complexity of support size estimation. We provide a doubly-logarithmic upper bound for the adaptive version of this problem, generalizing work of Ron and Tsur to our weaker model. We also establish a logarithmic lower bound for the non-adaptive version of this problem. This latter result carries on to the related problem of non-adaptive uniformity testing, an exponential improvement over previous results that resolves an open question of Chakraborty, Fischer, Goldhirsh, and Matsliah.
property testing
probability distributions
conditional samples
449-466
Regular Paper
Jayadev
Acharya
Jayadev Acharya
Clément L.
Canonne
Clément L. Canonne
Gautam
Kamath
Gautam Kamath
10.4230/LIPIcs.APPROX-RANDOM.2015.449
Jayadev Acharya, Clément L. Canonne, and Gautam Kamath. A chasm between identity and equivalence testing with conditional queries. ArXiV, abs/1411.7346, April 2015.
Jayadev Acharya, Hirakendu Das, Ashkan Jafarpour, Alon Orlitsky, and Shengjun Pan. Competitive closeness testing. In Proceedings of 24th COLT, pages 47-68, 2011.
Jayadev Acharya, Hirakendu Das, Ashkan Jafarpour, Alon Orlitsky, Shengjun Pan, and Ananda Theertha Suresh. Competitive classification and closeness testing. In Proceedings of 25th COLT, pages 1-18, 2012.
Tuğkan Batu, Eldar Fischer, Lance Fortnow, Ravi Kumar, Ronitt Rubinfeld, and Patrick White. Testing random variables for independence and identity. In Proceedings of FOCS, pages 442-451, 2001.
Tuğkan Batu, Lance Fortnow, Ronitt Rubinfeld, Warren D. Smith, and Patrick White. Testing closeness of discrete distributions. Journal of the ACM, 60(1):1-25, 2013.
Tuğkan Batu, Ravi Kumar, and Ronitt Rubinfeld. Sublinear algorithms for testing monotone and unimodal distributions. In Proceedings of STOC, pages 381-390, New York, NY, USA, 2004. ACM.
Arnab Bhattacharyya, Eldar Fischer, Ronitt Rubinfeld, and Paul Valiant. Testing monotonicity of distributions over general partial orders. In Proceedings of ITCS, pages 239-252, 2011.
Clément L. Canonne. A Survey on Distribution Testing: your data is Big, but is it Blue? Electronic Colloquium on Computational Complexity (ECCC), TR15-063, April 2015.
Clément L. Canonne, Dana Ron, and Rocco A. Servedio. Testing probability distributions using conditional samples. SIAM Journal on Computing, 44(3), 2015.
Clément L. Canonne and Ronitt Rubinfeld. Testing probability distributions underlying aggregated data. In Proceedings of ICALP, pages 283-295, 2014.
Sourav Chakraborty, Eldar Fischer, Yonatan Goldhirsh, and Arie Matsliah. On the power of conditional samples in distribution testing. In Proceedings of ITCS, pages 561-580, New York, NY, USA, 2013. ACM.
Siu-On Chan, Ilias Diakonikolas, Gregory Valiant, and Paul Valiant. Optimal algorithms for testing closeness of discrete distributions. In Proceedings of SODA, pages 1193-1203. Society for Industrial and Applied Mathematics (SIAM), 2014.
Moein Falahatgar, Ashkan Jafarpour, Alon Orlitsky, Venkatadheeraj Pichapathi, and Ananda Theertha Suresh. Faster algorithms for testing under conditional sampling. In Proceedings of 28th COLT, 2015.
Eldar Fischer. List of Open Problems in Sublinear Algorithms: Problem 66. http://sublinear.info/66. Suggested by Fischer at Bertinoro Workshop on Sublinear Algorithms 2014.
http://sublinear.info/66
Oded Goldreich and Dana Ron. On testing expansion in bounded-degree graphs. Electronic Colloquium on Computational Complexity (ECCC), TR00-020, March 2000.
Sudipto Guha, Andrew McGregor, and Suresh Venkatasubramanian. Streaming and sublinear approximation of entropy and information distances. In Proceedings of SODA, pages 733-742. Society for Industrial and Applied Mathematics (SIAM), 2006.
Piotr Indyk, Reut Levi, and Ronitt Rubinfeld. Approximating and Testing k-Histogram Distributions in Sub-linear Time. In Proceedings of PODS, pages 15-22, 2012.
Reut Levi, Dana Ron, and Ronitt Rubinfeld. Testing properties of collections of distributions. Theory of Computing, 9(8):295-347, 2013.
Rajeev Motwani and Prabhakar Raghavan. Randomized Algorithms. Cambridge University Press, New York, NY, USA, 1995.
Liam Paninski. A coincidence-based test for uniformity given very sparsely sampled discrete data. IEEE Transactions on Information Theory, 54(10):4750-4755, 2008.
Sofya Raskhodnikova, Dana Ron, Amir Shpilka, and Adam Smith. Strong lower bounds for approximating distributions support size and the distinct elements problem. SIAM Journal on Computing, 39(3):813-842, 2009.
Dana Ron and Gilad Tsur. The power of an example: Hidden set size approximation using group queries and conditional sampling. ArXiV, abs/1404.5568, 2014.
Ronitt Rubinfeld. Taming Big Probability Distributions. XRDS, 19(1):24-28, September 2012.
Ronitt Rubinfeld and Rocco A. Servedio. Testing monotone high-dimensional distributions. Random Structures and Algorithms, 34(1):24-44, January 2009.
L. Stockmeyer. On approximation algorithms for #P. SIAM Journal on Computing, 14(4):849-861, 1985.
Gregory Valiant and Paul Valiant. A CLT and tight lower bounds for estimating entropy. Electronic Colloquium on Computational Complexity (ECCC), TR10-179, 2010.
Gregory Valiant and Paul Valiant. Estimating the unseen: A sublinear-sample canonical estimator of distributions. Electronic Colloquium on Computational Complexity (ECCC), TR10-180, 2010.
Gregory Valiant and Paul Valiant. The power of linear estimators. In Proceedings of FOCS, pages 403-412, October 2011. See also [Valiant and Valiant, 2010] and [Valiant and Valiant, 2010].
Gregory Valiant and Paul Valiant. An automatic inequality prover and instance optimal identity testing. In Proceedings of FOCS, 2014.
Paul Valiant. Testing symmetric properties of distributions. SIAM Journal on Computing, 40(6):1927-1968, 2011.
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Harnessing the Bethe Free Energy
Gibbs measures induced by random factor graphs play a prominent role in computer science, combinatorics and physics. A key problem is to calculate the typical value of the partition function. According to the "replica symmetric cavity method", a heuristic that rests on non-rigorous considerations from statistical mechanics, in many cases this problem can be tackled by way of maximising a functional called the "Bethe free energy". In this paper we prove that the Bethe free energy upper-bounds the partition function in a broad class of models. Additionally, we provide a sufficient condition for this upper bound to be tight.
Belief Propagation
free energy
Gibbs measure
partition function
467-480
Regular Paper
Victor
Bapst
Victor Bapst
Amin
Coja-Oghlan
Amin Coja-Oghlan
10.4230/LIPIcs.APPROX-RANDOM.2015.467
Dimitris Achlioptas and Cristopher Moore. Random k-sat: Two moments suffice to cross a sharp threshold. SIAM J. Comput., 36(3):740-762, September 2006.
Dimitris Achlioptas, Assaf Naor, and Yuval Peres. Rigorous location of phase transitions in hard optimization problems. Nature, 435(7043):759-764, 06 2005.
Dimitris Achlioptas, Assaf Naor, and Yuval Peres. On the maximum satisfiability of random formulas. J. ACM, 54(2), April 2007.
Dimitris Achlioptas and Yuval Peres. The threshold for random k-sat is 2k (ln 2 - o(k)). In Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing, STOC'03, pages 223-231, New York, NY, USA, 2003. ACM.
H. A. Bethe. Statistical theory of superlattices. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 150(871):552-575, 1935.
Charles Bordenave and Pietro Caputo. Large deviations of empirical neighborhood distribution in sparse random graphs. Probability Theory and Related Fields, pages 1-73, 2014.
Amin Coja-Oghlan and Konstantinos Panagiotou. The asymptotic k-SAT threshold. arXiv:1310.2728, 2014.
Amin Coja-Oghlan and Lenka Zdeborová. The condensation transition in random hypergraph 2-coloring. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 241-250, 2012.
Amir Dembo, Andrea Montanari, Allan Sly, and Nike Sun. The replica symmetric solution for potts models on d-regular graphs. Communications in Mathematical Physics, 327(2):551-575, 2014.
Amir Dembo, Andrea Montanari, and Nike Sun. Factor models on locally tree-like graphs. Ann. Probab., 41(6):4162-4213, 11 2013.
Jian Ding, Allan Sly, and Nike Sun. Maximum independent sets on random regular graphs. arXiv:1310.4787, 2013.
Jian Ding, Allan Sly, and Nike Sun. Proof of the satisfiability conjecture for large k. arXiv:1411.0650, 2014.
Jian Ding, Allan Sly, and Nike Sun. Satisfiability threshold for random regular NAE-SAT. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC'14, pages 814-822, New York, NY, USA, 2014. ACM.
Silvio Franz and Michele Leone. Replica bounds for optimization problems and diluted spin systems. J. Stat. Phys., 111(3-4):535-564, 2003.
Alan Frieze and Nicholas C. Wormald. Random k-sat: A tight threshold for moderately growing k. In Proceedings of the Fifth International Symposium on Theory and Applications of Satisfiability Testing, pages 1-6, 2002.
Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. Inapproximability for antiferromagnetic spin systems in the tree non-uniqueness region. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC'14, pages 823-831, New York, NY, USA, 2014. ACM.
R.G. Gallager. Low-density Parity-check Codes. M.I.T. Press research monographs. M.I.T. Press, 1963.
H.O. Georgii. Gibbs Measures and Phase Transitions. De Gruyter studies in mathematics. De Gruyter, 2011.
Francesco Guerra. Broken replica symmetry bounds in the mean field spin glass model. Communications in Mathematical Physics, 233(1):1-12, 2003.
Svante Janson. Random regular graphs: Asymptotic distributions and contiguity. Combinatorics, Probability and Computing, 4:369-405, 12 1995.
Florent Krzakala, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Semerjian, and Lenka Zdeborová. Gibbs states and the set of solutions of random constraint satisfaction problems. P. Natl. Acad. Sci. USA, 104(25):10318-10323, 2007.
L. Lovász. Large Networks and Graph Limits. American Mathematical Society colloquium publications. American Mathematical Society, 2012.
M. Mézard and A. Montanari. Information, Physics and Computation. Oxford University Press, 2009.
M. Mézard, G. Parisi, and R. Zecchina. Analytic and algorithmic solution of random satisfiability problems. Science, 297:812-815, 2002.
Andrea Montanari and Devavrat Shah. Counting good truth assignments of random k-sat formulae. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'07, pages 1255-1264, Philadelphia, PA, USA, 2007.
Elchanan Mossel, Dror Weitz, and Nicholas Wormald. On the hardness of sampling independent sets beyond the tree threshold. Probability Theory and Related Fields, 143(3-4):401-439, 2009.
Ralph Neininger and Ludger Rüschendorf. A general limit theorem for recursive algorithms and combinatorial structures. Ann. Appl. Probab., 14(1):378-418, 02 2004.
Dmitry Panchenko and Michel Talagrand. Bounds for diluted mean-fields spin glass models. Probab. Theory Relat. Fields, 130(3):319-336, 2004.
Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1988.
Allan Sly and Nike Sun. The computational hardness of counting in two-spin models on d-regular graphs. 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 0:361-369, 2012.
Endre Szemerédi. Regular partitions of graphs. In Colloq. Internat. CNRS, volume 260, pages 399-401, 1978.
Jonathan S. Yedidia, W.T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized belief propagation algorithms. Information Theory, IEEE Transactions on, 51(7):2282-2312, July 2005.
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Internal Compression of Protocols to Entropy
We study internal compression of communication protocols to their internal entropy, which is the entropy of the transcript from the players' perspective. We provide two internal compression schemes with error. One of a protocol of Feige et al. for finding the first difference between two strings. The second and main one is an internal compression with error epsilon > 0 of a protocol with internal entropy H^{int} and communication complexity C to a protocol with communication at most order (H^{int}/epsilon)^2 * log(log(C)).
This immediately implies a similar compression to the internal information of public-coin protocols, which provides an exponential improvement over previously known public-coin compressions in the dependence on C. It further shows that in a recent protocol of Ganor, Kol and Raz, it is impossible to move the private randomness to be public without an exponential cost. To the best of our knowledge, No such example was previously known.
Communication complexity
Information complexity
Compression
Simulation
Entropy
481-496
Regular Paper
Balthazar
Bauer
Balthazar Bauer
Shay
Moran
Shay Moran
Amir
Yehudayoff
Amir Yehudayoff
10.4230/LIPIcs.APPROX-RANDOM.2015.481
Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci., 68(4):702-732, 2004.
Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. SIAM J. Comput., 42(3):1327-1363, 2013.
Mark Braverman. Interactive information complexity. In STOC, pages 505-524, 2012.
Mark Braverman and Ankit Garg. Public vs private coin in bounded-round information. In ICALP (1), pages 502-513, 2014.
Mark Braverman and Anup Rao. Information equals amortized communication. In FOCS, pages 748-757, 2011.
Mark Braverman, Anup Rao, Omri Weinstein, and Amir Yehudayoff. Direct product via round-preserving compression. In ICALP (1), pages 232-243, 2013.
Mark Braverman, Anup Rao, Omri Weinstein, and Amir Yehudayoff. Direct products in communication complexity. In FOCS, pages 746-755, 2013.
Joshua Brody, Harry Buhrman, Michal Koucký, Bruno Loff, Florian Speelman, and Nikolay K. Vereshchagin. Towards a reverse newman’s theorem in interactive information complexity. In IEEE Conference on Computational Complexity, pages 24-33, 2013.
Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, , and Andrew Yao. Informational complexity and the direct sum problem for simultaneous message complexity. Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, pages 270-278, 2001.
Thomas M. Cover and Joy A. Thomas. Elements of Information Theory. Wiley Interscience, 2006.
Martin Dietzfelbinger and Henning Wunderlich. A characterization of average case communication complexity. Inf. Process. Lett., 101(6):245-249, 2007.
R. M. Fano. The transmission of information. Technical Report 65, Research Laboratory for Electronics, MIT, Cambridge, MA, USA, 1949.
Uriel Feige, David Peleg, Prabhakar Raghavan, and Eli Upfal. Computing with noisy information. SIAM Journal on Computing, 23(5):1001-1018, 1994.
Abbas El Gamal and Alon Orlitsky. Interactive data compression. 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 0:100-108, 1984.
Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of information and communication. Electronic Colloquium on Computational Complexity, 2014.
Prahladh Harsha, Rahul Jain, David A. McAllester, and Jaikumar Radhakrishnan. The communication complexity of correlation. IEEE Transactions on Information Theory, 56(1):438-449, 2010.
David A. Huffman. A method for the construction of minimum-redundancy codes. Proceedings of the IRE, 40(9):1098-1101, September 1952.
Gillat Kol, Shay Moran, Amir Shpilka, and Amir Yehudayoff. Direct sum fails for zero error average communication. In ITCS, pages 517-522, 2014.
Eyal Kushilevitz and Noam Nisan. Communication complexity. Cambridge University Press, 1997.
Alon Orlitsky Moni Naor and Peter Shor. Three results on interactive communication. Information Theory, IEEE Transactions on, 39(5):1608-1615, Sep 1993.
Ilan Newman. Private vs. common random bits in communication complexity. Inf. Process. Lett., 39(2):67-71, 1991.
Alon Orlitsky. Average-case interactive communication. Information Theory, IEEE Transactions on, 38(5):1534-1547, Sep 1992.
Alon Orlitsky and Abbas El Gamal. Average and randomized communication complexity. IEEE Transactions on Information Theory, 36(1):3-16, 1990.
Denis Pankratov. Direct sum questions in classical communication complexity. PhD thesis, University of Chicago, 2012.
C.E. Shannon. A mathematical theory of communication. Bell System Technical Journal, 27:379-423, 623-656, 1948.
David Slepian and Rahul Jack K. Wolf. Noiseless coding of correlate information sources. IEEE Transactions on Information Theory, 19(4), July 1973.
Emanuele Viola. The communication complexity of addition. In SODA, pages 632-651, 2013.
Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (preliminary report). In STOC, pages 209-213, 1979.
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On Fortification of Projection Games
A recent result of Moshkovitz [Moshkovitz14] presented an ingenious method to provide a completely elementary proof of the Parallel Repetition Theorem for certain projection games via a construction called fortification. However, the construction used in [Moshkovitz14] to fortify arbitrary label cover instances using an arbitrary extractor is insufficient to prove parallel repetition. In this paper, we provide a fix by using a stronger graph that we call fortifiers. Fortifiers are graphs that have both l_1 and l_2 guarantees on induced distributions from large subsets.
We then show that an expander with sufficient spectral gap, or a bi-regular extractor with stronger parameters (the latter is also the construction used in an independent update [Moshkovitz15] of [Moshkovitz14] with an alternate argument), is a good fortifier. We also show that using a fortifier (in particular l_2 guarantees) is necessary for obtaining the robustness required for fortification.
Parallel Repetition
Fortification
497-511
Regular Paper
Amey
Bhangale
Amey Bhangale
Ramprasad
Saptharishi
Ramprasad Saptharishi
Girish
Varma
Girish Varma
Rakesh
Venkat
Rakesh Venkat
10.4230/LIPIcs.APPROX-RANDOM.2015.497
Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, May 1998.
Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70-122, January 1998.
Yonatan Bilu and Nathan Linial. Lifts, discrepancy and nearly optimal spectral gap*. Combinatorica, 26(5):495-519, 2006.
Mark Braverman and Ankit Garg. Small value parallel repetition for general games. Electronic Colloquium on Computational Complexity (ECCC), 21:95, 2014. To appear in STOC 2015.
Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 624-633, 2014.
Uriel Feige. On the success probability of the two provers in one-round proof systems. In Structure in Complexity Theory Conference, 1991., Proceedings of the Sixth Annual, pages 116-123, Jun 1991.
Uriel Feige and Joe Kilian. Two prover protocols: low error at affordable rates. In Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, 23-25 May 1994, Montréal, Québec, Canada, pages 172-183, 1994.
Lance Jeremy Fortnow. Complexity-theoretic aspects of interactive proof systems. PhD thesis, Massachusetts Institute of Technology, 1989.
Thomas Holenstein. Parallel repetition: Simplification and the no-signaling case. Theory of Computing, 5(1):141-172, 2009.
Dana Moshkovitz. Parallel repetition from fortification. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 414-423, 2014.
Dana Moshkovitz. Parallel repetition from fortification. http://people.csail.mit.edu/dmoshkov/papers/par-rep/final3.pdf, 2015.
http://people.csail.mit.edu/dmoshkov/papers/par-rep/final3.pdf
Jaikumar Radhakrishnan and Amnon Ta-Shma. Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM J. Discrete Math., 13(1):2-24, 2000.
Anup Rao. Parallel repetition in projection games and a concentration bound. SIAM J. Comput., 40(6):1871-1891, 2011.
Ran Raz. A parallel repetition theorem. SIAM J. Comput., 27(3):763-803, 1998.
Ran Raz. A counterexample to strong parallel repetition. SIAM J. Comput., 40(3):771-777, June 2011.
Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on, pages 3-13, 2000.
Oleg Verbitsky. Towards the parallel repetition conjecture. Theor. Comput. Sci., 157(2):277-282, May 1996.
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Learning Circuits with few Negations
Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, establishing nearly matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A.A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions).
Boolean functions
monotonicity
negations
PAC learning
512-527
Regular Paper
Eric
Blais
Eric Blais
Clément L.
Canonne
Clément L. Canonne
Igor C.
Oliveira
Igor C. Oliveira
Rocco A.
Servedio
Rocco A. Servedio
Li-Yang
Tan
Li-Yang Tan
10.4230/LIPIcs.APPROX-RANDOM.2015.512
K. Amano and A. Maruoka. On learning monotone Boolean functions under the uniform distribution. In International Conference on Algorithmic Learning Theory (ALT), pages 57-68, 2002.
K. Amano and A. Maruoka. A Superpolynomial Lower Bound for a Circuit Computing the Clique Function with At Most (1/6) log log n Negation Gates. SIAM Journal on Computing, 35(1):201-216, 2005.
