1st Symposium on Simplicity in Algorithms (SOSA 2018), SOSA 2018, January 7-10, 2018, New Orleans, LA, USA
SOSA 2018
January 7-10, 2018
New Orleans, LA, USA
Open Access Series in Informatics
OASIcs
https://www.dagstuhl.de/dagpub/2190-6807
https://dblp.org/db/series/oasics
2190-6807
Raimund
Seidel
Raimund Seidel
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
61
2018
978-3-95977-064-4
https://www.dagstuhl.de/dagpub/978-3-95977-064-4
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter
Table of Contents
Preface
Conference Organization
0:i-0:xii
Front Matter
Raimund
Seidel
Raimund Seidel
10.4230/OASIcs.SOSA.2018.0
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A Naive Algorithm for Feedback Vertex Set
Given a graph on n vertices and an integer k, the feedback vertex set problem asks for the deletion of at most k vertices to make the graph acyclic. We show that a greedy branching algorithm, which always branches on an undecided vertex with the largest degree, runs in single-exponential time, i.e., O(c^k n^2) for some constant c.
greedy algorithm
analysis of algorithms
branching algorithm
parameterized computation -- graph modification problem
1:1-1:9
Regular Paper
Yixin
Cao
Yixin Cao
10.4230/OASIcs.SOSA.2018.1
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A Note on Iterated Rounding for the Survivable Network Design Problem
In this note we consider the survivable network design problem (SNDP) in undirected graphs. We make two contributions. The first is a new counting argument in the iterated rounding based 2-approximation for edge-connectivity SNDP (EC-SNDP) originally due to Jain. The second contribution is to make some connections between hypergraphic version of SNDP (Hypergraph-SNDP) introduced by Zhao, Nagamochi and Ibaraki, and edge and node-weighted versions of EC-SNDP and element-connectivity SNDP (Elem-SNDP). One useful consequence is a 2-approximation for Elem-SNDP that avoids the use of set-pair based relaxation and analysis.
survivable network design
iterated rounding
approximation
element connectivity
2:1-2:10
Regular Paper
Chandra
Chekuri
Chandra Chekuri
Thapanapong
Rukkanchanunt
Thapanapong Rukkanchanunt
10.4230/OASIcs.SOSA.2018.2
Nikhil Bansal, Rohit Khandekar, and Viswanath Nagarajan. Additive guarantees for degree-bounded directed network design. SIAM Journal on Computing, 39(4):1413-1431, 2009.
Tanmoy Chakraborty, Julia Chuzhoy, and Sanjeev Khanna. Network design for vertex connectivity. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 167-176. ACM, 2008.
Chandra Chekuri, Alina Ene, and Ali Vakilian. Node-weighted network design in planar and minor-closed families of graphs. In Automata, Languages, and Programming, pages 206-217. Springer, 2012.
Chandra Chekuri, Alina Ene, and Ali Vakilian. Prize-collecting survivable network design in node-weighted graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 98-109. Springer, 2012.
J. Cheriyan, S. Vempala, and A. Vetta. Network design via iterative rounding of setpair relaxations. Combinatorica, 26(3):255-275, 2006.
Julia Chuzhoy and Sanjeev Khanna. An O(k³ log n)-approximation algorithm for vertex-connectivity survivable network design. Theory of Computing, 8:401-413, 2012. Preliminary version in Proc. of IEEE FOCS, 2009.
L. Fleischer, K. Jain, and D.P. Williamson. Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems. Journal of Computer and System Sciences, 72(5):838-867, 2006.
Takuro Fukunaga. Spider covers for prize-collecting network activation problem. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '15, pages 9-24. SIAM, 2015. URL: http://dl.acm.org/citation.cfm?id=2722129.2722131.
http://dl.acm.org/citation.cfm?id=2722129.2722131
M.X. Goemans, A.V. Goldberg, S. Plotkin, D.B. Shmoys, E. Tardos, and D.P. Williamson. Improved approximation algorithms for network design problems. In Proc. of ACM-SIAM SODA, pages 223-232, 1994.
K. Jain. A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica, 21(1):39-60, 2001. Preliminary version in FOCS 1998.
P. Klein and R. Ravi. A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithms, 19(1):104-115, 1995. Preliminary version in IPCO 1993.