D. Angluin. Queries and concept learning. Machine Learning, 2:319-342, 1988.
E. Blais and L-Y. Tan. Approximating Boolean functions with depth-2 circuits. In Conference on Computational Complexity (CCC), pages 74-85, 2013.
A. Blum, C. Burch, and J. Langford. On learning monotone Boolean functions. In Symposium on Foundations of Computer Science (FOCS), pages 408-415, 1998.
N. Bshouty and C. Tamon. On the Fourier spectrum of monotone functions. Journal of the ACM, 43(4):747-770, 1996.
V. Feldman, H. K. Lee, and R. A. Servedio. Lower bounds and hardness amplification for learning shallow monotone formulas. Journal of Machine Learning Research - Proceedings Track, 19:273-292, 2011.
O. Goldreich and R. Izsak. Monotone circuits: One-way functions versus pseudorandom generators. Theory of Computing, 8(1):231-238, 2012.
S. Guo and I. Komargodski. Negation-limited formulas. Technical Report 22(26), Electronic Colloquium on Computational Complexity (ECCC), 2015.
S. Guo, T. Malkin, I. C. Oliveira, and A. Rosen. The power of negations in cryptography. In Theory of Cryptography Conference (TCC), pages 36-65, 2015.
M. Kearns and L. Valiant. Cryptographic limitations on learning Boolean formulae and finite automata. Journal of the ACM, 41(1):67-95, 1994.
A. D. Korshunov. Monotone Boolean functions. Russian Mathematical Surveys (Uspekhi Matematicheskikh Nauk), 58(5):929-1001, 2003.
N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform and learnability. Journal of the ACM, 40(3):607-620, 1993.
A. A. Markov. On the inversion complexity of systems of functions. Doklady Akademii Nauk SSSR, 116:917-919, 1957. English translation in [Markov, 1958].
A. A. Markov. On the inversion complexity of a system of functions. Journal of the ACM, 5(4):331-334, October 1958.
H. Morizumi. Limiting negations in formulas. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 701-712, 2009.
H. Morizumi. Limiting negations in non-deterministic circuits. Theoretical Computer Science, 410(38-40):3988-3994, 2009.
E. Mossel and R. O'Donnell. On the noise sensitivity of monotone functions. Random Structures and Algorithms, 23(3):333-350, 2003.
R. O'Donnell. Computational applications of noise sensitivity. PhD thesis, MIT, June 2003.
R. O'Donnell and R. Servedio. Learning monotone decision trees in polynomial time. SIAM Journal on Computing, 37(3):827-844, 2007.
R. O'Donnell and K. Wimmer. KKL, Kruskal-Katona, and monotone nets. SIAM Journal on Computing, 42(6):2375-2399, 2013.
Ran Raz and Avi Wigderson. Monotone circuits for matching require linear depth. Journal of the ACM, 39(3):736-744, 1992.
A. Razborov. Lower bounds on the monotone complexity of some Boolean functions. Doklady Akademii Nauk SSSR, 281:798-801, 1985. English translation in: Soviet Mathematics Doklady 31:354-357, 1985.
B. Rossman. Correlation bounds against monotone NC¹. In Conference on Computational Complexity (CCC), 2015.
M. Santha and C. Wilson. Limiting negations in constant depth circuits. SIAM Journal on Computing, 22(2):294-302, 1993.
R. Servedio. On learning monotone DNF under product distributions. Information and Computation, 193(1):57-74, 2004.
S. Sung and K. Tanaka. Limiting Negations in Bounded-Depth Circuits: an Extension of Markov’s Theorem. In International Symposium on Algorithms and Computation (ISAAC), pages 108-116, 2003.
M. Talagrand. How much are increasing sets positively correlated? Combinatorica, 16(2):243-258, 1996.
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Dynamics for the Mean-field Random-cluster Model
The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and random spanning trees, but its dynamics have so far largely resisted analysis. In this paper we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, and identify a critical regime (lambda_s,lambda_S) of the model parameter lambda in which the dynamics undergoes an exponential slowdown. Namely, we prove that the mixing time is Theta(log n) if lambda is not in [lambda_s,lambda_S], and e^Omega(sqrt{n}) when lambda is in (lambda_s,lambda_S). These results hold for all values of the second model parameter q > 1. In addition, we prove that the local heat-bath dynamics undergoes a similar exponential slowdown in (lambda_s,lambda_S).
random-cluster model
random graphs
Markov chains
statistical physics
dynamics
528-543
Regular Paper
Antonio
Blanca
Antonio Blanca
Alistair
Sinclair
Alistair Sinclair
10.4230/LIPIcs.APPROX-RANDOM.2015.528
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, 153:511-542, 2012.
A. Blanca and A. Sinclair. Dynamics for the mean-field random-cluster model, 2014. Preprint. arXiv:1412.6180v1 [math.PR].
B. Bollobás, G. Grimmett, and S. Janson. The random-cluster model on the complete graph. Probability Theory and Related Fields, 104(3):283-317, 1996.
C. Borgs, J. Chayes, and P. Tetali. Swendsen-Wang algorithm at the Potts transition point. Probability Theory and Related Fields, 152:509-557, 2012.
C. Borgs, A. Frieze, J.H. Kim, P. Tetali, E. Vigoda, and V. Vu. Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics. Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS), pages 218-229, 1999.
L. Chayes and J. Machta. Graphical representations and cluster algorithms II. Physica A, 254:477-516, 1998.
C. Cooper, M.E. Dyer, A.M. Frieze, and R. Rue. Mixing properties of the Swendsen-Wang process on the complete graph and narrow grids. Journal of Mathematical Physics, 41:1499-1527, 2000.
P. Cuff, J. Ding, O. Louidor, E. Lubetzky, Y. Peres, and A. Sly. Glauber dynamics for the mean-field Potts model. Journal of Statistical Physics, 149(3):432-477, 2012.
R.G. Edwards and A.D. Sokal. Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Physical Review D, 38(6):2009-2012, 1988.
C.M. Fortuin and P.W. Kasteleyn. On the random-cluster model I. Introduction and relation to other models. Physica, 57(4):536-564, 1972.
A. Galanis, D. Štefankovič, and E. Vigoda. Swendsen-Wang Algorithm on the Mean-Field Potts Model, 2015. Preprint. arXiv:1502.06593v1 [cs.DM].
Q. Ge and D. Štefankovič. A graph polynomial for independent sets of bipartite graphs. Combinatorics, Probability and Computing, 21(5):695-714, 2012.
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 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2006.
M. Huber. A bounding chain for Swendsen-Wang. Random Structures & Algorithms, 22(1):43-59, 2003.
G. Kirchhoff. Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strome gefuhrt wird. Annalen der Physik und Chemie, pages 497-508, 1847.
D.A. Levin, Y. Peres, and E.L. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, 2008.
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), 2011.
M.J. Luczak and T. Łuczak. The phase transition in the cluster-scaled model of a random graph. Random Structures & Algorithms, 28(2):215-246, 2006.
R.H. Swendsen and J.S. Wang. Nonuniversal critical dynamics in Monte Carlo simulations. Physical Review Letters, 58:86-88, 1987.
M. Ullrich. Rapid mixing of Swendsen-Wang dynamics in two dimensions. Dissertationes Mathematicae, 502, 2014.
M. Ullrich. Swendsen-Wang is faster than single-bond dynamics. SIAM Journal on Discrete Mathematics, 28(1):37-48, 2014.
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Correlation in Hard Distributions in Communication Complexity
We study the effect that the amount of correlation in a bipartite distribution has on the communication complexity of a problem under that distribution. We introduce a new family of complexity measures that interpolates between the two previously studied extreme cases: the (standard) randomised communication complexity and the case of distributional complexity under product distributions.
- We give a tight characterisation of the randomised complexity of Disjointness under distributions with mutual information k, showing that it is Theta(sqrt(n(k+1))) for all 0 <= k <= n. This smoothly interpolates between the lower bounds of Babai, Frankl and Simon for the product distribution case (k=0), and the bound of Razborov for the randomised case. The upper bounds improve and generalise what was known for product distributions, and imply that any tight bound for Disjointness needs Omega(n) bits of mutual information in the corresponding distribution.
- We study the same question in the distributional quantum setting, and show a lower bound of Omega((n(k+1))^{1/4}), and an upper bound (via constructing communication protocols), matching up to a logarithmic factor.
- We show that there are total Boolean functions f_d that have distributional communication complexity O(log(n)) under all distributions of information up to o(n), while the (interactive) distributional complexity maximised over all distributions is Theta(log(d)) for n <= d <= 2^{n/100}. This shows, in particular, that the correlation needed to show that a problem is hard can be much larger than the communication complexity of the problem.
- We show that in the setting of one-way communication under product distributions, the dependence of communication cost on the allowed error epsilon is multiplicative in log(1/epsilon) - the previous upper bounds had the dependence of more than 1/epsilon. This result, for the first time, explains how one-way communication complexity under product distributions is stronger than PAC-learning: both tasks are characterised by the VC-dimension, but have very different error dependence (learning from examples, it costs more to reduce the error).
communication complexity; information theory
544-572
Regular Paper
Ralph Christian
Bottesch
Ralph Christian Bottesch
Dmitry
Gavinsky
Dmitry Gavinsky
Hartmut
Klauck
Hartmut Klauck
10.4230/LIPIcs.APPROX-RANDOM.2015.544
S. Aaronson and A. Ambainis. Quantum search of spatial regions. Theory of Computing, 1(1):47-79, 2005. Earlier version in FOCS'03. quant-ph/0303041.
Noga Alon, Shay Moran, and Amir Yehudayoff. Sign rank, VC dimension and spectral gaps. Electronic Colloquium on Computational Complexity (ECCC), 21:135, 2014.
A. Ambainis, A. Nayak, A. Ta-Shma, and U. V. Vazirani. Dense quantum coding and quantum finite automata. Journal of the ACM, 49(4):496-511, 2002. Earlier version in STOC'99.
L. Babai, P. Frankl, and J. Simon. Complexity classes in communication complexity theory. In Proceedings of 27th IEEE FOCS, pages 337-347, 1986.
Z. Bar-Yossef, T. S. Jayram, R. Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. In Proceedings of 43rd IEEE FOCS, pages 209-218, 2002.
H. Buhrman, R. Cleve, and A. Wigderson. Quantum vs. classical communication and computation. In Proceedings of 30th ACM STOC, pages 63-68, 1998. quant-ph/9802040.
B. Chor and O. Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM Journal on Computing, 17(2):230-261, 1988. Earlier version in FOCS'85.
T. M. Cover and J. A. Thomas. Elements of Information Theory. Wiley, 1991.
Ronald de Wolf. Quantum communication and complexity. Theoretical Computer Science, 287(1):337-353, 2002.
Prahladh Harsha, Rahul Jain, David A. McAllester, and Jaikumar Radhakrishnan. The communication complexity of correlation. IEEE Transactions on Information Theory, 56(1):438-449, 2010.
Johan Håstad and Avi Wigderson. The randomized communication complexity of set disjointness. Theory of Computing, 3(1):211-219, 2007.
R. Jain, H. Klauck, and A. Nayak. Direct product theorems for classical communication complexity via subdistribution bounds. In Proc. of 40th ACM STOC, pages 599-608, 2008.
R. Jain, J. Radhakrishnan, and P. Sen. Privacy and interaction in quantum communication complexity and a theorem about the relative entropy of quantum states. In Proceedings of 43rd IEEE FOCS, pages 429-438, 2002.
R. Jain and S. Zhang. New bounds on classical and quantum one-way communication complexity. Theoretical Computer Science, 410(26):2463-2477, 2009.
Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. A direct sum theorem in communication complexity via message compression. In ICALP, page 187, 2003.
B. Kalyanasundaram and G. Schnitger. The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics, 5(4):545-557, 1992. Earlier version in Structures'87.
Michael J. Kearns and Umesh V. Vazirani. An Introduction to Computational Learning Theory. MIT Press, 1994.
H. Klauck. On quantum and probabilistic communication: Las Vegas and one-way protocols. In Proceedings of 32nd ACM STOC, pages 644-651, 2000.
I. Kremer, N. Nisan, and D. Ron. On randomized one-round communication complexity. Computational Complexity, 8(1):21-49, 1999. Earlier version in STOC'95. Correction at http://www.eng.tau.ac.il/~ danar/Public/KNR-fix.ps.
E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge Univ. Press, 1997.
Marco Molinaro, David P. Woodruff, and Grigory Yaroslavtsev. Amplification of one-way information complexity via codes and noise sensitivity. In ICALP, pages 960-972, 2015.
A. Razborov. On the distributional complexity of disjointness. Theoretical Computer Science, 106(2):385-390, 1992.
A. Razborov. Quantum communication complexity of symmetric predicates. Izvestiya of the Russian Academy of Sciences, mathematics, 67(1):159-176, 2003. quant-ph/0204025.
Mert Saglam and Gábor Tardos. On the communication complexity of sparse set disjointness and exists-equal problems. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 678-687, 2013.
Alexander A. Sherstov. Communication complexity under product and nonproduct distributions. In Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, CCC, pages 64-70, 2008.
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Zero-One Laws for Sliding Windows and Universal Sketches
Given a stream of data, a typical approach in streaming algorithms is to design a sophisticated algorithm with small memory that computes a specific statistic over the streaming data. Usually, if one wants to compute a different statistic after the stream is gone, it is impossible. But what if we want to compute a different statistic after the fact? In this paper, we consider the following fascinating possibility: can we collect some small amount of specific data during the stream that is "universal," i.e., where we do not know anything about the statistics we will want to later compute, other than the guarantee that had we known the statistic ahead of time, it would have been possible to do so with small memory? This is indeed what we introduce (and show) in this paper with matching upper and lower bounds: we show that it is possible to collect universal statistics of polylogarithmic size, and prove that these universal statistics allow us after the fact to compute all other statistics that are computable with similar amounts of memory. We show that this is indeed possible, both for the standard unbounded streaming model and the sliding window streaming model.
Streaming Algorithms
Universality
Sliding Windows
573-590
Regular Paper
Vladimir
Braverman
Vladimir Braverman
Rafail
Ostrovsky
Rafail Ostrovsky
Alan
Roytman
Alan Roytman
10.4230/LIPIcs.APPROX-RANDOM.2015.573
List of open problems in sublinear algorithms: Problem 20. URL: http://sublinear.info/20.
http://sublinear.info/20
List of open problems in sublinear algorithms: Problem 30. URL: http://sublinear.info/30.
http://sublinear.info/30
N. Alon, Y. Matias, and M. Szegedy. The space complexity of approximating the frequency moments. In STOC, 1996.
Z. Bar-Yossef, T. S. Jayram, R. Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. In FOCS, 2002.
Z. Bar-Yossef, T. S. Jayram, R. Kumar, D. Sivakumar, and L. Trevisan. Counting distinct elements in a data stream. In RANDOM. Springer, 2002.
L. Bhuvanagiri, S. Ganguly, D. Kesh, and C. Saha. Simpler algorithm for estimating frequency moments of data streams. In SODA, 2006.
V. Braverman, R. Gelles, and R. Ostrovsky. How to catch l₂-heavy-hitters on sliding windows. In COCOON. Springer, 2013.
V. Braverman and R. Ostrovsky. Smooth histograms for sliding windows. In FOCS, 2007.
V. Braverman and R. Ostrovsky. Zero-one frequency laws. In STOC, 2010.
V. Braverman and R. Ostrovsky. Generalizing the layering method of Indyk and Woodruff: Recursive sketches for frequency-based vectors on streams. In APPROX-RANDOM. Springer, 2013.
A. Chakrabarti, S. Khot, and X. Sun. Near-optimal lower bounds on the multi-party communication complexity of set disjointness. In CCC, 2003.
D. Coppersmith and R. Kumar. An improved data stream algorithm for frequency moments. In SODA, 2004.
G. Cormode, M. Datar, P. Indyk, and S. Muthukrishnan. Comparing data streams using hamming norms (how to zero in). IEEE TKDE, 15(3):529-540, 2003.
G. Cormode and S. Muthukrishnan. What’s hot and what’s not: tracking most frequent items dynamically. ACM TODS, 30(1):249-278, 2005.
M. Datar, A. Gionis, P. Indyk, and R. Motwani. Maintaining stream statistics over sliding windows. In SODA, 2002.
J. Feigenbaum, S. Kannan, M. Strauss, and M. Viswanathan. An approximate l₁-difference algorithm for massive data streams. In FOCS, 1999.
P. Flajolet and G. Nigel Martin. Probabilistic counting algorithms for data base applications. JCSS, 31(2):182-209, 1985.
S. Ganguly and G. Cormode. On estimating frequency moments of data streams. In APPROX-RANDOM. Springer, 2007.
P. B. Gibbons and S. Tirthapura. Distributed streams algorithms for sliding windows. In SPAA, 2002.
L. Golab, D. DeHaan, E. D. Demaine, A. Lopez-Ortiz, and J. I. Munro. Identifying frequent items in sliding windows over on-line packet streams. In IMC, 2003.
R. Hung, L. K. Lee, and H. F. Ting. Finding frequent items over sliding windows with constant update time. IPL, 110(7):257-260, 2010.
P. Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation. JACM, 53(3):307-323, 2006.
P. Indyk and D. Woodruff. Tight lower bounds for the distinct elements problem. In FOCS, 2003.
P. Indyk and D. Woodruff. Optimal approximations of the frequency moments of data streams. In STOC, 2005.
C. Jin, W. Qian, C. Sha, J. X. Yu, and A. Zhou. Dynamically maintaining frequent items over a data stream. In CIKM, 2003.
D. M. Kane, J. Nelson, and D. Woodruff. On the exact space complexity of sketching and streaming small norms. In SODA, 2010.
D. M. Kane, J. Nelson, and D. Woodruff. An optimal algorithm for the distinct elements problem. In PODS, 2010.
Y. Li, H. Nguy\sbox0ê\sbox2~\ooalign\hidewidthåise\dimexpr\ht0-\ht2+.3ex\box2 \hidewidthŗêŗn, and D. Woodruff. Turnstile streaming algorithms might as well be linear sketches. In STOC, 2014.
R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, 1995.
G. Nie and Z. Lu. Approximate frequency counts in sliding window over data stream. In CCECE, 2005.
D. Woodruff. Optimal space lower bounds for all frequency moments. In SODA, 2004.
L. Zhang and Y. Guan. Frequency estimation over sliding windows. In ICDE, 2008.
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Universal Sketches for the Frequency Negative Moments and Other Decreasing Streaming Sums
Given a stream with frequency vector f in n dimensions, we characterize the space necessary for approximating the frequency negative moments Fp, where p<0, in terms of n, the accuracy, and the L_1 length of the vector f. To accomplish this, we actually prove a much more general result. Given any nonnegative and nonincreasing function g, we characterize the space necessary for any streaming algorithm that outputs a (1 +/- eps)-approximation to the sum of the coordinates of the vector f transformed by g. The storage required is expressed in the form of the solution to a relatively simple nonlinear optimization problem, and the algorithm is universal for (1 +/- eps)-approximations to any such sum where the applied function is nonnegative, nonincreasing, and has the same or smaller space complexity as g. This partially answers an open question of Nelson (IITK Workshop Kanpur, 2009).
data streams
frequency moments
negative moments
591-605
Regular Paper
Vladimir
Braverman
Vladimir Braverman
Stephen R.
Chestnut
Stephen R. Chestnut
10.4230/LIPIcs.APPROX-RANDOM.2015.591
Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. In Symposium on the Theory of Computing, pages 20-29, 1996.
Alexandr Andoni, Robert Krauthgamer, and Krzysztof Onak. Streaming algorithms via precision sampling. In IEEE Foundations of Computer Science, pages 363-372, 2011.
Alexandr Andoni, Huy L. Nguy\cftilen, Yury Polyanskiy, and Yihong Wu. Tight lower bound for linear sketches of moments. In Automata, languages, and programming, volume 7965 of Lec. Notes in Comput. Sci., pages 25-32. Springer, Heidelberg, 2013.
Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. J. Comput. System Sci., 68(4):702-732, 2004.
Lakshminath Bhuvanagiri, Sumit Ganguly, Deepanjan Kesh, and Chandan Saha. Simpler algorithm for estimating frequency moments of data streams. In ACM-SIAM Symposium on Discrete Algorithms, pages 708-713, 2006.
Vladimir Braverman, Jonathan Katzman, Charles Seidell, and Gregory Vorsanger. Approximating large frequency moments with O(n^1-2/k) bits. arXiv preprint arXiv:1401.1763, 2014.
Vladimir Braverman and Rafail Ostrovsky. Recursive sketching for frequency moments. arXiv preprint arXiv:1011.2571, 2010.
Vladimir Braverman and Rafail Ostrovsky. Zero-one frequency laws. In ACM Symposium on the Theory of Computing, pages 281-290, 2010.
Vladimir Braverman, Rafail Ostrovsky, and Alan Roytman. Universal streaming. arXiv preprint arXiv:1408.2604, 2014.
Peter S. Bullen. Handbook of means and their inequalities. Springer Science & Business Media, 2003.
Amit Chakrabarti, Graham Cormode, and Andrew McGregor. A near-optimal algorithm for computing the entropy of a stream. In ACM-SIAM Symposium on Discrete Algorithms, pages 328-335. Society for Industrial and Applied Mathematics, 2007.
Amit Chakrabarti, Khanh Do Ba, and S. Muthukrishnan. Estimating entropy and entropy norm on data streams. Internet Math., 3(1):63-78, 2006.
Amit Chakrabarti, Subhash Khot, and Xiaodong Sun. Near-optimal lower bounds on the multi-party communication complexity of set disjointness. In IEEE Conference on Computational Complexity, pages 107-117, 2003.
Moses Charikar, Kevin Chen, and Martin Farach-Colton. Finding frequent items in data streams. In Automata, languages and programming, volume 2380 of Lec. Notes in Comput. Sci., pages 693-703. Springer, Berlin, 2002.
Don Coppersmith and Ravi Kumar. An improved data stream algorithm for frequency moments. In ACM-SIAM Symposium on Discrete Algorithms, pages 151-156, 2004.
Sumit Ganguly. Estimating frequency moments of data streams using random linear combinations. In Approximation, Randomization, and Combinatorial Optimization, pages 369-380. Springer, 2004.
Sumit Ganguly. Polynomial estimators for high frequency moments. arXiv preprint arXiv:1104.4552, 2011.
Edwin L Grab and I Richard Savage. Tables of the expected value of 1/X for positive bernoulli and poisson variables. J. Am. Stat. Assoc., 49(265):169-177, 1954.
André Gronemeier. Asymptotically optimal lower bounds on the NIH-multi-party information complexity of the AND-function and disjointness. In Symposium on Theoretical Aspects of Computer Science, 2009.
Sudipto Guha, Piotr Indyk, and Andrew McGregor. Sketching information divergences. In Learning theory, volume 4539 of Lec. Notes in Comput. Sci., pages 424-438. Springer, Berlin, 2007.
Nicholas JA Harvey, Jelani Nelson, and Krzysztof Onak. Sketching and streaming entropy via approximation theory. In IEEE Symposium on Foundations of Computer Science, pages 489-498, 2008.
Piotr Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation. J. of the ACM, 53(3):307-323, 2006.
Piotr Indyk and David Woodruff. Optimal approximations of the frequency moments of data streams. In ACM Symposium on the Theory of Computing, pages 202-208, 2005.
C.~Matthew Jones and Anatoly A. Zhigljavsky. Approximating the negative moments of the Poisson distribution. Statistics & Probability letters, 66(2):171-181, 2004.
Hossein Jowhari, Mert Sağlam, and Gábor Tardos. Tight bounds for lp samplers, finding duplicates in streams, and related problems. In ACM Symposium on Principles of Database Systems, pages 49-58, 2011.
Daniel M. Kane, Jelani Nelson, and David P. Woodruff. On the exact space complexity of sketching and streaming small norms. In ACM-SIAM Symposium on Discrete Algorithms, pages 1161-1178, 2010.
Daniel M Kane, Jelani Nelson, and David P Woodruff. An optimal algorithm for the distinct elements problem. In ACM Symposium on Principles of Database Systems, pages 41-52, 2010.
Eyal Kushilevitz and Noam Nisan. Communication complexity. Cambridge University Press, Cambridge, 1997.
Ping Li. Estimators and tail bounds for dimension reduction in l_α (0 < α\leq2) using stable random projections. In ACM-SIAM Symposium on Discrete Algorithms, pages 10-19, 2008.
Yi Li and David P Woodruff. A tight lower bound for high frequency moment estimation with small error. In Approximation, Randomization, and Combinatorial Optimization, pages 623-638. Springer, 2013.
W. Mendenhall and H. Lehman, E.Ȧn approximation to the negative moments of the positive binomial useful in life testing. Technometrics, 2(2):227-242, 1960.
Morteza Monemizadeh and David P Woodruff. 1-pass relative-error l_p-sampling with applications. In ACM-SIAM Symposium on Discrete Algorithms, pages 1143-1160, 2010.
Jelani Nelson. List of open problems in sublinear algorithms: Problem 30. URL: http://sublinear.info/30.
http://sublinear.info/30
Robert F Reilly and Robert P Schweihs. The handbook of business valuation and intellectual property analysis. McGraw Hill, 2004.
Frederick F Stephan. The expected value and variance of the reciprocal and other negative powers of a positive Bernoullian variate. Ann. Math. Stat., 16(1):50-61, 1945.
David P Woodruff. Data streams and applications in computer science. Bulletin of EATCS, 3(114), 2014.
Marko Znidaric. Asymptotic expansion for inverse moments of binomial and Poisson distributions. arXiv preprint math/0511226, 2005.
Marko Žnidarič and Martin Horvat. Exponential complexity of an adiabatic algorithm for an NP-complete problem. Phys. Rev. A, 73(2), 2006.
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Dependent Random Graphs and Multi-Party Pointer Jumping
We initiate a study of a relaxed version of the standard Erdos-Renyi random graph model, where each edge may depend on a few other edges. We call such graphs "dependent random graphs". Our main result in this direction is a thorough understanding of the clique number of dependent random graphs. We also obtain bounds for the chromatic number. Surprisingly, many of the standard properties of random graphs also hold in this relaxed setting. We show that with high probability, a dependent random graph will contain a clique of size ((1-o(1))log(n))/log(1/p), and the chromatic number will be at most (nlog(1/(1-p)))/log(n). We expect these results to be of independent interest. As an application and second main result, we give a new communication protocol for the k-player Multi-Party Pointer Jumping problem (MPJk) in the number-on-the-forehead (NOF) model. Multi-Party Pointer Jumping is one of the canonical NOF communication problems, yet even for three players, its communication complexity is not well understood. Our protocol for MPJ3 costs O((n * log(log(n)))/log(n)) communication, improving on a bound from [BrodyChakrabarti08]. We extend our protocol to the non-Boolean pointer jumping problem, achieving an upper bound which is o(n) for any k >= 4 players. This is the first o(n) protocol and improves on a bound of Damm, Jukna, and Sgall, which has stood for almost twenty years.
random graphs
communication complexity
number-on-the-forehead model
pointer jumping
606-624
Regular Paper
Joshua
Brody
Joshua Brody
Mario
Sanchez
Mario Sanchez
10.4230/LIPIcs.APPROX-RANDOM.2015.606
Noga Alon and Asaf Nussboim. k-wise independent random graphs. In Proc. 49th Annual IEEE Symposium on Foundations of Computer Science, pages 813-822, 2008.
Noga Alon and Joel H. Spencer. The Probabilistic Method. Wiley-Interscience, New York, NY, 2000.
László Babai, Thomas P. Hayes, and Peter G. Kimmel. The cost of the missing bit: Communication complexity with help. Combinatorica, 21(4):455-488, 2001.
Richard Beigel and Jun Tarui. On ACC. Comput. Complexity, 4:350-366, 1994.
Béla Bollobás. The chromatic number of random graphs. Combinatorica, 8(1):49-55, 1988.
Béla Bollobás. Random graphs. Springer, 1998.
Béla Bollobás and Paul Erdős. Cliques in random graphs. Mathematical Proceedings of the Cambridge Philosophical Society, 80:419-427, 11 1976.
Joshua Brody. The maximum communication complexity of multi-party pointer jumping. In Proc. 24th Annual IEEE Conference on Computational Complexity, pages 379-386, 2009.
Joshua Brody and Amit Chakrabarti. Sublinear communication protocols for multi-party pointer jumping and a related lower bound. In Proc. 25th International Symposium on Theoretical Aspects of Computer Science, pages 145-156, 2008.
Amit Chakrabarti. Lower bounds for multi-player pointer jumping. In Proc. 22nd Annual IEEE Conference on Computational Complexity, pages 33-45, 2007.
Ashok K. Chandra, Merrick L. Furst, and Richard J. Lipton. Multi-party protocols. In Proc. 15th Annual ACM Symposium on the Theory of Computing, pages 94-99, 1983.
Carsten Damm, Stasys Jukna, and Jiří Sgall. Some bounds on multiparty communication complexity of pointer jumping. Comput. Complexity, 7(2):109-127, 1998. Preliminary version in Proc. 13th International Symposium on Theoretical Aspects of Computer Science , pages 643-654, 1996.
Devdatt P. Dubhashi and Alessandro Panconesi. Concentration of measure for the analysis of randomized algorithms. Cambridge University Press, 2009.
Paul Erdős and Alfréd Rényi. On random graphs i. Publ. Math. Debrecen, 6:290-297, 1959.
Andre Gronemeier. Nof-multiparty information complexity bounds for pointer jumping. In Proc. 31st International Symposium on Mathematical Foundations of Computer Science, pages 459-470. Springer, 2006.
Johan Håstad and Mikael Goldmann. On the power of small-depth threshold circuits. Comput. Complexity, 1:113-129, 1991.
Svante Janson. Large deviations for sums of partly dependent random variables. Random Structures & Algorithms, 24(3):234-248, 2004.
Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, Cambridge, 1997.
Pavel Pudlák, Vojtěch Rödl, and Jiří Sgall. Boolean circuits, tensor ranks and communication complexity. SIAM J. Comput., 26(3):605-633, 1997.
Emanuele Viola and Avi Wigderson. One-way multi-party communication lower bound for pointer jumping with applications. In Proc. 48th Annual IEEE Symposium on Foundations of Computer Science, pages 427-437, 2007.
Ryan Williams. Nonuniform acc circuit lower bounds. J. ACM, 61(1):32, 2014.
Andrew C. Yao. On ACC and threshold circuits. In Proc. 31st Annual IEEE Symposium on Foundations of Computer Science, pages 619-627, 1990.
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Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness
Low-degree polynomial approximations to the sign function underlie pseudorandom generators for halfspaces, as well as algorithms for agnostically learning halfspaces. We study the limits of these constructions by proving inapproximability results for the sign function. First, we investigate the derandomization of Chernoff-type concentration inequalities. Schmidt et al. (SIAM J. Discrete Math. 1995) showed that a tail bound of delta can be established for sums of Bernoulli random variables with only O(log(1/delta))-wise independence. We show that their results are tight up to constant factors. Secondly, the “polynomial regression” algorithm of Kalai et al. (SIAM J. Comput. 2008) shows that halfspaces can be efficiently learned with respect to log-concave distributions on R^n in the challenging agnostic learning model. The power of this algorithm relies on the fact that under log-concave distributions, halfspaces can be approximated arbitrarily well by low-degree polynomials. In contrast, we exhibit a large class of non-log-concave distributions under which polynomials of any degree cannot approximate the sign function to within arbitrarily low error.
Polynomial Approximations
Pseudorandomness
Concentration
Learning Theory
Halfspaces
625-644
Regular Paper
Mark
Bun
Mark Bun
Thomas
Steinke
Thomas Steinke
10.4230/LIPIcs.APPROX-RANDOM.2015.625
Scott Aaronson and Yaoyun Shi. Quantum lower bounds for the collision and the element distinctness problems. J. ACM, 51(4):595-605, 2004.
Noga Alon, László Babai, and Alon Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7(4):567-583, 1986.
Noga Alon and Asaf Nussboim. K-wise independent random graphs. In Foundations of Computer Science, 2008. FOCS'08. IEEE 49th Annual IEEE Symposium on, pages 813-822. IEEE, 2008.
Louay M. J. Bazzi. Polylogarithmic independence can fool DNF formulas. SIAM J. Comput., 38(6):2220-2272, March 2009.
Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. J. ACM, 48(4):778-797, 2001.
Richard Beigel. The polynomial method in circuit complexity. In Structure in Complexity Theory Conference, pages 82-95, 1993.
Richard Beigel. Perceptrons, PP, and the polynomial hierarchy. Computational Complexity, 4:339-349, 1994.
M. Bellare and J. Rompel. Randomness-efficient oblivious sampling. In FOCS, pages 276-287, Nov 1994.
Itai Benjamini, Ori Gurel-Gurevich, and Ron Peled. On k-wise independent distributions and boolean functions. arXiv preprint arXiv:1201.3261, 2012.
S. N. Bernstein. Le problème de l'approximation des fonctions continues sur tout l'axe réel et l'une de ses applications. Bull. Math. Soc. France, 52:399-410, 1924.
Eric Blais, Ryan O'Donnell, and Karl Wimmer. Polynomial regression under arbitrary product distributions. Machine Learning, 80(2-3):273-294, 2010.
Aline Bonami. Étude des coefficients de fourier des fonctions de l^p(g). Annales de l'institut Fourier, 20(2):335-402, 1970.
Mark Braverman. Polylogarithmic independence fools AC0 circuits. J. ACM, 57(5):28:1-28:10, June 2008.
Mark Braverman, Anup Rao, Ran Raz, and Amir Yehudayoff. Pseudorandom generators for regular branching programs. FOCS, pages 40-47, 2010.
Joshua Brody and Elad Verbin. The coin problem and pseudorandomness for branching programs. In FOCS, pages 30-39, 2010.
Lennart Carleson. Bernstein’s approximation problem. Proc. Amer. Math. Soc., 2:953-961, 1951.
E.W. Cheney. Introduction to Approximation Theory. AMS Chelsea Publishing Series. AMS Chelsea Pub., 1982.
Dana Dachman-Soled, Vitaly Feldman, Li-Yang Tan, Andrew Wan, and Karl Wimmer. Approximate resilience, monotonicity, and the complexity of agnostic learning. CoRR, abs/1405.5268, 2014. To appear in SODA 2015.
Amit Daniely, Nati Linial, and Shai Shalev-Shwartz. The complexity of learning halfspaces using generalized linear methods. CoRR, abs/1211.0616, 2014.
Anindya De, Omid Etesami, Luca Trevisan, and Madhur Tulsiani. Improved pseudorandom generators for depth 2 circuits. In Maria Serna, Ronen Shaltiel, Klaus Jansen, and Jose Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 6302 of Lecture Notes in Computer Science, pages 504-517, 2010.
Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, and Emanuele Viola. Bounded independence fools halfspaces. In In Proc. 50th Annual Symposium on Foundations of Computer Science (FOCS), pages 171-180, 2009.
Ilias Diakonikolas, Rocco A. Servedio, Li-Yang Tan, and Andrew Wan. A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functions. In Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity, CCC'10, pages 211-222, Washington, DC, USA, 2010. IEEE Computer Society.
H. Ehlich and K. Zeller. Schwankung von polynomen zwischen gitterpunkten. Mathematische Zeitschrift, 86:41-44, 1964.
Vitaly Feldman, Parikshit Gopalan, Subhash Khot, and Ashok Kumar Ponnuswami. New results for learning noisy parities and halfspaces. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS'06, pages 563-574, Washington, DC, USA, 2006. IEEE Computer Society.
Vitaly Feldman and Pravesh Kothari. Agnostic learning of disjunctions on symmetric distributions. CoRR, abs/1405.6791, 2014.
Parikshit Gopalan, Adam Tauman Kalai, and Adam R. Klivans. Agnostically learning decision trees. In STOC, pages 527-536, 2008.
Parikshit Gopalan, Daniel Kane, and Raghu Meka. Pseudorandomness for concentration bounds and signed majorities. CoRR, abs/1411.4584, 2014.
Parikshit Gopalan, Ryan O'Donnell, Yi Wu, and David Zuckerman. Fooling functions of halfspaces under product distributions. In Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity, CCC'10, pages 223-234, Washington, DC, USA, 2010. IEEE Computer Society.
Parikshit Gopalan and Jaikumar Radhakrishnan. Finding duplicates in a data stream. In Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 402-411. Society for Industrial and Applied Mathematics, 2009.
V. Guruswami and P. Raghavendra. Hardness of learning halfspaces with noise. In Proceedings of FOCS'06, pages 543-552, 2006.
Nicholas J. A. Harvey, Jelani Nelson, and Krzysztof Onak. Sketching and streaming entropy via approximation theory. In FOCS, pages 489-498, 2008.
Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58(301):pp. 13-30, 1963.
Russell Impagliazzo, Noam Nisan, and Avi Wigderson. Pseudorandomness for network algorithms. In STOC, pages 356-364, 1994.
S. Izumi and T. Kawata. Quasi-analytic class and closure of tⁿ in the interval (-∞, ∞). Tohoku Math. J., 43:267-273, 1937.
Adam Tauman Kalai, Adam R. Klivans, Yishay Mansour, and Rocco A. Servedio. Agnostically learning halfspaces. SIAM J. Comput., 37(6):1777-1805, 2008.
Daniel M. Kane, Adam Klivans, and Raghu Meka. Learning halfspaces under log-concave densities: Polynomial approximations and moment matching. In COLT, pages 522-545, 2013.
Daniel M Kane and Jelani Nelson. A derandomized sparse Johnson-Lindenstrauss transform. arXiv preprint arXiv:1006.3585, 2010.
Michael Kearns, Robert E. Schapire, and Linda M. Sellie. Toward efficient agnostic learning. In Machine Learning, pages 341-352. ACM Press, 1994.
Adam R. Klivans, Philip M. Long, and Alex K. Tang. Baum’s algorithm learns intersections of halfspaces with respect to log-concave distributions. In Irit Dinur, Klaus Jansen, Joseph Naor, and José Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 5687 of Lecture Notes in Computer Science, pages 588-600. Springer Berlin Heidelberg, 2009.
Adam R. Klivans, Ryan O'Donnell, and Rocco A. Servedio. Learning geometric concepts via gaussian surface area. In FOCS, pages 541-550, 2008.
Adam R. Klivans and Rocco A. Servedio. Learning DNF in time 2^õ(n^1/3). J. Comput. Syst. Sci., 68(2):303-318, 2004.
Adam R. Klivans and Alexander A. Sherstov. Lower bounds for agnostic learning via approximate rank. Computational Complexity, 19(4):581-604, 2010.
Michal Koucký, Prajakta Nimbhorkar, and Pavel Pudlák. Pseudorandom generators for group products. In STOC, pages 263-272, 2011.
Wee Sun Lee, Peter L. Bartlett, and Robert C. Williamson. On efficient agnostic learning of linear combinations of basis functions. In Proceedings of the Eighth Annual Conference on Computational Learning Theory, COLT'95, pages 369-376, New York, NY, USA, 1995. ACM.
Doron Lubinsky. A survey of weighted polynomial approximation with exponential weights. Surveys in Approximation Theory, 3:1-105, 2007.
Michael Luby and Avi Wigderson. Pairwise independence and derandomization. Citeseer, 1995.
A. A. Markov. On a question of D. I. Mendeleev. Zapiski Imperatorskoi Akademii Nauk,, 62:1-24, 1890.
Raghu Meka and David Zuckerman. Pseudorandom generators for polynomial threshold functions. In Proceedings of the Forty-second ACM Symposium on Theory of Computing, STOC'10, pages 427-436, New York, NY, USA, 2010. ACM.
Marvin Minsky and Seymour Papert. Perceptrons: An Introduction to Computational Geometry. MIT Press, Cambridge MA, 1972.
Michael Mitzenmacher and Salil Vadhan. Why simple hash functions work: Exploiting the entropy in a data stream. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'08, pages 746-755, Philadelphia, PA, USA, 2008. Society for Industrial and Applied Mathematics.
Elchanan Mossel and Mesrob I Ohannessian. On the impossibility of learning the missing mass. arXiv preprint arXiv:1503.03613, 2015.
Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM J. Computing, 22:838-856, 1993.
P. Nevai and V. Totik. Sharp Nikolskii inequalities with exponential weights. Analysis Mathematica, 13(4):261-267, 1987.
Paul Nevai. Géza Freud, orthogonal polynomials and Christoffel functions. A case study. Journal of Approximation Theory, 48(1):3-167, 1986.