Lap Chi Lau, Ramamoorthi Ravi, and Mohit Singh. Iterative methods in combinatorial optimization, volume 46. Cambridge University Press, 2011.
Viswanath Nagarajan, R Ravi, and Mohit Singh. Simpler analysis of lp extreme points for traveling salesman and survivable network design problems. Operations Research Letters, 38(3):156-160, 2010.
Z. Nutov. Approximating minimum cost connectivity problems via uncrossable bifamilies and spider-cover decompositions. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 417-426. IEEE, 2009.
David P Williamson and David B Shmoys. The design of approximation algorithms. Cambridge university press, 2011.
Liang Zhao, Hiroshi Nagamochi, and Toshihide Ibaraki. A note on approximating the survivable network design problem in hypergraphs. IEICE TRANSACTIONS on Information and Systems, 85(2):322-326, 2002.
Liang Zhao, Hiroshi Nagamochi, and Toshihide Ibaraki. A primal-dual approximation algorithm for the survivable network design problem in hypergraphs. Discrete applied mathematics, 126(2):275-289, 2003. Preliminary version appeared in Proc. of STACS, 2001.
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Congestion Minimization for Multipath Routing via Multiroute Flows
Congestion minimization is a well-known routing problem for which
there is an O(log n/loglog n)-approximation via randomized rounding due to Raghavan and Thompson. Srinivasan formally introduced the low-congestion multi-path routing problem as a generalization of the (single-path) congestion minimization problem. The goal is to route multiple disjoint paths for each pair, for the sake of fault tolerance. Srinivasan developed a dependent randomized scheme for a special case of the multi-path problem when the input consists of a given set of disjoint paths for each pair and the goal is to select a given subset of them. Subsequently Doerr gave a different dependentrounding scheme and derandomization. Doerr et al. considered the problem where the paths have to be chosen, and applied the dependent rounding technique and evaluated it experimentally. However, their algorithm does not maintain the required disjointness property without which the problem easily reduces to the standard congestion minimization problem.
In this note we show a simple algorithm that solves the problem
correctly without the need for dependent rounding --- standard
independent rounding suffices. This is made possible via the notion
of multiroute flows originally suggested by Kishimoto et al. One advantage of the simpler rounding is an improved bound on the congestion when the path lengths are short.
multipath routing
congestion minimization
multiroute flows
3:1-3:12
Regular Paper
Chandra
Chekuri
Chandra Chekuri
Mark
Idleman
Mark Idleman
10.4230/OASIcs.SOSA.2018.3
Charu C. Aggarwal and James B. Orlin. On multiroute maximum flows in networks. Networks, 39(1):43-52, 2002. URL: http://dx.doi.org/10.1002/net.10008.
http://dx.doi.org/10.1002/net.10008
Stephen Baum and Leslie E Trotter Jr. Integer rounding and polyhedral decomposition for totally unimodular systems. Optimization and Operations Research, 157:15-23, 1978.
Graham Brightwell, Gianpaolo Oriolo, and F Bruce Shepherd. Reserving resilient capacity in a network. SIAM journal on discrete mathematics, 14(4):524-539, 2001.
Chandra Chekuri, Alina Ene, and Ali Vakilian. Prize-collecting survivable network design in node-weighted graphs. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 98-109, 2012.
Chandra Chekuri, Jan Vondrak, and Rico Zenklusen. Dependent randomized rounding via exchange properties of combinatorial structures. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Symposium on, pages 575-584. IEEE, 2010.
Julia Chuzhoy, Venkatesan Guruswami, Sanjeev Khanna, and Kunal Talwar. Hardness of routing with congestion in directed graphs. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 165-178. ACM, 2007.
Benjamin Doerr. Randomly rounding rationals with cardinality constraints and derandomizations. STACS 2007, pages 441-452, 2007.
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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.
Bernhard Haeupler, Barna Saha, and Aravind Srinivasan. New constructive aspects of the lovász local lemma. Journal of the ACM (JACM), 58(6):28, 2011.
Mark Idleman. Approximation algorithms for the minimum congestion routing problem via k-route flows. Master’s thesis, University of Illinois, July 2017.