Paul Nevai and Vilmos Totik. Weighted polynomial inequalities. Constructive Approximation, 2(1):113-127, 1986.
N. Nisan and M. Szegedy. On the degree of boolean functions as real polynomials. Computational Complexity, 4:301-313, 1994.
Noam Nisan. RL ⊂ SC. In STOC, pages 619-623, 1992.
Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, New York, NY, USA, 2014.
Ramamohan Paturi. On the degree of polynomials that approximate symmetric boolean functions (preliminary version). In STOC, pages 468-474, 1992.
Omer Reingold. Undirected connectivity in log-space. J. ACM, 55(4):17:1-17:24, September 2008.
Omer Reingold, Thomas Steinke, and Salil Vadhan. Pseudorandomness for regular branching programs via fourier analysis. In APPROX-RANDOM, pages 655-670, 2013.
T. J. Rivlin and E. W. Cheney. A comparison of uniform approximations on an interval and a finite subset thereof. SIAM J. Numer. Anal., 3(2):311-320, 1966.
Sushant Sachdeva and Nisheeth K. Vishnoi. Faster algorithms via approximation theory. Foundations and Trends in Theoretical Computer Science, 9(2):125-210, 2014.
J. Schmidt, A. Siegel, and A. Srinivasan. Chernoff-Hoeffding bounds for applications with limited independence. SIAM J. Discrete Mathematics, 8(2):223-250, 1995.
Alexander A. Sherstov. Communication lower bounds using dual polynomials. Bulletin of the EATCS, 95:59-93, 2008.
Alexander A. Sherstov. Separating AC^0 from depth-2 majority circuits. SIAM J. Comput., 38(6):2113-2129, 2009.
Leslie G. Valiant. A theory of the learnable. Commun. ACM, 27(11):1134-1142, 1984.
Karl Wimmer. Agnostically learning under permutation invariant distributions. In Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS'10, pages 113-122, Washington, DC, USA, 2010. IEEE Computer Society.
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Tighter Connections between Derandomization and Circuit Lower Bounds
We tighten the connections between circuit lower bounds and derandomization for each of the following three types of derandomization:
- general derandomization of promiseBPP (connected to Boolean circuits),
- derandomization of Polynomial Identity Testing (PIT) over fixed finite fields (connected to arithmetic circuit lower bounds over the same field), and
- derandomization of PIT over the integers (connected to arithmetic circuit lower bounds over the integers).
We show how to make these connections uniform equivalences, although at the expense of using somewhat less common versions of complexity classes and for a less studied notion of inclusion.
Our main results are as follows:
1. We give the first proof that a non-trivial (nondeterministic subexponential-time) algorithm for PIT over a fixed finite field yields arithmetic circuit lower bounds.
2. We get a similar result for the case of PIT over the integers, strengthening a result of Jansen and Santhanam [JS12] (by removing the need for advice).
3. We derive a Boolean circuit lower bound for NEXP intersect coNEXP from the assumption of sufficiently strong non-deterministic derandomization of promiseBPP (without advice), as well as from the assumed existence of an NP-computable non-empty property of Boolean functions useful for proving superpolynomial circuit lower bounds (in the sense of natural proofs of [RR97]); this strengthens the related results of [IKW02].
4. Finally, we turn all of these implications into equivalences for appropriately defined promise classes and for a notion of robust inclusion/separation (inspired by [FS11]) that lies between the classical "almost everywhere" and "infinitely often" notions.
derandomization
circuit lower bounds
polynomial identity testing
promise BPP
hardness vs. randomness
645-658
Regular Paper
Marco L.
Carmosino
Marco L. Carmosino
Russell
Impagliazzo
Russell Impagliazzo
Valentine
Kabanets
Valentine Kabanets
Antonina
Kolokolova
Antonina Kolokolova
10.4230/LIPIcs.APPROX-RANDOM.2015.645
S. Aaronson and D. van Melkebeek. On circuit lower bounds from derandomization. Theory of Computing, 7(1):177-184, 2011.
M. Ajtai and A. Wigderson. Deterministic simulation of probabilistic constant depth circuits. In Proceedings of the Twenty-Sixth Annual IEEE Symposium on Foundations of Computer Science, pages 11-19, 1985.
B. Aydinlioglu and D. van Melkebeek. Nondeterministic circuit lower bounds from mildly de-randomizing arthur-merlin games. In Proceedings of the 27th Conference on Computational Complexity, CCC 2012, Porto, Portugal, June 26-29, 2012, pages 269-279. IEEE, 2012.
L. Babai, L. Fortnow, and C. Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1:3-40, 1991.
L. Babai, L. Fortnow, N. Nisan, and A. Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3:307-318, 1993.
L. Babai and S. Moran. Arthur-merlin games: a randomized proof system, and a hierarchy of complexity classes. Journal of Computer and System Sciences, 36(2):254-276, 1988.
M. Blum and S. Micali. How to generate cryptographically strong sequences of pseudo-random bits. SIAM Journal on Computing, 13:850-864, 1984.
R.A. DeMillo and R.J. Lipton. A probabilistic remark on algebraic program testing. Information Processing Letters, 7:193-195, 1978.
L. Fortnow and R. Santhanam. Robust simulations and significant separations. In Automata, Languages and Programming - 38th International Colloquium, ICALP, Proceedings, Part I, pages 569-580, 2011.
J. Heintz and C.-P. Schnorr. Testing polynomials which are easy to compute. L'Enseignement Mathématique, 30:237-254, 1982.
R. Impagliazzo, V. Kabanets, and A. Wigderson. In search of an easy witness: Exponential time vs. probabilistic polynomial time. Journal of Computer and System Sciences, 65(4):672-694, 2002.
R. Impagliazzo and A. Wigderson. P=BPP if E requires exponential circuits: Derandomizing the XOR Lemma. In STOC'97, pages 220-229, 1997.
M.J. Jansen and R. Santhanam. Stronger lower bounds and randomness-hardness trade-offs using associated algebraic complexity classes. In Christoph Dürr and Thomas Wilke, editors, STACS, volume 14 of LIPIcs, pages 519-530. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2012.
V. Kabanets and R. Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004.
E. Kaltofen. Factorization of polynomials given by straight-line programs. In S. Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 375-412. JAI Press, Greenwich, CT, 1989.
R. Kannan. Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control, 55(1-3):40-56, 1982.
R.M. Karp and R.J. Lipton. Turing machines that take advice. L'Enseignement Mathématique, 28(3-4):191-209, 1982.
J. Kinne, D. van Melkebeek, and R. Shaltiel. Pseudorandom generators, typically-correct derandomization, and circuit lower bounds. Computational Complexity, 21(1):3-61, 2012.
N. Nisan and A. Wigderson. Hardness vs. randomness. Journal of Computer and System Sciences, 49:149-167, 1994.
Alexander A. Razborov and Steven Rudich. Natural proofs. J. Comput. Syst. Sci., 55(1):24-35, 1997.
J.T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the Association for Computing Machinery, 27(4):701-717, 1980.
R. Shaltiel and C. Umans. Simple extractors for all min-entropies and a new pseudorandom generator. Journal of the Association for Computing Machinery, 52(2):172-216, 2005.
A. Shamir. IP=PSPACE. Journal of the Association for Computing Machinery, 39(4):869-877, 1992.
M. Sudan, L. Trevisan, and S. Vadhan. Pseudorandom generators without the XOR lemma. Journal of Computer and System Sciences, 62(2):236-266, 2001.
L. Trevisan and S.P. Vadhan. Pseudorandomness and average-case complexity via uniform reductions. Computational Complexity, 16(4):331-364, 2007.
C. Umans. Pseudo-random generators for all hardnesses. Journal of Computer and System Sciences, 67(2):419-440, 2003.
R. Williams. Nonuniform acc circuit lower bounds. Journal of the ACM (JACM), 61(1):2, 2014.
A.C. Yao. Theory and applications of trapdoor functions. In Proceedings of the Twenty-Third Annual IEEE Symposium on Foundations of Computer Science, pages 80-91, 1982.
R.E. Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of an International Symposium on Symbolic and Algebraic Manipulation (EUROSAM'79), LNCS, pages 216-226, 1979.
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Average Distance Queries through Weighted Samples in Graphs and Metric Spaces: High Scalability with Tight Statistical Guarantees
The average distance from a node to all other nodes in a graph, or from a query point in a metric space to a set of points, is a fundamental quantity in data analysis. The inverse of the average distance, known as the (classic) closeness centrality of a node, is a popular importance measure in the study of social networks. We develop novel structural insights on the sparsifiability of the distance relation via weighted sampling. Based on that, we present highly practical algorithms with strong statistical guarantees for fundamental problems. We show that the average distance (and hence the centrality) for all nodes in a graph can be estimated using O(epsilon^{-2}) single-source distance computations. For a set V of n points in a metric space, we show that after preprocessing which uses O(n) distance computations we can compute a weighted sample S subset of V of size O(epsilon^{-2}) such that the average distance from any query point v to V can be estimated from the distances from v to S. Finally, we show that for a set of points V in a metric space, we can estimate the average pairwise distance using O(n+epsilon^{-2}) distance computations. The estimate is based on a weighted sample of O(epsilon^{-2}) pairs of points, which is computed using O(n) distance computations. Our estimates are unbiased with normalized mean square error (NRMSE) of at most epsilon. Increasing the sample size by a O(log(n)) factor ensures that the probability that the relative error exceeds epsilon is polynomially small.
Closeness Centrality; Average Distance; Metric Space; Weighted Sampling
659-679
Regular Paper
Shiri
Chechik
Shiri Chechik
Edith
Cohen
Edith Cohen
Haim
Kaplan
Haim Kaplan
10.4230/LIPIcs.APPROX-RANDOM.2015.659
A. Abboud, F. Grandoni, and V. Vassilevska Williams. Subcubic equivalences between graph centrality problems, APSP and diameter. In SODA. ACM-SIAM, 2015.
K. Barhum, O. Goldreich, and A. Shraibman. On approximating the average distance between points. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 4627 of Lecture Notes in Computer Science. Springer, 2007.
A. Bavelas. A mathematical model for small group structures. Human Organization, 7:16-30, 1948.
A. Bavelas. Communication patterns in task oriented groups. Journal of the Acoustical Society of America, 22:271-282, 1950.
M. A. Beauchamp. An improved index of centrality. Behavioral Science, 10:161-163, 1965.
E. Cohen, D. Delling, T. Pajor, and R. F. Werneck. Computing classic closeness centrality, at scale. In COSN. ACM, 2014.
E. Cohen, D. Delling, T. Pajor, and R. F. Werneck. Sketch-based influence maximization and computation: Scaling up with guarantees. In CIKM, 2014.
E. Cohen, N. Duffield, C. Lund, M. Thorup, and H. Kaplan. Efficient stream sampling for variance-optimal estimation of subset sums. SIAM J. Comput., 40(5), 2011.
M. B. Cohen and R. Peng. 𝓁_p row sampling by lewis weights. In STOC. ACM, 2015.
T. M. Cover and P. E. Hart. Nearest neighbor pattern classification. IEEE Transactions on Information Theory, 13(1):21-27, 1967.
D. Eppstein and J. Wang. Fast approximation of centrality. In SODA, pages 228-229, 2001.
M. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. Journal of the ACM, 34(3):596-615, 1987.
L. C. Freeman. A set of measures of centrality based on betweeness. Sociometry, 40:35-41, 1977.
L. C. Freeman. Centrality in social networks: Conceptual clarification. Social Networks, 1, 1979.
O. Goldreich and D. Ron. Approximating average parameters of graphs. Random Struct. Algorithms, 32(4):473-493, 2008.
J. Hamidzadeh, R. Monsefi, and H. S. Yazdi. DDC: distance-based decision classifier. Neural Computing and Applications, 21(7), 2012.
P. Indyk. Sublinear time algorithms for metric space problems. In STOC. ACM, 1999.
P. Indyk. High-dimensional Computational Geometry. PhD thesis, Stanford University, 2000.
K. Okamoto, W. Chen, and X. Li. Ranking of closeness centrality for large-scale social networks. In Proc. 2nd Annual International Workshop on Frontiers in Algorithmics, FAW. Springer-Verlag, 2008.
G. Sabidussi. The centrality index of a graph. Psychometrika, 31(4):581-603, 1966.
M. Talagrand. Embedding subspaces of l₁ into lⁿ₁. Proc. of the American Math. Society, 108(2):363-369, 1990.
M. Thorup. Quick k-median, k-center, and facility location for sparse graphs. SIAM J. Comput., 34(2):405-432, 2004.
V. Vassilevska Williams and R. Williams. Subcubic equivalences between path, matrix and triangle problems. In FOCS. IEEE, 2010.
S. Wasserman and K. Faust, editors. Social Network Analysis: Methods and Applications. Cambridge University Press, 1994.
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Two Structural Results for Low Degree Polynomials and Applications
In this paper, two structural results concerning low degree polynomials over finite fields are given. The first states that over any finite field F, for any polynomial f on n variables with degree d > log(n)/10, there exists a subspace of F^n with dimension at least d n^(1/(d-1)) on which f is constant. This result is shown to be tight. Stated differently, a degree d polynomial cannot compute an affine disperser for dimension smaller than the stated dimension. Using a recursive argument, we obtain our second structural result, showing that any degree d polynomial f induces a partition of F^n to affine subspaces of dimension n^(1/(d-1)!), such that f is constant on each part.
We extend both structural results to more than one polynomial. We further prove an analog of the first structural result to sparse polynomials (with no restriction on the degree) and to functions that are close to low degree polynomials. We also consider the algorithmic aspect of the two structural results.
Our structural results have various applications, two of which are:
* Dvir [CC 2012] introduced the notion of extractors for varieties, and gave explicit constructions of such extractors over large fields. We show that over any finite field any affine extractor is also an extractor for varieties with related parameters. Our reduction also holds for dispersers, and we conclude that Shaltiel's affine disperser [FOCS 2011] is a disperser for varieties over the binary field.
* Ben-Sasson and Kopparty [SIAM J. C 2012] proved that any degree 3 affine disperser over a prime field is also an affine extractor with related parameters. Using our structural results, and based on the work of Kaufman and Lovett [FOCS 2008] and Haramaty and Shpilka [STOC 2010], we generalize this result to any constant degree.
low degree polynomials
affine extractors
affine dispersers
extractors for varieties
dispersers for varieties
680-709
Regular Paper
Gil
Cohen
Gil Cohen
Avishay
Tal
Avishay Tal
10.4230/LIPIcs.APPROX-RANDOM.2015.680
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The Minimum Bisection in the Planted Bisection Model
In the planted bisection model a random graph G(n,p_+,p_-) with n vertices is created by partitioning the vertices randomly into two classes of equal size (up to plus or minus 1). Any two vertices that belong to the same class are linked by an edge with probability p_+ and any two that belong to different classes with probability (p_-) <(p_+) independently. The planted bisection model has been used extensively to benchmark graph partitioning algorithms. If (p_+)=2(d_+)/n and (p_-)=2(d_-)/n for numbers 0 <= (d_-) <(d_+) that remain fixed as n tends to infinity, then with high probability the "planted" bisection (the one used to construct the graph) will not be a minimum bisection. In this paper we derive an asymptotic formula for the minimum bisection width under the assumption that (d_+)-(d_-) > c * sqrt((d_+)ln(d_+)) for a certain constant c>0.
Random graphs
minimum bisection
planted bisection
belief propagation.
710-725
Regular Paper
Amin
Coja-Oghlan
Amin Coja-Oghlan
Oliver
Cooley
Oliver Cooley
Mihyun
Kang
Mihyun Kang
Kathrin
Skubch
Kathrin Skubch
10.4230/LIPIcs.APPROX-RANDOM.2015.710
E. Abbe, A. Bandeira, and G. Hall. Exact recovery in the stochastic block model. arXiv preprint arXiv:1405.3267, 2014.
D. Aldous and J. Steele. The objective method: probabilistic combinatorial optimization and local weak convergence. In Probability on discrete structures, pages 1-72. Springer, 2004.
S. Arora, S. Rao, and U. Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM (JACM), 56:5, 2009.
V. Bapst, A. Coja-Oghlan, S. Hetterich, F. Rassmann, and D. Vilenchik. The condensation phase transition in random graph coloring. arXiv preprint arXiv:1404.5513, 2014.
B. Bollobás and A. Scott. Max cut for random graphs with a planted partition. Combinatorics, Probability and Computing, 13:451-474, 2004.
R. Boppana. Eigenvalues and graph bisection: An average-case analysis. In Proc. 28th Foundations of Computer Science, pages 280-285. IEEE, 1987.
T. Bui, S. Chaudhuri, T. Leighton, and M. Sipser. Graph bisection algorithms with good average case behavior. Combinatorica, 7:171-191, 1987.
T. Carson and R. Impagliazzo. Hill-climbing finds random planted bisections. In Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, pages 903-909. Society for Industrial and Applied Mathematics, 2001.
A. Coja-Oghlan. A spectral heuristic for bisecting random graphs. In Random Structures and Algorithms, pages 351-398, 2006.
A. Coja-Oghlan, O. Cooley, M. Kang, and K. Skubch. How does the core sit inside the mantle? arXiv preprint arXiv:1503.09030, 2015.
A. Condon and R. Karp. Algorithms for graph partitioning on the planted partition model. Random Structures and Algorithms, 18:116-140, 2001.
A. Decelle, F. Krzakala, C. Moore, and L. Zdeborová. Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Physical Review E, 84:066106, 2011.
A. Dembo, A. Montanari, and S. Sen. Extremal cuts of sparse random graphs. arXiv preprint arXiv:1503.03923, 2015.
T. Dimitriou and R. Impagliazzo. Go with the winners for graph bisection. In Proc. 9th SODA, pages 510-520. ACM/SIAM, 1998.
M. Dyer and A. Frieze. The solution of some random np-hard problems in polynomial expected time. Journal of Algorithms, 10:451-489, 1989.
U. Feige and J. Kilian. Heuristics for semirandom graph problems. Journal of Computer and System Sciences, 63:639-671, 2001.
U. Feige and R. Krauthgamer. A polylogarithmic approximation of the minimum bisection. SIAM Journal on Computing, 31:1090-1118, 2002.
M. Garey, D. Johnson, and L. Stockmeyer. Some simplified np-complete graph problems. Theoretical computer science, 1:237-267, 1976.
M. Goemans and D. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM (JACM), 42:1115-1145, 1995.
P. Holland, K. Laskey, and S. Leinhardt. Stochastic blockmodels: First steps. Social networks, 5:109-137, 1983.
S. Janson, T. Łuczak, and A. Ruciński. Random graphs. John Wiley & Sons, 2000.
M. Jerrum and G. Sorkin. The metropolis algorithm for graph bisection. Discrete Applied Mathematics, 82:155-175, 1998.
A. Juels. Topics in black-box combinatorial function optimization. PhD thesis, UC Berkeley, 1996.
R. Karp. Reducibility among combinatorial problems. Springer, 1972.
M. Karpinski. Approximability of the minimum bisection problem: An algorithmic challenge. In Proc. 27th Mathematical Foundations of Computer Science, pages 59-67. Springer, 2002.
S. Khot. Ruling out ptas for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM Journal on Computing, 36:1025-1071, 2006.
L. Kučera. Expected complexity of graph partitioning problems. Discrete Applied Mathematics, 57:193-212, 1995.
M. Luczak and C. McDiarmid. Bisecting sparse random graphs. Random Structures and Algorithms, 18:31-38, 2001.
K. Makarychev, Y. Makarychev, and A. Vijayaraghavan. Approximation algorithms for semi-random partitioning problems. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 367-384. ACM, 2012.
L. Massoulié. Community detection thresholds and the weak ramanujan property. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 694-703. ACM, 2014.
F. McSherry. Spectral partitioning of random graphs. In Proc. 42nd Foundations of Computer Science, pages 529-537. IEEE, 2001.
M. Mézard and A. Montanari. Information, physics, and computation. Oxford University Press, 2009.
E. Mossel, J. Neeman, and A. Sly. Stochastic block models and reconstruction. arXiv preprint arXiv:1202.1499, 2012.
E. Mossel, J. Neeman, and A. Sly. Belief propagation, robust reconstruction, and optimal recovery of block models. arXiv preprint arXiv:1309.1380, 2013.
E. Mossel, J. Neeman, and A. Sly. A proof of the block model threshold conjecture. arXiv preprint arXiv:1311.4115, 2013.
E. Mossel, J. Neeman, and A. Sly. Consistency thresholds for the planted bisection model. arXiv preprint arXiv:1407.1591 v2, 2014.
R. Neininger and L. Rüschendorf. A general limit theorem for recursive algorithms and combinatorial structures. The Annals of Applied Probability, 14:378-418, 2004.
H. Räcke. Optimal hierarchical decompositions for congestion minimization in networks. In Proc. 40th ACM symposium on Theory of computing, pages 255-264. ACM, 2008.
M. Talagrand. The parisi formula. Annals of Mathematics, 163:221-263, 2006.
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Local Convergence of Random Graph Colorings
Let G=G(n,m) be a random graph whose average degree d=2m/n is below the k-colorability threshold. If we sample a k-coloring Sigma of G uniformly at random, what can we say about the correlations between the colors assigned to vertices that are far apart? According to a prediction from statistical physics, for average degrees below the so-called condensation threshold d_c, the colors assigned to far away vertices are asymptotically independent [Krzakala et al: PNAS 2007]. We prove this conjecture for k exceeding a certain constant k_0. More generally, we determine the joint distribution of the k-colorings that Sigma induces locally on the bounded-depth neighborhoods of a fixed number of vertices.
Random graph
Galton-Watson tree
phase transitions
graph coloring
Gibbs distribution
convergence
726-737
Regular Paper
Amin
Coja-Oghlan
Amin Coja-Oghlan
Charilaos
Efthymiou
Charilaos Efthymiou
Nor
Jaafari
Nor Jaafari
10.4230/LIPIcs.APPROX-RANDOM.2015.726
Dimitris Achlioptas and Amin Coja-Oghlan. Algorithmic barriers from phase transitions. In Foundations of Computer Science, 2008. FOCS'08. IEEE 49th Annual IEEE Symposium on, pages 793-802. IEEE, 2008.
Dimitris Achlioptas and Ehud Friedgut. A sharp threshold for k-colorability. Random Structures Algorithms, 14(1):63-70, 1999.
Dimitris Achlioptas and Michael Molloy. The analysis of a list-coloring algorithm on a random graph. In Foundations of Computer Science, 1997. Proceedings., 38th Annual Symposium on, pages 204-212. IEEE, 1997.
Dimitris Achlioptas and Assaf Naor. The two possible values of the chromatic number of a random graph. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, pages 587-593. ACM, 2004.
Noga Alon and Michael Krivelevich. The concentration of the chromatic number of random graphs. Combinatorica, 17(3):303-313, 1997.
Victor Bapst, Amin Coja-Oghlan, and Charilaos Efthymiou. Planting colourings silently. arXiv preprint arXiv:1411.0610, 2014.
Victor Bapst, Amin Coja-Oghlan, Samuel Hetterich, Felicia Raßmann, and Dan Vilenchik. The condensation phase transition in random graph coloring. arXiv preprint arXiv:1404.5513, 2014.
Nayantara Bhatnagar, Allan Sly, and Prasad Tetali. Decay of correlations for the hardcore model on the d-regular random graph. arXiv preprint arXiv:1405.6160, 2014.
Béla Bollobás. The chromatic number of random graphs. Combinatorica, 8(1):49-55, 1988.
Amin Coja-Oghlan. Upper-bounding the k-colorability threshold by counting covers. arXiv preprint arXiv:1305.0177, 2013.
Amin Coja-Oghlan and Dan Vilenchik. Chasing the k-colorability threshold. In Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium on, pages 380-389. IEEE, 2013.
Martin Dyer and Alan Frieze. Randomly coloring random graphs. Random Structures & Algorithms, 36(3):251-272, 2010.
Martin Dyer, Alistair Sinclair, Eric Vigoda, and Dror Weitz. Mixing in time and space for lattice spin systems: A combinatorial view. Random Structures & Algorithms, 24(4):461-479, 2004.
Charilaos Efthymiou. MCMC sampling colourings and independent sets of G(n, d/n) near uniqueness threshold. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 305-316. SIAM, 2014.
Charilaos Efthymiou. Reconstruction/non-reconstruction thresholds for colourings of general Galton-watson trees. arXiv preprint arXiv:1406.3617, 2014.
Charilaos Efthymiou. Switching colouring of G(n, d/n) for sampling up to Gibbs uniqueness threshold. In Algorithms-ESA 2014, pages 371-381. Springer, 2014.
Paul Erdős. Graph theory and probability. canad. J. Math, 11:34G38, 1959.
Paul Erdős and A Rényi. On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci, 5:17-61, 1960.
Andreas Galanis, Qi Ge, Daniel Štefankovič, Eric Vigoda, and Linji Yang. Improved inapproximability results for counting independent sets in the hard-core model. Random Structures & Algorithms, 45(1):78-110, 2014.
Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. Inapproximability for antiferromagnetic spin systems in the tree non-uniqueness region. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 823-831. ACM, 2014.
Hans-Otto Georgii. Gibbs measures and phase transitions, volume 9. Walter de Gruyter, 2011.
Antoine Gerschenfeld and Andrea Montanari. Reconstruction for models on random graphs. In Foundations of Computer Science, 2007. FOCS'07. 48th Annual IEEE Symposium on, pages 194-204. IEEE, 2007.
Geoffrey R Grimmett and Colin JH McDiarmid. On colouring random graphs. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 77, pages 313-324. Cambridge Univ Press, 1975.
Michael Krivelevich and Benny Sudakov. Coloring random graphs. Information Processing Letters, 67(2):71-74, 1998.
Florent Krzakała, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Semerjian, and Lenka Zdeborová. Gibbs states and the set of solutions of random constraint satisfaction problems. Proceedings of the National Academy of Sciences, 104(25):10318-10323, 2007.
Tomasz Łuczak. The chromatic number of random graphs. Combinatorica, 11(1):45-54, 1991.
Tomasz Łuczak. A note on the sharp concentration of the chromatic number of random graphs. Combinatorica, 11(3):295-297, 1991.
Fabio Martinelli, Alistair Sinclair, and Dror Weitz. Fast mixing for independent sets, colorings, and other models on trees. Random Structures & Algorithms, 31(2):134-172, 2007.
David W Matula. Expose-and-merge exploration and the chromatic number of a random graph. Combinatorica, 7(3):275-284, 1987.
Marc Mézard and Andrea Montanari. Information, physics, and computation. Oxford University Press, 2009.
Marc Mézard, Giorgio Parisi, and Riccardo Zecchina. Analytic and algorithmic solution of random satisfiability problems. Science, 297(5582):812-815, 2002.
Michael Molloy. The freezing threshold for k-colourings of a random graph. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 921-930. ACM, 2012.
Andrea Montanari, Ricardo Restrepo, and Prasad Tetali. Reconstruction and clustering in random constraint satisfaction problems. SIAM Journal on Discrete Mathematics, 25(2):771-808, 2011.
Jaroslav Nešetřil. A combinatorial classic—sparse graphs with high chromatic number. In Erdős Centennial, pages 383-407. Springer, 2013.
Olivier Rivoire, Giulio Biroli, Olivier C Martin, and Marc Mézard. Glass models on bethe lattices. The European Physical Journal B-Condensed Matter and Complex Systems, 37(1):55-78, 2004.
Eli Shamir and Joel Spencer. Sharp concentration of the chromatic number on random graphs G(n,p). Combinatorica, 7(1):121-129, 1987.
Allan Sly. Computational transition at the uniqueness threshold. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 287-296. IEEE, 2010.
Yitong Yin and Chihao Zhang. Sampling colorings almost uniformly in sparse random graphs. arXiv preprint arXiv:1503.03351, 2015.
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Towards Resistance Sparsifiers
We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a (1+epsilon)-resistance sparsifier of size ~O(n/epsilon), and conjecture this bound holds for all graphs on n nodes. In comparison, spectral sparsification is a strictly stronger notion and requires Omega(n/epsilon^2) edges even on the complete graph.
Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein (JMLR, 2014) leads to the aforementioned resistance sparsifiers.
edge sparsification
spectral sparsifier
graph expansion
effective resistance
commute time
738-755
Regular Paper
Michael
Dinitz
Michael Dinitz
Robert
Krauthgamer
Robert Krauthgamer
Tal
Wagner
Tal Wagner
10.4230/LIPIcs.APPROX-RANDOM.2015.738
Alexandr Andoni, Robert Krauthgamer, and David P. Woodruff. The sketching complexity of graph cuts. CoRR, abs/1403.7058, 2014. URL: http://arxiv.org/abs/1403.7058.
http://arxiv.org/abs/1403.7058
Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM J. Comput., 41(6):1704-1721, 2012. URL: http://dx.doi.org/10.1137/090772873.
http://dx.doi.org/10.1137/090772873
A. A. Benczúr and D. R. Karger. Approximating s-t minimum cuts in Õ(n^2) time. In 28th Annual ACM Symposium on Theory of Computing, pages 47-55. ACM, 1996. URL: http://dx.doi.org/10.1145/237814.237827.
http://dx.doi.org/10.1145/237814.237827
Béla Csaba, Daniela Kühn, Allan Lo, Deryk Osthus, and Andrew Treglown. Proof of the 1-factorization and Hamilton decomposition conjectures. ArXiv e-prints, abs/1401.4159, 2014. URL: http://arxiv.org/abs/1401.4159.
http://arxiv.org/abs/1401.4159
M. M. Deza and M. Laurent. Geometry of cuts and metrics. Springer-Verlag, Berlin, 1997.
Michael Kapralov and Rina Panigrahy. Spectral sparsification via random spanners. In 3rd Innovations in Theoretical Computer Science Conference, pages 393-398. ACM, 2012. URL: http://dx.doi.org/10.1145/2090236.2090267.
http://dx.doi.org/10.1145/2090236.2090267
Rohit Khandekar, Subhash A. Khot, Lorenzo Orecchia, and Nisheeth K. Vishnoi. On a cut-matching game for the sparsest cut problem. Technical Report UCB/EECS-2007-177, EECS Department, University of California, Berkeley, 2007. URL: http://www.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-177.html.
http://www.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-177.html
Rohit Khandekar, Satish Rao, and Umesh V. Vazirani. Graph partitioning using single commodity flows. J. ACM, 56(4), 2009. URL: http://dx.doi.org/10.1145/1538902.1538903.
http://dx.doi.org/10.1145/1538902.1538903
Daniela Kühn and Deryk Osthus. Decompositions of complete uniform hypergraphs into hamilton berge cycles. J. Comb. Theory, Ser. A, 126:128-135, 2014. URL: http://dx.doi.org/10.1016/j.jcta.2014.04.010.
http://dx.doi.org/10.1016/j.jcta.2014.04.010
L. Lovász. Random walks on graphs: a survey. In Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993), volume 2 of Bolyai Soc. Math. Stud., pages 353-397. János Bolyai Math. Soc., Budapest, 1996.
D. Peleg and A. A. Schäffer. Graph spanners. J. Graph Theory, 13(1):99-116, 1989. URL: http://dx.doi.org/10.1002/jgt.3190130114.
http://dx.doi.org/10.1002/jgt.3190130114
L. Perkovic and B. Reed. Edge coloring regular graphs of high degree. In Discrete Math.,165/166, pages 567-578, 1997.
D. A. Spielman and S.-H. Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In 36th Annual ACM Symposium on Theory of Computing, pages 81-90. ACM, 2004. URL: http://dx.doi.org/10.1145/1007352.1007372.
http://dx.doi.org/10.1145/1007352.1007372
Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. SIAM J. Comput., 40(6):1913-1926, December 2011. URL: http://dx.doi.org/10.1137/080734029.
http://dx.doi.org/10.1137/080734029
Daniel A. Spielman and Shang-Hua Teng. Spectral sparsification of graphs. SIAM J. Comput., 40(4):981-1025, July 2011. URL: http://dx.doi.org/10.1137/08074489X.
http://dx.doi.org/10.1137/08074489X
Ulrike von Luxburg, Agnes Radl, and Matthias Hein. Hitting and commute times in large random neighborhood graphs. Journal of Machine Learning Research, 15(1):1751-1798, 2014. URL: http://jmlr.org/papers/v15/vonluxburg14a.html.
http://jmlr.org/papers/v15/vonluxburg14a.html
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Reconstruction/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees
The broadcasting models on trees arise in many contexts such as discrete mathematics, biology, information theory, statistical physics and computer science. In this work, we consider the k-colouring model. A basic question here is whether the assignment at the root affects the distribution of the colourings at the vertices at distance h from the root. This is the so-called reconstruction problem. For the case where the underlying tree is d -ary it is well known that d/ln(d) is the reconstruction threshold. That is, for k=(1+epsilon)*d/ln(d) we have non-reconstruction while for k=(1-epsilon)*d/ln(d) we have reconstruction.
Here, we consider the largely unstudied case where the underlying tree is chosen according to a predefined distribution. In particular, we consider the well-known Galton-Watson trees. The corresponding model arises naturally in many contexts such as
the theory of spin-glasses and its applications on random Constraint Satisfaction Problems (rCSP). The study on rCSP focuses on Galton-Watson trees with offspring distribution B(n,d/n), i.e. the binomial with parameters n and d/n, where d is fixed. Here we consider a broader version of the problem, as we assume general offspring distribution which includes B(n,d/n) as a special case.
Our approach relates the corresponding bounds for (non)reconstruction to certain concentration properties of the offspring distribution. This allows to derive reconstruction thresholds for a very wide family of offspring distributions, which includes B(n,d/n). A very interesting corollary is that for distributions with expected offspring d, we get reconstruction threshold d/ln(d) under weaker concentration conditions than what we have in B(n,d/n).
Furthermore, our reconstruction threshold for the random colorings of Galton-Watson with offspring B(n,d/n), implies the reconstruction threshold for the random colourings of G(n,d/n).
Random Colouring
Reconstruction Problem
Galton-Watson Tree
756-774
Regular Paper
Charilaos
Efthymiou
Charilaos Efthymiou
10.4230/LIPIcs.APPROX-RANDOM.2015.756
D. Achlioptas and A. Coja-Oghlan. Algorithmic barriers from phase transitions. In Foundations of Computer Science, 2008. FOCS'08. IEEE 49th Annual IEEE Symposium on, pages 793-802, Oct 2008.
Nayantara Bhatnagar, Allan Sly, and Prasad Tetali. Reconstruction threshold for the hardcore model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 13th International Workshop, APPROX 2010, and 14th International Workshop, RANDOM 2010, Barcelona, Spain, September 1-3, 2010. Proceedings, pages 434-447, 2010.
Nayantara Bhatnagar, Juan Carlos Vera, Eric Vigoda, and Dror Weitz. Reconstruction for colorings on trees. SIAM J. Discrete Math., 25(2):809-826, 2011.
Christian Borgs, Jennifer T. Chayes, Elchanan Mossel, and Sébastien Roch. The kesten-stigum reconstruction bound is tight for roughly symmetric binary channels. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 518-530, 2006.
J. T. Chayes, L. Chayes, James P. Sethna, and D. J. Thouless. A mean field spin glass with short-range interactions. Comm. Math. Phys., 106(1):41-89, 1986.
Amin Coja-Oghlan. A better algorithm for random k-sat. SIAM J. Comput., 39(7):2823-2864, 2010.
Amin Coja-Oghlan, Charilaos Efthymiou, and Nor Jaafari. Local convergence of random graph colorings. CoRR, abs/1501.06301, 2015.
Constantinos Daskalakis, Elchanan Mossel, and Sébastien Roch. Optimal phylogenetic reconstruction. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 159-168, 2006.
Martin E. Dyer, Abraham D. Flaxman, Alan M. Frieze, and Eric Vigoda. Randomly coloring sparse random graphs with fewer colors than the maximum degree. Random Struct. Algorithms, 29(4):450-465, 2006.
Charilaos 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 2012, Kyoto, Japan, January 17-19, 2012, pages 272-280, 2012.
Charilaos Efthymiou. MCMC sampling colourings and independent sets of G(n, d/n) near uniqueness threshold. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 305-316, 2014.
Charilaos Efthymiou. Reconstruction/non-reconstruction thresholds for colourings of general galton-watson trees. CoRR, abs/1406.3617, 2014.
Charilaos Efthymiou. Switching colouring of g(n, d/n) for sampling up to gibbs uniqueness threshold. In Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, pages 371-381, 2014.
William Evans, Claire Kenyon, Yuval Peres, and Leonard J. Schulman. Broadcasting on trees and the ising model. Ann. Appl. Probab., 10(2):410-433, 05 2000.
Antoine Gerschenfeld and Andrea Montanari. Reconstruction for models on random graphs. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2007), October 20-23, 2007, Providence, RI, USA, Proceedings, pages 194-204, 2007.
Geoffrey R Grimmett and Colin JH McDiarmid. On colouring random graphs. Mathematical Proceedings of the Cambridge Philosophical Society Mathematical Proceedings of the Cambridge Philosophical Society, 77(02):311-324, 1975.
Florent Krzakała, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Semerjian, and Lenka Zdeborová. Gibbs states and the set of solutions of random constraint satisfaction problems. Proceedings of the National Academy of Sciences, 104(25):10318-10323, 2007.
Marc Mézard, Giorgio Parisi, and Riccardo Zecchina. Analytic and algorithmic solution of random satisfiability problems. Science, 297(5582):812-815, 2002.
Michael Molloy. The freezing threshold for k-colourings of a random graph. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19-22, 2012, pages 921-930, 2012.
Andrea Montanari, Ricardo Restrepo, and Prasad Tetali. Reconstruction and clustering in random constraint satisfaction problems. SIAM J. Discrete Math., 25(2):771-808, 2011.
E. Mossel. Phase transitions in phylogeny. Trans. Amer. Math. Soc., 356(6):2379-2404 (electronic), 2004.
E. Mossel and Y. Peres. Information flow on trees. Ann. Appl. Probab., 13(3):817-844, 2003.
Elchanan Mossel. Reconstruction on trees: Beating the second eigenvalue. Ann. Appl. Probab., 11(1):285-300, 02 2001.
Rajeev Motwani and Prabhakar Raghavan. Randomized Algorithms. Cambridge University Press,, 1995.
Guilhem Semerjian. On the freezing of variables in random constraint satisfaction problems. Journal of Statistical Physics, 130(2):251- 293, January 2008.
Allan Sly. Reconstruction of random colourings. Communications in Mathematical Physics, 288(3):943-961, 2009.
Yitong Yin and Chihao Zhang. Sampling colorings almost uniformly in sparse random graphs. CoRR, abs/1503.03351, 2015.
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A Randomized Online Quantile Summary in O(1/epsilon * log(1/epsilon)) Words
A quantile summary is a data structure that approximates to epsilon-relative error the order statistics of a much larger underlying dataset.
In this paper we develop a randomized online quantile summary for the cash register data input model and comparison data domain model that uses O((1/epsilon) log(1/epsilon)) words of memory. This improves upon the previous best upper bound of O((1/epsilon) (log(1/epsilon))^(3/2)) by Agarwal et al. (PODS 2012). Further, by a lower bound of Hung and Ting (FAW 2010) no deterministic summary for the comparison model can outperform our randomized summary in terms of space complexity. Lastly, our summary has the nice property that O((1/epsilon) log(1/epsilon)) words suffice to ensure that the success probability is 1 - exp(-poly(1/epsilon)).
order statistics
data stream
streaming algorithm
775-785
Regular Paper
David
Felber
David Felber
Rafail
Ostrovsky
Rafail Ostrovsky
10.4230/LIPIcs.APPROX-RANDOM.2015.775
Pankaj K. Agarwal, Graham Cormode, Zengfeng Huang, Jeff Phillips, Zhewei Wei, and Ke Yi. Mergeable summaries. In Proceedings of the 31st Symposium on Principles of Database Systems, PODS'12, pages 23-34, New York, NY, USA, 2012. ACM.
Pankaj K Agarwal, Graham Cormode, Zengfeng Huang, Jeff M Phillips, Zhewei Wei, and Ke Yi. Mergeable summaries. ACM Transactions on Database Systems (TODS), 38(4):26, 2013.
Michael Greenwald and Sanjeev Khanna. Space-efficient online computation of quantile summaries. ACM SIGMOD Record, 30(2):58-66, 2001.
Regant YS Hung and Hingfung F Ting. An$$1 omega ($$1 frac 1$$1 varepsilon$$1 log$$1 frac 1$$1 varepsilon) space lower bound for finding ε-approximate quantiles in a data stream. In Frontiers in Algorithmics, pages 89-100. Springer, 2010.