Wataru Kishimoto. A method for obtaining the maximum multiroute flows in a network. Networks, 27(4):279-291, 1996. URL: http://dx.doi.org/10.1002/(SICI)1097-0037(199607)27:4<279::AID-NET3>3.0.CO;2-D.
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Aravind Srinivasan. An extension of the lovász local lemma, and its applications to integer programming. SIAM Journal on Computing, 36(3):609-634, 2006.
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Better and Simpler Error Analysis of the Sinkhorn-Knopp Algorithm for Matrix Scaling
Given a non-negative real matrix A, the matrix scaling problem is to determine if it is possible to scale the rows and columns so that each row and each column sums to a specified target value for it.
The matrix scaling problem arises in many algorithmic applications, perhaps most notably as a preconditioning step in solving linear system of equations. One of the most natural and by now classical approach to matrix scaling is the Sinkhorn-Knopp algorithm (also known as the RAS method) where one alternately scales either all rows or all columns to meet the target values. In addition to being extremely simple and natural, another appeal of this procedure is that it easily lends itself to parallelization. A central question is to understand the rate of convergence of the Sinkhorn-Knopp algorithm.
Specifically, given a suitable error metric to measure deviations from target values, and an error bound epsilon, how quickly does the Sinkhorn-Knopp algorithm converge to an error below epsilon? While there are several non-trivial convergence results known about the Sinkhorn-Knopp algorithm, perhaps somewhat surprisingly, even for natural error metrics such as ell_1-error or ell_2-error, this is not entirely understood.
In this paper, we present an elementary convergence analysis for the Sinkhorn-Knopp algorithm that improves upon the previous best bound. In a nutshell, our approach is to show (i) a simple bound on the number of iterations needed so that the KL-divergence between the current row-sums and the target row-sums drops below a specified threshold delta, and (ii) then show that for a suitable choice of delta, whenever KL-divergence is below delta, then the ell_1-error or the ell_2-error is below epsilon. The well-known Pinsker's inequality immediately allows us to translate a bound on the KL divergence to a bound on ell_1-error. To bound the ell_2-error in terms of the KL-divergence, we establish a new inequality, referred to as (KL vs ell_1/ell_2) inequality in the paper. This new inequality is a strengthening of the Pinsker's inequality that we believe is of independent interest. Our analysis of ell_2-error significantly improves upon the best previous convergence bound for ell_2-error.
The idea of studying Sinkhorn-Knopp convergence via KL-divergence is not new and has indeed been previously explored. Our contribution is an elementary, self-contained presentation of this approach and an interesting new inequality that yields a significantly stronger convergence guarantee for the extensively studied ell_2-error.
Matrix Scaling
Entropy Minimization
KL Divergence Inequalities
4:1-4:11
Regular Paper
Deeparnab
Chakrabarty
Deeparnab Chakrabarty
Sanjeev
Khanna
Sanjeev Khanna
10.4230/OASIcs.SOSA.2018.4
Scott Aaronson. Quantum computing and hidden variables. Phys. Rev. A, 71:032325, Mar 2005. URL: http://dx.doi.org/10.1103/PhysRevA.71.032325.
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Approximation Schemes for 0-1 Knapsack
We revisit the standard 0-1 knapsack problem. The latest polynomial-time approximation scheme by Rhee (2015) with approximation factor 1+eps has running time near O(n+(1/eps)^{5/2}) (ignoring polylogarithmic factors), and is randomized. We present a simpler algorithm which achieves the same result and is deterministic.
With more effort, our ideas can actually lead to an improved time bound near O(n + (1/eps)^{12/5}), and still further improvements for small n.
knapsack problem
approximation algorithms
optimization
(min,+)-convolution
5:1-5:12
Regular Paper
Timothy M.
Chan
Timothy M. Chan
10.4230/OASIcs.SOSA.2018.5
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David Bremner, Timothy M. Chan, Erik D. Demaine, Jeff Erickson, Ferran Hurtado, John Iacono, Stefan Langerman, Mihai Patrascu, and Perouz Taslakian. Necklaces, convolutions, and X+Y. Algorithmica, 69(2):294-314, 2014. URL: http://dx.doi.org/10.1007/s00453-012-9734-3.