Gurmeet Singh Manku, Sridhar Rajagopalan, and Bruce G Lindsay. Approximate medians and other quantiles in one pass and with limited memory. ACM SIGMOD Record, 27(2):426-435, 1998.
Gurmeet Singh Manku, Sridhar Rajagopalan, and Bruce G Lindsay. Random sampling techniques for space efficient online computation of order statistics of large datasets. ACM SIGMOD Record, 28(2):251-262, 1999.
J. I. Munro and M. S. Paterson. Selection and sorting with limited storage. In Proceedings of the 19th Annual Symposium on Foundations of Computer Science, SFCS'78, pages 253-258, Washington, DC, USA, 1978. IEEE Computer Society.
Problem 2: Quantiles - open problems in sublinear algorithms (suggested by graham cormode at kanpur 2006), 2006. URL: http://sublinear.info/2.
http://sublinear.info/2
Nisheeth Shrivastava, Chiranjeeb Buragohain, Divyakant Agrawal, and Subhash Suri. Medians and beyond: new aggregation techniques for sensor networks. In Proceedings of the 2nd international conference on Embedded networked sensor systems, pages 239-249. ACM, 2004.
Vladimir N Vapnik and A Ya Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability & Its Applications, 16(2):264-280, 1971.
Lu Wang, Ge Luo, Ke Yi, and Graham Cormode. Quantiles over data streams: an experimental study. In Proceedings of the 2013 ACM SIGMOD International Conference on Management of Data, pages 737-748. ACM, 2013.
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On Constant-Size Graphs That Preserve the Local Structure of High-Girth Graphs
Let G=(V,E) be an undirected graph with maximum degree d. The k-disc of a vertex v is defined as the rooted subgraph that is induced by all vertices whose distance to v is at most k. The k-disc frequency vector of G, freq(G), is a vector indexed by all isomorphism types of k-discs. For each such isomorphism type Gamma, the k-disc frequency vector counts the fraction of vertices that have k-disc isomorphic to Gamma. Thus, the frequency vector freq(G) of G captures the local structure of G. A natural question is whether one can construct a much smaller graph H such that H has a similar local structure. N. Alon proved that for any epsilon>0 there always exists a graph H whose size is independent of |V| and whose frequency vector satisfies ||freq(G) - freq(G)||_1 <= epsilon. However, his proof is only existential and neither gives an explicit bound on the size of H nor an efficient algorithm. He gave the open problem to find such explicit bounds. In this paper, we solve this problem for the special case of high girth graphs. We show how to efficiently compute a graph H with the above properties when G has girth at least 2k+2 and we give explicit bounds on the size of H.
local graph structure
k-disc frequency vector
graph property testing
786-799
Regular Paper
Hendrik
Fichtenberger
Hendrik Fichtenberger
Pan
Peng
Pan Peng
Christian
Sohler
Christian Sohler
10.4230/LIPIcs.APPROX-RANDOM.2015.786
D. Aldous and R. Lyons. Processes on Unimodular Random Networks. Electronic Journal of Probability, 12(54):1454-1508, 2007.
A. Benczúr and D. Karger. Approximating s-t Minimum Cuts In Õ(n²) Time. In Proceedings of the 28th annual ACM Symposium on Theory of Computing, pages 47-55. ACM, 1996.
I. Benjamini, O. Schramm, and A. Shapira. Every Minor-Closed Property of Sparse Graphs Is Testable. Advances in Mathematics, 223(6):2200-2218, 2010.
P. Chew. There Are Planar Graphs Almost as Good as the Complete Graph. Journal of Computer and System Sciences, 39(2):205-219, 1989.
G. Elek. On the Limit of Large Girth Graph Sequences. Combinatorica, 30(5):553-563, 2010.
J. Friedman. A proof of Alon’s second eigenvalue conjecture and related problems. American Mathematical Society, 2008.
O. Goldreich and D. Ron. Property Testing in Bounded Degree Graphs. Algorithmica, 32:302-343, 2002.
A. Hassidim, J. Kelner, H. Nguyen, and K. Onak. Local Graph Partitions for Approximation and Testing. In Proceedings of the 50th annual IEEE Symposium on Foundations of Computer Science, pages 22-31. IEEE, 2009.
P. Indyk, A. McGregor, I. Newman, and K. Onak. Bertinoro workshop on sublinear algorithms 2011. http://sublinear.info/42. In Open Problems in Data Streams, Property Testing, and Related Topics, 2011.
http://sublinear.info/42
L. Lovász. Very Large Graphs. arXiv preprint arXiv:0902.0132, 2009.
L. Lovász. Large Networks and Graph Limits. American Mathematical Society, 2012.
B. McKay, N. Wormald, and B. Wysocka. Short Cycles in Random Regular Graphs. Electronic Journal of Combinatorics, 11(1):66, 2004.
I. Newman and C. Sohler. Every Property of Hyperfinite Graphs Is Testable. SIAM Journal on Computing, 42(3):1095-1112, 2013.
D. Spielman and S. Teng. Spectral Sparsification of Graphs. SIAM Journal on Computing, 40(4):981-1025, 2011.
E. Szemerédi. Regular Partitions of Graphs. Technical report, DTIC Document, 1975.
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Dimension Expanders via Rank Condensers
An emerging theory of "linear algebraic pseudorandomness: aims to understand the linear algebraic analogs of fundamental Boolean pseudorandom objects where the rank of subspaces plays the role of the size of subsets. In this work, we study and highlight the interrelationships between several such algebraic objects such as subspace designs, dimension expanders, seeded rank condensers, two-source rank condensers, and rank-metric codes. In particular, with the recent construction of near-optimal subspace designs by Guruswami and Kopparty as a starting point, we construct good (seeded) rank condensers (both lossless and lossy versions), which are a small collection of linear maps F^n to F^t for t<<n such that for every subset of F^n of small rank, its rank is preserved (up to a constant factor in the lossy case) by at least one of the maps.
We then compose a tensoring operation with our lossy rank condenser to construct constant-degree dimension expanders over polynomially large fields. That is, we give a constant number of explicit linear maps A_i from F^n to F^n such that for any subspace V of F^n of dimension at most n/2, the dimension of the span of the A_i(V) is at least (1+Omega(1)) times the dimension of V. Previous constructions of such constant-degree dimension expanders were based on Kazhdan's property T (for the case when F has characteristic zero) or monotone expanders (for every field F); in either case the construction was harder than that of usual vertex expanders. Our construction, on the other hand, is simpler.
For two-source rank condensers, we observe that the lossless variant (where the output rank is the product of the ranks of the two sources) is equivalent to the notion of a linear rank-metric code. For the lossy case, using our seeded rank condensers, we give a reduction of the general problem to the case when the sources have high (n^Omega(1)) rank. When the sources have constant rank, combining this with an "inner condenser" found by brute-force leads to a two-source rank condenser with output length nearly matching the probabilistic constructions.
dimension expanders
rank condensers
rank-metric codes
subspace designs
Wronskians
800-814
Regular Paper
Michael A.
Forbes
Michael A. Forbes
Venkatesan
Guruswami
Venkatesan Guruswami
10.4230/LIPIcs.APPROX-RANDOM.2015.800
Manindra Agrawal, Chandan Saha, and Nitin Saxena. Quasi-polynomial hitting-set for set-depth-Δ formulas. In Proceedings of the \nth\intcalcSub20131968 Annual ACM Symposium on Theory of Computing (STOC 2013), pages 321-330, 2013. Full version at http://arxiv.org/abs/1209.2333.
Boaz Barak, Russell Impagliazzo, Amir Shpilka, and Avi Wigderson. Personal Communication to Dvir-Shpilka [Dvir and Shpilka, 2011], 2004.
Boaz Barak, Guy Kindler, Ronen Shaltiel, Benny Sudakov, and Avi Wigderson. Simulating independence: New constructions of condensers, Ramsey graphs, dispersers, and extractors. J. ACM, 57(4), 2010. Preliminary version in the \nth\intcalcSub20051968 Annual ACM Symposium on Theory of Computing (STOC 2005).
Jean Bourgain and Amir Yehudayoff. Expansion in SL₂(ℝ) and monotone expanders. Geometric and Functional Analysis, 23(1):1-41, 2013. Preliminary version in the \nth\intcalcSub20121968 Annual ACM Symposium on Theory of Computing (STOC 2012). This work is the full version of [Bourgain, 2009].
Michael R. Capalbo, Omer Reingold, Salil P. Vadhan, and Avi Wigderson. Randomness conductors and constant-degree lossless expanders. In Proceedings of the \nth\intcalcSub20021968 Annual ACM Symposium on Theory of Computing (STOC 2002), pages 659-668, 2002.
Ho Yee Cheung, Tsz Chiu Kwok, and Lap Chi Lau. Fast matrix rank algorithms and applications. J. ACM, 60(5):31, 2013. Preliminary version in the \nth\intcalcSub20121968 Annual ACM Symposium on Theory of Computing (STOC 2012).
Zeev Dvir and Amir Shpilka. Locally decodable codes with two queries and polynomial identity testing for depth 3 circuits. SIAM J. Comput., 36(5):1404-1434, 2007. Preliminary version in the \nth\intcalcSub20051968 Annual ACM Symposium on Theory of Computing (STOC 2005).
Michael A. Forbes, Ramprasad Saptharishi, and Amir Shpilka. Hitting sets for multilinear read-once algebraic branching programs, in any order. In Proceedings of the \nth\intcalcSub20141968 Annual ACM Symposium on Theory of Computing (STOC 2014), pages 867-875, 2014. Full version at http://arxiv.org/abs/1309.5668.
Michael A. Forbes and Amir Shpilka. On identity testing of tensors, low-rank recovery and compressed sensing. In Proceedings of the \nth\intcalcSub20121968 Annual ACM Symposium on Theory of Computing (STOC 2012), pages 163-172, 2012. Full version at http://arxiv.org/abs/1111.0663.
Ariel Gabizon and Ran Raz. Deterministic extractors for affine sources over large fields. Combinatorica, 28(4):415-440, 2008. Preliminary version in the \nth\intcalcSub20051959 Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005).
Venkatesan Guruswami. Linear-algebraic list decoding of folded reed-solomon codes. In Proceedings of the \ifnumcomp2011>1995\nth\intcalcSub20111985 Annual IEEE Conference on Computational Complexity (CCC 2011)\nth\intcalcSub20111985 Annual Structure in Complexity Theory Conference (CCC 2011), pages 77-85, 2011. The full version of this paper is merged into Guruswami-Wang [Guruswami and Wang, 2013].
Venkatesan Guruswami and Swastik Kopparty. Explicit subspace designs. Combinatorica, pages 1-25, 2014. Preliminary version in the \nth\intcalcSub20131959 Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013).
Venkatesan Guruswami and Carol Wang. Linear-algebraic list decoding for variants of reed-solomon codes. IEEE Transactions on Information Theory, 59(6):3257-3268, 2013. Preliminary versions appeared in Proceedings of the \ifnumcomp2011>1995\nth\intcalcSub20111985 Annual IEEE Conference on Computational Complexity (CCC 2011)\nth\intcalcSub20111985 Annual Structure in Complexity Theory Conference (CCC 2011) and Proceedings of the \nth\intcalcSub20111996 International Workshop on Randomization and Computation (RANDOM 2011).
Venkatesan Guruswami and Carol Wang. Evading subspaces over large fields and explicit list-decodable rank-metric codes. In Proceedings of the \nth\intcalcSub20141996 International Workshop on Randomization and Computation (RANDOM 2014), pages 748-761, 2014. Full version at http://arxiv.org/abs/1311.7084.
Venkatesan Guruswami and Chaoping Xing. Folded codes from function field towers and improved optimal rate list decoding. In Proceedings of the \nth\intcalcSub20121968 Annual ACM Symposium on Theory of Computing (STOC 2012), pages 339-350, 2012. Full version at http://arxiv.org/abs/1204.4209.
Venkatesan Guruswami and Chaoping Xing. List decoding Reed-Solomon, algebraic-geometric, and Gabidulin subcodes up to the Singleton bound. In Proceedings of the \nth\intcalcSub20131968 Annual ACM Symposium on Theory of Computing (STOC 2013), pages 843-852, 2013. Full version in the http://eccc.hpi-web.de/report/2012/146/.
Aram W. Harrow. Quantum expanders from any classical cayley graph expander. Quantum Information & Computation, 8(8-9):715-721, 2008.
Zohar S. Karnin and Amir Shpilka. Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in. Combinatorica, 31(3):333-364, 2011. Preliminary version in the \ifnumcomp2008>1995\nth\intcalcSub20081985 Annual IEEE Conference on Computational Complexity (CCC 2008)\nth\intcalcSub20081985 Annual Structure in Complexity Theory Conference (CCC 2008).
Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, and Saket Saurabh. Deterministic truncation of linear matroids. arXiv, 1404.4506, 2014.
Alexander Lubotzky and Efim Zelmanov. Dimension expanders. J. Algebra, 319(2):730-738, 2008.
Dániel Marx. A parameterized view on matroid optimization problems. Theor. Comput. Sci., 410(44):4471-4479, 2009. Preliminary version in the \nth\intcalcSub20061973 International Colloquium on Automata, Languages and Programming (ICALP 2006).
Pavel Pudlák and Vojtěch Rödl. Pseudorandom sets and explicit constructions of Ramsey graphs. In Complexity of computations and proofs, volume 13 of Quad. Mat., pages 327-346. Dept. Math., Seconda Univ. Napoli, Caserta, 2004.
Ran Raz. Extractors with weak random seeds. In Proceedings of the \nth\intcalcSub20051968 Annual ACM Symposium on Theory of Computing (STOC 2005), pages 11-20, 2005. Full version in the http://eccc.hpi-web.de/report/2004/099/.
Ran Raz and Amir Shpilka. Deterministic polynomial identity testing in non-commutative models. Comput. Complex., 14(1):1-19, April 2005. Preliminary version in the \ifnumcomp2004>1995\nth\intcalcSub20041985 Annual IEEE Conference on Computational Complexity (CCC 2004)\nth\intcalcSub20041985 Annual Structure in Complexity Theory Conference (CCC 2004).
Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):2070-388, 2010.
Salil P. Vadhan. Pseudorandomness. Foundations and Trends in Theoretical Computer Science, 7(1-3):1-336, 2012.
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Swendsen-Wang Algorithm on the Mean-Field Potts Model
We study the q-state ferromagnetic Potts model on the n-vertex complete graph known as the mean-field (Curie-Weiss) model. We analyze the Swendsen-Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single-site Glauber dynamics. The case q=2 (the Swendsen-Wang algorithm for the ferromagnetic Ising model) undergoes a slow-down at the uniqueness/non-uniqueness critical temperature for the infinite Delta-regular tree (Long et al., 2014) but yet still has polynomial mixing time at all (inverse) temperatures beta>0 (Cooper et al., 2000). In contrast for q>=3 there are two critical temperatures 0<beta_u<beta_rc that are relevant, these two critical points relate to phase transitions in the infinite tree. We prove that the mixing time of the Swendsen-Wang algorithm for the ferromagnetic Potts model on the n-vertex complete graph satisfies: (i) O(log n) for beta<beta_u, (ii) O(n^(1/3)) for beta=beta_u, (iii) exp(n^(Omega(1))) for beta_u<beta<beta_rc, and (iv) O(log n) for beta>=beta_rc. These results complement refined results of Cuff et al. (2012) on the mixing time of the Glauber dynamics for the ferromagnetic Potts model. The most interesting aspect of our analysis is at the critical temperature beta=beta_u, which requires a delicate choice of a potential function to balance the conflating factors for the slow drift away from a fixed point (which is repulsive but not Jacobian repulsive): close to the fixed point the variance from the percolation step dominates and sufficiently far from the fixed point the dynamics of the size of the dominant color class takes over.
Ferromagnetic Potts model
Swendsen-Wang dynamics
mixing time
mean-field analysis
phase transition.
815-828
Regular Paper
Andreas
Galanis
Andreas Galanis
Daniel
Štefankovic
Daniel Štefankovic
Eric
Vigoda
Eric Vigoda
10.4230/LIPIcs.APPROX-RANDOM.2015.815
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Decomposing Overcomplete 3rd Order Tensors using Sum-of-Squares Algorithms
Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to n^{\lfloor p/2 \rceil} for a p-th order tensor. Previously no efficient algorithm can decompose 3rd order tensors when the rank is super-linear in the dimension. Using ideas from sum-of-squares hierarchy, we give the first quasi-polynomial time algorithm that can decompose a random 3rd order tensor decomposition when the rank is as large as n^{3/2}/poly log n.
We also give a polynomial time algorithm for certifying the injective norm of random low rank tensors. Our tensor decomposition algorithm exploits the relationship between injective norm and the tensor components. The proof relies on interesting tools for decoupling random variables to prove better matrix concentration bounds.
sum of squares
overcomplete tensor decomposition
829-849
Regular Paper
Rong
Ge
Rong Ge
Tengyu
Ma
Tengyu Ma
10.4230/LIPIcs.APPROX-RANDOM.2015.829
Boris Alexeev, Michael A Forbes, and Jacob Tsimerman. Tensor rank: Some lower and upper bounds. In Computational Complexity (CCC), 2011 IEEE 26th Annual Conference on, pages 283-291. IEEE, 2011.
A. Anandkumar, D. P. Foster, D. Hsu, S. M. Kakade, and Y. K. Liu. Two SVDs Suffice: Spectral Decompositions for Probabilistic Topic Modeling and Latent Dirichlet Allocation. to appear in the special issue of Algorithmica on New Theoretical Challenges in Machine Learning, July 2013.
A. Anandkumar, R. Ge, D. Hsu, and S. M. Kakade. A Tensor Spectral Approach to Learning Mixed Membership Community Models. In Conference on Learning Theory (COLT), June 2013.
A. Anandkumar, R. Ge, D. Hsu, S. M. Kakade, and M. Telgarsky. Tensor Methods for Learning Latent Variable Models. J. of Machine Learning Research, 15:2773-2832, 2014.
A. Anandkumar, D. Hsu, and S. M. Kakade. A Method of Moments for Mixture Models and Hidden Markov Models. In Proc. of Conf. on Learning Theory, June 2012.
Anima Anandkumar, Rong Ge, and Majid Janzamin. Guaranteed Non-Orthogonal Tensor Decomposition via Alternating Rank-1 Updates. arXiv preprint arXiv:1402.5180, Feb. 2014.
Joseph Anderson, Mikhail Belkin, Navin Goyal, Luis Rademacher, and James Voss. The more, the merrier: the blessing of dimensionality for learning large gaussian mixtures. arXiv preprint arXiv:1311.2891, 2013.
Boaz Barak, Fernando G.S.L. Brandao, Aram W. Harrow, Jonathan Kelner, David Steurer, and Yuan Zhou. Hypercontractivity, sum-of-squares proofs, and their applications. In Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC'12, pages 307-326, New York, NY, USA, 2012. ACM.
Boaz Barak, Jonathan A. Kelner, and David Steurer. Rounding sum-of-squares relaxations. In STOC, pages 31-40, 2014.
Boaz Barak, Jonathan A. Kelner, and David Steurer. Dictionary learning and tensor decomposition via the sum-of-squares method. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC'15, 2015.
Boaz Barak and Ankur Moitra. Tensor prediction, rademacher complexity and random 3-XOR. http://arxiv.org/abs/1501.06521, 2015.
http://arxiv.org/abs/1501.06521
Boaz Barak and David Steurer. Sum-of-squares proofs and the quest toward optimal algorithms. In Proceedings of International Congress of Mathematicians (ICM), 2014. To appear.
Aditya Bhaskara, Moses Charikar, Ankur Moitra, and Aravindan Vijayaraghavan. Smoothed analysis of tensor decompositions. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 594-603. ACM, 2014.
Joseph T. Chang. Full reconstruction of Markov models on evolutionary trees: Identifiability and consistency. Mathematical Biosciences, 137:51-73, 1996.
Pierre Comon. Tensor: a partial survey. Signal Processing Magazine, page 11, 2014.
Lieven De Lathauwer, Joséphine Castaing, and Jean-François Cardoso. Fourth-order cumulant-based blind identification of underdetermined mixtures. Signal Processing, IEEE Transactions on, 55(6):2965-2973, 2007.
Ignat Domanov and Lieven De Lathauwer. Canonical polyadic decomposition of third-order tensors: relaxed uniqueness conditions and algebraic algorithm. arXiv preprint arXiv:1501.07251, 2015.
Ignat Domanov and Lieven De Lathauwer. Canonical polyadic decomposition of third-order tensors: reduction to generalized eigenvalue decomposition. SIAM Journal on Matrix Analysis and Applications, 35(2):636-660, 2014.
Rong Ge, Qingqing Huang, and Sham M. Kakade. Learning mixtures of gaussians in high dimensions. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC'15, 2015.
Leonid Gurvits. Classical deterministic complexity of edmonds' problem and quantum entanglement. In Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing, STOC'03, pages 10-19, New York, NY, USA, 2003. ACM.
Aram W Harrow and Ashley Montanaro. Testing product states, quantum merlin-arthur games and tensor optimization. Journal of the ACM (JACM), 60(1):3, 2013.
Johan Håstad. Tensor rank is np-complete. Journal of Algorithms, 11(4):644-654, 1990.
Christopher J. Hillar and Lek-Heng Lim. Most tensor problems are NP hard. arXiv preprint arXiv:0911.1393, 2009.
J.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.
Jean B Lasserre. Global optimization with polynomials and the problem of moments. SIAM Journal on Optimization, 11(3):796-817, 2001.
Elchanan Mossel and Sébastian Roch. Learning nonsingular phylogenies and hidden Markov models. Annals of Applied Probability, 16(2):583-614, 2006.
Pablo A Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, 2000.
Victor H. de la Pena and S. J. Montgomery-Smith. Decoupling inequalities for the tail probabilities of multivariate u-statistics. The Annals of Probability, 23(2):pp. 806-816, 1995.
Volker Strassen. Vermeidung von divisionen. Journal für die reine und angewandte Mathematik, 264:184-202, 1973.
Joel A Tropp. User-friendly tail bounds for sums of random matrices. Foundations of Computational Mathematics, 12(4):389-434, 2012.
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Negation-Limited Formulas
We give an efficient structural decomposition theorem for formulas that depends on their negation complexity and demonstrate its power with the following applications.
We prove that every formula that contains t negation gates can be shrunk using a random restriction to a formula of size O(t) with the shrinkage exponent of monotone formulas. As a result, the shrinkage exponent of formulas that contain a constant number of negation gates is equal to the shrinkage exponent of monotone formulas.
We give an efficient transformation of formulas with t negation gates to circuits with log(t) negation gates. This transformation provides a generic way to cast results for negation-limited circuits to the setting of negation-limited formulas. For example, using a result of Rossman (CCC'15), we obtain an average-case lower bound for formulas of polynomial-size on n variables with n^{1/2-epsilon} negations.
In addition, we prove a lower bound on the number of negations required to compute one-way permutations by polynomial-size formulas.
Negation complexity
De Morgan formulas
Shrinkage
850-866
Regular Paper
Siyao
Guo
Siyao Guo
Ilan
Komargodski
Ilan Komargodski
10.4230/LIPIcs.APPROX-RANDOM.2015.850
Miklós Ajtai, János Komlós, and Endre Szemerédi. An o(n log n) sorting network. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, STOC, pages 1-9, 1983.
Kazuyuki Amano and Akira Maruoka. A superpolynomial lower bound for a circuit computing the clique function with at most (1/6)log log n negation gates. SIAM J. Comput., 35(1):201-216, 2005.
Benny Applebaum, Yuval Ishai, and Eyal Kushilevitz. Cryptography in NC⁰. SIAM J. Comput., 36(4):845-888, 2006.
Robert Beals, Tetsuro Nishino, and Keisuke Tanaka. On the complexity of negation-limited boolean networks. SIAM J. Comput., 27(5):1334-1347, 1998.
Eric Blais, Clément L. Canonne, Igor Carboni Oliveira, Rocco A. Servedio, and Li-Yang Tan. Learning circuits with few negations. To appear in Proceedings of the 19th International Workshop on Randomization and Computation, RANDOM, 2015. Available at URL: http://arxiv.org/abs/1410.8420.
http://arxiv.org/abs/1410.8420
Nader H. Bshouty and Christino Tamon. On the Fourier spectrum of monotone functions. J. ACM, 43(4):747-770, 1996.
Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel, and David Zuckerman. Mining circuit lower bound proofs for meta-algorithms. In Proceedings of the 29th Conference on Computational Complexity, CCC, pages 262-273, 2014.
Ruiwen Chen, Valentine Kabanets, and Nitin Saurabh. An improved deterministic #SAT algorithm for small De Morgan formulas. In Proceedings of the 39th International Symposium on Mathematical Foundations of Computer Science, MFCS, pages 165-176, 2014.
Moshe Dubiner and Uri Zwick. How do read-once formulae shrink? Combinatorics, Probability & Computing, 3:455-469, 1994.
Michael. J. Fischer. The complexity of negation-limited networks-a brief survey. Automata Theory and Formal Languages, 33:71-82, 1975.
Oded Goldreich and Rani Izsak. Monotone circuits: One-way functions versus pseudorandom generators. Theory of Computing, 8(1):231-238, 2012.
Mika Göös and Toniann Pitassi. Communication lower bounds via critical block sensitivity. In Proceedings of the 46th Annual Symposium on Theory of Computing, STOC, pages 847-856, 2014.
Siyao Guo, Tal Malkin, Igor Carboni Oliveira, and Alon Rosen. The power of negations in cryptography. In Proceedings of the 12th Theory of Cryptography Conference, TCC, pages 36-65, 2015.
Danny Harnik and Ran Raz. Higher lower bounds on monotone size. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, STOC, pages 378-387, 2000.
Johan Håstad. One-way permutations in NC0. Inf. Process. Lett., 26(3):153-155, 1987.
Johan Håstad. The shrinkage exponent of de Morgan formulas is 2. SIAM J. Comput., 27(1):48-64, 1998.
Johan Håstad, Alexander A. Razborov, and Andrew Chi-Chih Yao. On the shrinkage exponent for read-once formulae. Theor. Comput. Sci., 141(1&2):269-282, 1995.
Russell Impagliazzo and Valentine Kabanets. Fourier concentration from shrinkage. In Proceedings of the 29th Conference on Computational Complexity, CCC, pages 321-332, 2014.
Russell Impagliazzo, Raghu Meka, and David Zuckerman. Pseudorandomness from shrinkage. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 111-119, 2012.
Russell Impagliazzo and Noam Nisan. The effect of random restrictions on formula size. Random Struct. Algorithms, 4(2):121-134, 1993.
Kazuo Iwama, Hiroki Morizumi, and Jun Tarui. Negation-limited complexity of parity and inverters. Algorithmica, 54(2):256-267, 2009.
Stasys Jukna. Boolean Function Complexity - Advances and Frontiers, volume 27 of Algorithms and combinatorics. Springer, 2012.
Ilan Komargodski, Ran Raz, and Avishay Tal. Improved average-case lower bounds for DeMorgan formula size. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 588-597, 2013.
Andrey A. Markov. On the inversion complexity of a system of functions. J. ACM, 5(4):331-334, 1958.
Hiroki Morizumi. Limiting negations in formulas. In Proceedings of the 36th International Colloquium on Automata, Languages and Programming, ICALP, pages 701-712, 2009.
Moni Naor and Omer Reingold. Number-theoretic constructions of efficient pseudo-random functions. J. ACM, 51(2):231-262, 2004.
Eduard I. Nechiporuk. On the complexity of schemes in some bases containing nontrivial elements with zero weights. Problemy Kibernetiki, 8:123-160, 1962. In Russian.
Noam Nisan. Pseudorandom bits for constant depth circuits. Combinatorica, 11(1):63-70, 1991.
Noam Nisan and Avi Wigderson. Hardness vs randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994.
Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014.
Mike Paterson and Uri Zwick. Shrinkage of de Morgan formulae under restriction. Random Struct. Algorithms, 4(2):135-150, 1993.
Ran Raz and Avi Wigderson. Probabilistic communication complexity of boolean relations (extended abstract). In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, FOCS, pages 562-567, 1989.
Benjamin Rossman. Correlation bounds against monotone NC¹. In Proceedings of the 30th Conference on Computational Complexity, CCC, pages 392-411, 2015.
Miklos Santha and Christopher B. Wilson. Polynomial size constant depth circuits with a limited number of negations. In Proceedings of the 8th Annual Symposium on Theoretical Aspects of Computer Science, STACS, pages 228-237, 1991.
Bella A. Subbotovskaya. Realizations of linear function by formulas using +,⋅,-. Doklady Akademii Nauk SSSR, 136:3:553-555, 1961. In Russian.
Shao Chin Sung and Keisuke Tanaka. Limiting negations in bounded-depth circuits: An extension of markov’s theorem. Inf. Process. Lett., 90(1):15-20, 2004.
Avishay Tal. Shrinkage of De Morgan formulae by spectral techniques. In Proceedings of the 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 551-560, 2014.
Michel Talagrand. How much are increasing sets positively correlated? Combinatorica, 16(2):243-258, 1996.
Keisuke Tanaka, Tetsuro Nishino, and Robert Beals. Negation-limited circuit complexity of symmetric functions. Inf. Process. Lett., 59(5):273-279, 1996.
Leslie G. Valiant. Short monotone formulae for the majority function. J. Algorithms, 5(3):363-366, 1984.
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Deletion Codes in the High-noise and High-rate Regimes
The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst-case deletions, with a focus on constructing efficiently encodable and decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following trade-offs (for any epsilon > 0):
(1) Codes that can correct a fraction 1-epsilon of deletions with rate poly(eps) over an alphabet of size poly(1/epsilon); (2) Binary codes of rate 1-O~(sqrt(epsilon)) that can correct a fraction eps of deletions; and
(3) Binary codes that can be list decoded from a fraction (1/2-epsilon) of deletions with rate poly(epsion)
Our work is the first to achieve the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets, and correcting a constant fraction of bit deletions with rate aproaching 1. The above results bring our understanding of deletion code constructions in these regimes to a similar level as worst-case errors.
algorithmic coding theory
deletion codes
list decoding
probabilistic method
explicit constructions
867-880
Regular Paper
Venkatesan
Guruswami
Venkatesan Guruswami
Carol
Wang
Carol Wang
10.4230/LIPIcs.APPROX-RANDOM.2015.867
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Communication with Partial Noiseless Feedback
We introduce the notion of one-way communication schemes with partial noiseless feedback. In this setting, Alice wishes to communicate a message to Bob by using a communication scheme that involves sending a sequence of bits over a channel while receiving feedback bits from Bob for delta fraction of the transmissions. An adversary is allowed to corrupt up to a constant fraction of Alice's transmissions, while the feedback is always uncorrupted. Motivated by questions related to coding for interactive communication, we seek to determine the maximum error rate, as a function of 0 <= delta <= 1, such that Alice can send a message to Bob via some protocol with delta fraction of noiseless feedback. The case delta = 1 corresponds to full feedback, in which the result of Berlekamp ['64] implies that the maximum tolerable error rate is 1/3, while the case delta = 0 corresponds to no feedback, in which the maximum tolerable error rate is 1/4, achievable by use of a binary error-correcting code.
In this work, we show that for any delta in (0,1] and gamma in [0, 1/3), there exists a randomized communication scheme with noiseless delta-feedback, such that the probability of miscommunication is low, as long as no more than a gamma fraction of the rounds are corrupted. Moreover, we show that for any delta in (0, 1] and gamma < f(delta), there exists a deterministic communication scheme with noiseless delta-feedback that always decodes correctly as long as no more than a gamma fraction of rounds are corrupted. Here f is a monotonically increasing, piecewise linear, continuous function with f(0) = 1/4 and f(1) = 1/3. Also, the rate of communication in both cases is constant (dependent on delta and gamma but independent of the input length).
Communication with feedback
Interactive communication
Coding theory Digital
881-897
Regular Paper
Bernhard
Haeupler
Bernhard Haeupler
Pritish
Kamath
Pritish Kamath
Ameya
Velingker
Ameya Velingker
10.4230/LIPIcs.APPROX-RANDOM.2015.881
Elwyn R. Berlekamp. Block Coding with Noiseless Feedback. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 1964.
Zvika Brakerski and Yael Tauman Kalai. Efficient interactive coding against adversarial noise. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 160-166, 2012.
Zvika Brakerski, Yael Tauman Kalai, and Moni Naor. Fast interactive coding against adversarial noise. J. ACM, 61(6):35, 2014.
Zvika Brakerski and Moni Naor. Fast algorithms for interactive coding. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 443-456, 2013.
Mark Braverman and Klim Efremenko. List and unique coding for interactive communication in the presence of adversarial noise. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 236-245, 2014.
Mark Braverman and Anup Rao. Toward coding for maximum errors in interactive communication. IEEE Transactions on Information Theory, 60(11):7248-7255, 2014.
Klim Efremenko, Ran Gelles, and Bernhard Haeupler. Maximal noise in interactive communication over erasure channels and channels with feedback. In Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, Rehovot, Israel, January 11-13, 2015, pages 11-20, 2015.
Ran Gelles and Bernhard Haeupler. Capacity of interactive communication over erasure channels and channels with feedback. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1296-1311, 2015.
Ran Gelles, Ankur Moitra, and Amit Sahai. Efficient coding for interactive communication. IEEE Transactions on Information Theory, 60(3):1899-1913, 2014.
Mohsen Ghaffari and Bernhard Haeupler. Optimal error rates for interactive coding II: efficiency and list decoding. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 394-403, 2014.
Mohsen Ghaffari, Bernhard Haeupler, and Madhu Sudan. Optimal error rates for interactive coding I: adaptivity and other settings. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 794-803, 2014.
E. N. Gilbert. A comparison of signalling alphabets. Bell System Technical Journal, 31:504-522, 1952.
Bernhard Haeupler. Interactive channel capacity revisited. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 226-235, 2014.
Gillat Kol and Ran Raz. Interactive channel capacity. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 715-724, 2013.
Morris Plotkin. Binary codes with specified minimum distance. IRE Transactions on Information Theory, 6(4):445-450, 1960.
Leonard J. Schulman. Communication on noisy channels: A coding theorem for computation. In 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, Pennsylvania, USA, 24-27 October 1992, pages 724-733, 1992.
Leonard J. Schulman. Deterministic coding for interactive communication. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, May 16-18, 1993, San Diego, CA, USA, pages 747-756, 1993.
Leonard J. Schulman. Coding for interactive communication. IEEE Transactions on Information Theory, 42(6):1745-1756, 1996.
Claude E. Shannon. The zero error capacity of a noisy channel. IRE Transactions on Information Theory, 2(3):8-19, 1956.
Joel Spencer and Peter Winkler. Three thresholds for a liar. Combinatorics, Probability & Computing, 1:81-93, 1992.
R. R. Varshamov. Estimate of the number of signals in error correcting codes. Dokl. Acad. Nauk SSSR, 117:739-741, 1957.
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Spectral Norm of Random Kernel Matrices with Applications to Privacy
Kernel methods are an extremely popular set of techniques used for many important machine learning and data analysis applications. In addition to having good practical performance, these methods are supported by a well-developed theory. Kernel methods use an implicit mapping of the input data into a high dimensional feature space defined by a kernel function, i.e., a function returning the inner product between the images of two data points in the feature space. Central to any kernel method is the kernel matrix, which is built by evaluating the kernel function on a given sample dataset.
In this paper, we initiate the study of non-asymptotic spectral properties of random kernel matrices. These are n x n random matrices whose (i,j)th entry is obtained by evaluating the kernel function on x_i and x_j, where x_1,..,x_n are a set of n independent random high-dimensional vectors. Our main contribution is to obtain tight upper bounds on the spectral norm (largest eigenvalue) of random kernel matrices constructed by using common kernel functions such as polynomials and Gaussian radial basis.
As an application of these results, we provide lower bounds on the distortion needed for releasing the coefficients of kernel ridge regression under attribute privacy, a general privacy notion which captures a large class of privacy definitions. Kernel ridge regression is standard method for performing non-parametric regression that regularly outperforms traditional regression approaches in various domains. Our privacy distortion lower bounds are the first for any kernel technique, and our analysis assumes realistic scenarios for the input, unlike all previous lower bounds for other release problems which only hold under very restrictive input settings.
Random Kernel Matrices
Spectral Norm
Subguassian Distribution
Data Privacy
Reconstruction Attacks
898-914
Regular Paper
Shiva Prasad
Kasiviswanathan
Shiva Prasad Kasiviswanathan
Mark
Rudelson
Mark Rudelson
10.4230/LIPIcs.APPROX-RANDOM.2015.898
Olivier Bousquet, Ulrike von Luxburg, and G Rätsch. Advanced Lectures on Machine Learning. In ML Summer Schools 2003, 2004.
Xiuyuan Cheng and Amit Singer. The Spectrum of Random Inner-Product Kernel Matrices. Random Matrices: Theory and Applications, 2(04), 2013.
Krzysztof Choromanski and Tal Malkin. The Power of the Dinur-Nissim Algorithm: Breaking Privacy of Statistical and Graph Databases. In PODS, pages 65-76. ACM, 2012.
Anindya De. Lower Bounds in Differential Privacy. In TCC, pages 321-338, 2012.
Irit Dinur and Kobbi Nissim. Revealing Information while Preserving Privacy. In PODS, pages 202-210. ACM, 2003.
Yen Do and Van Vu. The Spectrum of Random Kernel Matrices: Universality Results for Rough and Varying Kernels. Random Matrices: Theory and Applications, 2(03), 2013.
Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating Noise to Sensitivity in Private Data Analysis. In TCC, volume 3876 of LNCS, pages 265-284. Springer, 2006.
Cynthia Dwork, Frank McSherry, and Kunal Talwar. The Price of Privacy and the Limits of LP Decoding. In STOC, pages 85-94. ACM, 2007.
Cynthia Dwork and Sergey Yekhanin. New Efficient Attacks on Statistical Disclosure Control Mechanisms. In CRYPTO, pages 469-480. Springer, 2008.
Arthur E Hoerl and Robert W Kennard. Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, 12(1):55-67, 1970.
Prateek Jain and Abhradeep Thakurta. Differentially Private Learning with Kernels. In ICML, pages 118-126, 2013.
Lei Jia and Shizhong Liao. Accurate Probabilistic Error Bound for Eigenvalues of Kernel Matrix. In Advances in Machine Learning, pages 162-175. Springer, 2009.
Noureddine El Karoui. The Spectrum of Kernel Random Matrices. The Annals of Statistics, pages 1-50, 2010.
Shiva Prasad Kasiviswanathan, Mark Rudelson, and Adam Smith. The Power of Linear Reconstruction Attacks. In SODA, pages 1415-1433, 2013.
Shiva Prasad Kasiviswanathan, Mark Rudelson, Adam Smith, and Jonathan Ullman. The Price of Privately Releasing Contingency Tables and the Spectra of Random Matrices with Correlated Rows. In STOC, pages 775-784, 2010.
Gert RG Lanckriet, Nello Cristianini, Peter Bartlett, Laurent El Ghaoui, and Michael I Jordan. Learning the Kernel Matrix with Semidefinite Programming. The Journal of Machine Learning Research, 5:27-72, 2004.
James Mercer. Functions of Positive and Negative Type, and their Connection with the Theory of Integral Equations. Philosophical transactions of the royal society of London. Series A, containing papers of a mathematical or physical character, pages 415-446, 1909.
Martin M Merener. Polynomial-time Attack on Output Perturbation Sanitizers for Real-valued Databases. Journal of Privacy and Confidentiality, 2(2):5, 2011.
S. Muthukrishnan and Aleksandar Nikolov. Optimal Private Halfspace Counting via Discrepancy. In STOC, pages 1285-1292, 2012.
Mark Rudelson. Recent Developments in Non-asymptotic Theory of Random Matrices. Modern Aspects of Random Matrix Theory, 72:83, 2014.
Craig Saunders, Alexander Gammerman, and Volodya Vovk. Ridge Regression Learning Algorithm in Dual Variables. In ICML, pages 515-521, 1998.
Bernhard Schölkopf, Ralf Herbrich, and Alex J Smola. A Generalized Representer Theorem. In COLT, pages 416-426, 2001.
Bernhard Scholkopf and Alexander J Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, 2001.
John Shawe-Taylor and Nello Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004.
John Shawe-Taylor, Christopher KI Williams, Nello Cristianini, and Jaz Kandola. On the Eigenspectrum of the Gram matrix and the Generalization Error of Kernel-PCA. Information Theory, IEEE Transactions on, 51(7):2510-2522, 2005.
Vikas Sindhwani, Minh Ha Quang, and Aurélie C Lozano. Scalable Matrix-valued Kernel Learning for High-dimensional Nonlinear Multivariate Regression and Granger Causality. arXiv preprint arXiv:1210.4792, 2012.
Roman Vershynin. Introduction to the Non-asymptotic Analysis of Random Matrices. arXiv preprint arXiv:1011.3027, 2010.
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Separating Decision Tree Complexity from Subcube Partition Complexity
The subcube partition model of computation is at least as powerful as decision trees but no separation between these models was known. We show that there exists a function whose deterministic subcube partition complexity is asymptotically smaller than its randomized decision tree complexity, resolving an open problem of Friedgut, Kahn, and Wigderson (2002). Our lower bound is based on the information-theoretic techniques first introduced to lower bound the randomized decision tree complexity of the recursive majority function.
We also show that the public-coin partition bound, the best known lower bound method for randomized decision tree complexity subsuming other general techniques such as block sensitivity, approximate degree, randomized certificate complexity, and the classical adversary bound, also lower bounds randomized subcube partition complexity. This shows that all these lower bound techniques cannot prove optimal lower bounds for randomized decision tree complexity, which answers an open question of Jain and Klauck (2010) and Jain, Lee, and Vishnoi (2014).
Decision tree complexity
query complexity
randomized algorithms
subcube partition complexity
915-930
Regular Paper
Robin
Kothari
Robin Kothari
David
Racicot-Desloges
David Racicot-Desloges
Miklos
Santha
Miklos Santha
10.4230/LIPIcs.APPROX-RANDOM.2015.915
S. Aaronson. Quantum certificate complexity. SIAM Journal on Computing, 35(4):804-824, 2006.
S. Aaronson. Lower bounds for local search by quantum arguments. Journal of Computer and System Sciences, 74(3):313-332, 2008.
A. Ambainis, K. Balodis, A. Belovs, T. Lee, M. Santha, and J. Smotrovs. Separations in query complexity based on pointer functions. arXiv preprint http://arxiv.org/abs/arXiv:1506.04719, 2015.
Y. Brandman, A. Orlitsky, and J. Hennessy. A spectral lower bound technique for the size of decision trees and two-level AND/OR circuits. IEEE Transactons on Computers, 39(2):282-287, February 1990.
H. Buhrman and R. de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002.
S. Chakraborty, R. Kulkarni, S. V. Lokam, and N. Saurabh. Upper bounds on Fourier entropy. Electronic Colloquium on Computational Complexity (ECCC), 20:52, 2013.
E. Friedgut, J. Kahn, and A. Wigderson. Computing graph properties by randomized subcube partitions. In Randomization and Approximation Techniques in Computer Science, volume 2483 of Lecture Notes in Computer Science, pages 105-113. Springer Berlin Heidelberg, 2002.
M. Göös, T. Pitassi, and T. Watson. Deterministic communication vs. partition number. To appear in the Proceedings of the 56th IEEE Symposium on Foundations of Computer Science (FOCS), 2015.
R. Jain and H. Klauck. The partition bound for classical communication complexity and query complexity. In Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity, CCC'10, pages 247-258, 2010.
R. Jain, T. Lee, and N. Vishnoi. A quadratically tight partition bound for classical communication complexity and query complexity. arXiv preprint http://arxiv.org/abs/arXiv:1401.4512, 2014.
T. S. Jayram, R. Kumar, and D. Sivakumar. Two applications of information complexity. In Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing (STOC), STOC'03, pages 673-682, New York, NY, USA, 2003. ACM.
S. Jukna. Boolean Function Complexity: Advances and Frontiers. Algorithms and Combinatorics. Springer, 2012.
E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, 2006.
I. Landau, A. Nachmias, Y. Peres, and S. Vanniasegaram. The lower bound for evaluating a recursive ternary majority function: an entropy-free proof. Undergraduate Research Reports, Department of Statistics, University of California, Berkeley, 2006.
S. Laplante and F. Magniez. Lower bounds for randomized and quantum query complexity using Kolmogorov arguments. SIAM Journal on Computing, 38(1):46-62, 2008.
A. Montanaro. A composition theorem for decision tree complexity. Chicago Journal of Theoretical Computer Science, 2014(6), July 2014.
N. Nisan. CREW PRAMs and decision trees. SIAM Journal on Computing, 20(6):999-1007, 1991.
N. Nisan and M. Szegedy. On the degree of Boolean functions as real polynomials. Computational Complexity, 15(4):557-565, 1995.
A. L. Rosenberg. On the time required to recognize properties of graphs: a problem. SIGACT News, 5(4):15-16, 1973.
M. Saks and A. Wigderson. Probabilistic boolean decision trees and the complexity of evaluating game trees. Proceedings of the 27th IEEE Symposium on Foundations of Computer Science (FOCS), pages 29-38, 1986.
P. Savický. On determinism versus unambiguous nondeterminism for decision trees. ECCC, TR02-009, 2002.
R. Špalek and M. Szegedy. All quantum adversary methods are equivalent. Theory of Computing, 2(1):1-18, 2006.
A. Tal. Properties and applications of boolean function composition. In Proc. of the 4th Conf. on Innovations in Theoretical Computer Science, ITCS'13, pages 441-454, 2013.
A. Yao. Probabilistic computations: Toward a unified measure of complexity. Proc. of the 18th IEEE Symp. on Foundations of Computer Science (FOCS), pages 222-227, 1977.
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Distance-based Species Tree Estimation: Information-Theoretic Trade-off between Number of Loci and Sequence Length under the Coalescent
We consider the reconstruction of a phylogeny from multiple genes under the multispecies coalescent. We establish a connection with the sparse signal detection problem, where one seeks to distinguish between a distribution and a mixture of the distribution and a sparse signal. Using this connection, we derive an information-theoretic trade-off between the number of genes needed for an accurate reconstruction and the sequence length of the genes.
phylogenetic reconstruction
multispecies coalescent
sequence length requirement.
931-942
Regular Paper
Elchanan
Mossel
Elchanan Mossel
Sebastien
Roch
Sebastien Roch
10.4230/LIPIcs.APPROX-RANDOM.2015.931
Elizabeth S. Allman, James H. Degnan, and John A. Rhodes. Identifying the rooted species tree from the distribution of unrooted gene trees under the coalescent. Journal of Mathematical Biology, 62(6):833-862, 2011.
Christian N.K. Anderson, Liang Liu, Dennis Pearl, and Scott V. Edwards. Tangled trees: The challenge of inferring species trees from coalescent and noncoalescent genes. In Maria Anisimova, editor, Evolutionary Genomics, volume 856 of Methods in Molecular Biology, pages 3-28. Humana Press, 2012.
Alexandr Andoni, Constantinos Daskalakis, Avinatan Hassidim, and Sébastien Roch. Global alignment of molecular sequences via ancestral state reconstruction (extended abstract). In ICS, pages 358-369, 2010.
Anand Bhaskar and Yun S. Song. Descartes' rule of signs and the identifiability of population demographic models from genomic variation data. Ann. Statist., 42(6):2469-2493, 2014.
T. Tony Cai, X. Jessie Jeng, and Jiashun Jin. Optimal detection of heterogeneous and heteroscedastic mixtures. J. R. Stat. Soc. Ser. B Stat. Methodol., 73(5):629-662, 2011.
T.T. Cai and Yihong Wu. Optimal detection of sparse mixtures against a given null distribution. Information Theory, IEEE Transactions on, 60(4):2217-2232, April 2014.
L. Cayon, J. Jin, and A. Treaster. Higher criticism statistic: detecting and identifying non-gaussianity in the wmap first-year data. Monthly Notices of the Royal Astronomical Society, 362(3):826-832, 2005.
T. M. Cover and J. A. Thomas. Elements of information theory. Wiley Series in Telecommunications. John Wiley & Sons Inc., New York, 1991. A Wiley-Interscience Publication.
M. Cryan, L. A. Goldberg, and P. W. Goldberg. Evolutionary trees can be learned in polynomial time. SIAM J. Comput., 31(2):375-397, 2002. short version, Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS 98), pages 436-445, 1998.
Gautam Dasarathy, Robert D. Nowak, and Sébastien Roch. New sample complexity bounds for phylogenetic inference from multiple loci. In 2014 IEEE International Symposium on Information Theory, Honolulu, HI, USA, June 29 - July 4, 2014, pages 2037-2041, 2014.
Constantinos Daskalakis, Elchanan Mossel, and Sébastien Roch. Evolutionary trees and the ising model on the bethe lattice: a proof of steel’s conjecture. Probability Theory and Related Fields, 149:149-189, 2011. 10.1007/s00440-009-0246-2.
Constantinos Daskalakis, Elchanan Mossel, and Sébastien Roch. Phylogenies without branch bounds: Contracting the short, pruning the deep. SIAM J. Discrete Math., 25(2):872-893, 2011.
Constantinos Daskalakis and Sébastien Roch. Alignment-free phylogenetic reconstruction. In RECOMB, pages 123-137, 2010.
Michael DeGiorgio and James H Degnan. Fast and consistent estimation of species trees using supermatrix rooted triples. Molecular Biology and Evolution, 27(3):552-69, March 2010.
J. H. Degnan and N. A. Rosenberg. Discordance of species trees with their most likely gene trees. PLoS Genetics, 2(5), May 2006.
James H. Degnan, Michael DeGiorgio, David Bryant, and Noah A. Rosenberg. Properties of consensus methods for inferring species trees from gene trees. Systematic Biology, 58(1):35-54, 2009.
James H. Degnan and Noah A. Rosenberg. Gene tree discordance, phylogenetic inference and the multispecies coalescent. Trends in Ecology and Evolution, 24(6):332-340, 2009.
Frederic Delsuc, Henner Brinkmann, and Herve Philippe. Phylogenomics and the reconstruction of the tree of life. Nat Rev Genet, 6(5):361-375, 05 2005.
R. L. Dobrusin. A statistical problem arising in the theory of detection of signals in the presence of noise in a multi-channel system and leading to stable distribution laws. Theory of Probability & Its Applications, 3(2):161-173, 1958.
David Donoho and Jiashun Jin. Higher criticism for detecting sparse heterogeneous mixtures. Ann. Statist., 32(3):962-994, 06 2004.
Richard Durrett. Probability models for DNA sequence evolution. Probability and its Applications (New York). Springer, New York, second edition, 2008.
P. L. Erdös, M. A. Steel;, L. A. Székely, and T. A. Warnow. A few logs suffice to build (almost) all trees (part 1). Random Struct. Algor., 14(2):153-184, 1999.
P. L. Erdös, M. A. Steel;, L. A. Székely, and T. A. Warnow. A few logs suffice to build (almost) all trees (part 2). Theor. Comput. Sci., 221:77-118, 1999.
J. Felsenstein. Inferring Phylogenies. Sinauer, New York, New York, 2004.
Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The elements of statistical learning. Springer Series in Statistics. Springer, New York, second edition, 2009. Data mining, inference, and prediction.
Yu. I. Ingster. Some problems of hypothesis testing leading to infinitely divisible distributions. Math. Methods Statist., 6(1):47-69, 1997.
X. Jessie Jeng, T. Tony Cai, and Hongzhe Li. Optimal sparse segment identification with application in copy number variation analysis. J. Amer. Statist. Assoc., 105(491):1156-1166, 2010.
T. H. Jukes and C. Cantor. Mammalian protein metabolism. In H. N. Munro, editor, Evolution of protein molecules, pages 21-132. Academic Press, 1969.
Junhyong Kim, Elchanan Mossel, Miklos Z. Racz, and Nathan Ross. Can one hear the shape of a population history? Theoretical Population Biology, 100(0):26-38, 2015.
Martin Kulldorff, Richard Heffernan, Jessica Hartman, Renato Assuncao, and Farzad Mostashari. A space time permutation scan statistic for disease outbreak detection. PLoS Med, 2(3):e59, 02 2005.
Liang Liu, Lili Yu, Laura Kubatko, Dennis K. Pearl, and Scott V. Edwards. Coalescent methods for estimating phylogenetic trees. Molecular Phylogenetics and Evolution, 53(1):320-328, 2009.
Liang Liu, Lili Yu, and Dennis K. Pearl. Maximum tree: a consistent estimator of the species tree. Journal of Mathematical Biology, 60(1):95-106, 2010.
Wayne P. Maddison. Gene trees in species trees. Systematic Biology, 46(3):523-536, 1997.
E. Mossel. On the impossibility of reconstructing ancestral data and phylogenies. J. Comput. Biol., 10(5):669-678, 2003.
E. Mossel. Phase transitions in phylogeny. Trans. Amer. Math. Soc., 356(6):2379-2404, 2004.
E. Mossel and S. Roch. Distance-based species tree estimation: information-theoretic trade-off between number of loci and sequence length under the coalescent. ArXiv e-print 1504.05289, 2015.
Elchanan Mossel and Sébastien Roch. Learning nonsingular phylogenies and hidden Markov models. In STOC'05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 366-375, New York, 2005. ACM.
Elchanan Mossel and Sébastien Roch. Incomplete lineage sorting: Consistent phylogeny estimation from multiple loci. IEEE/ACM Trans. Comput. Biology Bioinform., 7(1):166-171, 2010.
Elchanan Mossel, Sébastien Roch, and Allan Sly. On the inference of large phylogenies with long branches: How long is too long? Bulletin of Mathematical Biology, 73:1627-1644, 2011. 10.1007/s11538-010-9584-6.
Raphaël Mourad, Christine Sinoquet, Nevin Lianwen Zhang, Tengfei Liu, and Philippe Leray. A survey on latent tree models and applications. J. Artif. Intell. Res. (JAIR), 47:157-203, 2013.
Simon Myers, Charles Fefferman, and Nick Patterson. Can one learn history from the allelic spectrum? Theoretical Population Biology, 73(3):342-348, 2008.
Luay Nakhleh. Computational approaches to species phylogeny inference and gene tree reconciliation. Trends in ecology & evolution, 28(12):10.1016/j.tree.2013.09.004, 12 2013.
Bruce Rannala and Ziheng Yang. Bayes estimation of species divergence times and ancestral population sizes using DNA sequences from multiple loci. Genetics, 164(4):1645-1656, 2003.
Sebastien Roch. Toward extracting all phylogenetic information from matrices of evolutionary distances. Science, 327(5971):1376-1379, 2010.
Sebastien Roch. An analytical comparison of multilocus methods under the multispecies coalescent: The three-taxon case. In Pacific Symposium in Biocomputing 2013, pages 297-306, 2013.
Sebastien Roch and Mike Steel. Likelihood-based tree reconstruction on a concatenation of alignments can be positively misleading. Theoretical Population Biology, 2015. To appear.
Sebastien Roch and Tandy Warnow. On the robustness to gene tree estimation error (or lack thereof) of coalescent-based species tree methods. Systematic Biology, 2015. In press.
C. Semple and M. Steel. Phylogenetics, volume 22 of Mathematics and its Applications series. Oxford University Press, 2003.
M. A. Steel and L. A. Székely. Inverting random functions. II. Explicit bounds for discrete maximum likelihood estimation, with applications. SIAM J. Discrete Math., 15(4):562-575 (electronic), 2002.
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Deterministically Factoring Sparse Polynomials into Multilinear Factors and Sums of Univariate Polynomials
We present the first efficient deterministic algorithm for factoring sparse polynomials that split into multilinear factors
and sums of univariate polynomials. Our result makes partial progress towards the resolution of the classical question posed by von zur Gathen and Kaltofen in [von zur Gathen/Kaltofen, J. Comp. Sys. Sci., 1985] to devise an efficient deterministic algorithm for factoring (general) sparse polynomials. We achieve our goal by introducing essential factorization schemes which can be thought of as a relaxation of the regular factorization notion.
Derandomization
Multivariate Polynomial Factorization
Sparse polynomials
943-958
Regular Paper
Ilya
Volkovich
Ilya Volkovich
10.4230/LIPIcs.APPROX-RANDOM.2015.943
M. Ben-Or and P. Tiwari. A deterministic algorithm for sparse multivariate polynominal interpolation. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC), pages 301-309, 1988.
A. M. Cuyt and W. Lee. Sparse interpolation of multivariate rational functions. Theor. Comput. Sci., 412(16):1445-1456, 2011.
S. Gao, E. Kaltofen, and A. G. B. Lauder. Deterministic distinct-degree factorization of polynomials over finite fields. J. Symb. Comput., 38(6):1461-1470, 2004.
J. von zur Gathen. Who was who in polynomial factorization:. In ISSAC, page 2, 2006.
J. von zur Gathen and J. Gerhard. Modern computer algebra. Cambridge University Press, 1999.
J. von zur Gathen and E. Kaltofen. Factoring sparse multivariate polynomials. Journal of Computer and System Sciences, 31(2):265-287, 1985.
E. Grigorescu, K. Jung, and R. Rubinfeld. A local decision test for sparse polynomials. Inf. Process. Lett., 110(20):898-901, 2010.
A. Gupta, N. Kayal, and S. V. Lokam. Reconstruction of depth-4 multilinear circuits with top fanin 2. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC), pages 625-642, 2012. Full version at http://eccc.hpi-web.de/report/2011/153.
V. Guruswami and M. Sudan. Improved decoding of reed-solomon codes and algebraic-geometry codes. IEEE Transactions on Information Theory, 45(6):1757-1767, 1999.
V. Kabanets and R. Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004.
E. Kaltofen. Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization. SIAM J. on computing, 14(2):469-489, 1985.
E. Kaltofen. Factorization of polynomials given by straight-line programs. In S. Micali, editor, Randomness in Computation, volume 5 of Advances in Computing Research, pages 375-412. JAI Press Inc., Greenwhich, Connecticut, 1989.
E. Kaltofen. Polynomial factorization: a success story. In ISSAC, pages 3-4, 2003.
E. Kaltofen and P. Koiran. On the complexity of factoring bivariate supersparse (lacunary) polynomials. In ISSAC, pages 208-215, 2005.
E. Kaltofen and B. M. Trager. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. of Symbolic Computation, 9(3):301-320, 1990.
Z. S. Karnin, P. Mukhopadhyay, A. Shpilka, and I. Volkovich. Deterministic identity testing of depth 4 multilinear circuits with bounded top fan-in. SIAM J. on Computing, 42(6):2114-2131, 2013.
N. Kayal. Derandomizing some number-theoretic and algebraic algorithms. PhD thesis, Indian Institute of Technology, Kanpur, India, 2007.
A. Klivans and D. Spielman. Randomness efficient identity testing of multivariate polynomials. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pages 216-223, 2001.
S. Kopparty, S. Saraf, and A. Shpilka. Equivalence of polynomial identity testing and deterministic multivariate polynomial factorization. In Proceedings of the 29th Annual IEEE Conference on Computational Complexity (CCC), pages 169-180, 2014.
C. Saha, R. Saptharishi, and N. Saxena. A case of depth-3 identity testing, sparse factorization and duality. Computational Complexity, 22(1):39-69, 2013.
S. Saraf and I. Volkovich. Blackbox identity testing for depth-4 multilinear circuits. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC), pages 421-430, 2011. Full version at http://eccc.hpi-web.de/report/2011/046.
N. Saxena. Diagonal circuit identity testing and lower bounds. In Automata, Languages and Programming, 35th International Colloquium, pages 60-71, 2008. Full version at http://eccc.hpi-web.de/eccc-reports/2007/TR07-124/index.html.
V. Shoup. A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic. In ISSAC, pages 14-21, 1991.
A. Shpilka and I. Volkovich. Improved polynomial identity testing for read-once formulas. In APPROX-RANDOM, pages 700-713, 2009. Full version at http://eccc.hpi-web.de/report/2010/011.
A. Shpilka and I. Volkovich. On the relation between polynomial identity testing and finding variable disjoint factors. In Automata, Languages and Programming, 37th International Colloquium (ICALP), pages 408-419, 2010. Full version at http://eccc.hpi-web.de/report/2010/036.
A. Shpilka and I. Volkovich. On reconstruction and testing of read-once formulas. Theory of Computing, 10:465-514, 2014.
A. Shpilka and I. Volkovich. Read-once polynomial identity testing. Computational Complexity, 2015.
A. Shpilka and A. Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):207-388, 2010.
M. Sudan. Decoding of reed solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997.
R. Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposium on Symbolic and Algebraic Computation, pages 216-226, 1979.
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