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Timothy M. Chan and Ryan Williams. Deterministic apsp, orthogonal vectors, and more: Quickly derandomizing razborov-smolensky. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1246-1255. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch87.
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http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.22
Oscar H. Ibarra and Chul E. Kim. Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM, 22(4):463-468, 1975. URL: http://dx.doi.org/10.1145/321906.321909.
http://dx.doi.org/10.1145/321906.321909
Klaus Jansen and Stefan Erich Julius Kraft. A faster FPTAS for the unbounded knapsack problem. In Zsuzsanna Lipták and William F. Smyth, editors, Combinatorial Algorithms - 26th International Workshop, IWOCA 2015, Verona, Italy, October 5-7, 2015, Revised Selected Papers, volume 9538 of Lecture Notes in Computer Science, pages 274-286. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-29516-9_23.
http://dx.doi.org/10.1007/978-3-319-29516-9_23
Hans Kellerer, Renata Mansini, Ulrich Pferschy, and Maria Grazia Speranza. An efficient fully polynomial approximation scheme for the subset-sum problem. J. Comput. Syst. Sci., 66(2):349-370, 2003. URL: http://dx.doi.org/10.1016/S0022-0000(03)00006-0.
http://dx.doi.org/10.1016/S0022-0000(03)00006-0
Hans Kellerer and Ulrich Pferschy. A new fully polynomial time approximation scheme for the knapsack problem. J. Comb. Optim., 3(1):59-71, 1999. URL: http://dx.doi.org/10.1023/A:1009813105532.
http://dx.doi.org/10.1023/A:1009813105532
Hans Kellerer and Ulrich Pferschy. Improved dynamic programming in connection with an FPTAS for the knapsack problem. J. Comb. Optim., 8(1):5-11, 2004. URL: http://dx.doi.org/10.1023/B:JOCO.0000021934.29833.6b.
http://dx.doi.org/10.1023/B:JOCO.0000021934.29833.6b
Marvin Künnemann, Ramamohan Paturi, and Stefan Schneider. On the fine-grained complexity of one-dimensional dynamic programming. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 21:1-21:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.21.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.21
Eugene L. Lawler. Fast approximation algorithms for knapsack problems. Math. Oper. Res., 4(4):339-356, 1979. URL: http://dx.doi.org/10.1287/moor.4.4.339.
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http://dx.doi.org/10.1016/0377-2217(84)90286-8
Donguk Rhee. Faster fully polynomial approximation schemes for knapsack problems. Master’s thesis, MIT, 2015. URL: https://dspace.mit.edu/bitstream/handle/1721.1/98564/920857251-MIT.pdf.
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http://dx.doi.org/10.1145/321864.321873
Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In David B. Shmoys, editor, Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 664-673. ACM, 2014. URL: http://dx.doi.org/10.1145/2591796.2591811.
http://dx.doi.org/10.1145/2591796.2591811
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Counting Solutions to Polynomial Systems via Reductions
This paper provides both positive and negative results for counting solutions to systems of polynomial equations over a finite field. The general idea is to try to reduce the problem to counting solutions to a single polynomial, where the task is easier. In both cases, simple methods are utilized that we expect will have wider applicability (far beyond algebra).
First, we give an efficient deterministic reduction from approximate counting for a system of (arbitrary) polynomial equations to approximate counting for one equation, over any finite field. We apply this reduction to give a deterministic poly(n,s,log p)/eps^2 time algorithm for approximately counting the fraction of solutions to a system of s quadratic n-variate polynomials over F_p (the finite field of prime order p) to within an additive eps factor, for any prime p. Note that uniform random sampling would already require Omega(s/eps^2) time, so our algorithm behaves as a full derandomization of uniform sampling. The approximate-counting algorithm yields efficient approximate counting for other well-known problems, such as 2-SAT, NAE-3SAT, and 3-Coloring. As a corollary, there is a deterministic algorithm (with analogous running time) for producing solutions to such systems which have at least eps p^n solutions.
Second, we consider the difficulty of exactly counting solutions to a single polynomial of constant degree, over a finite field. (Note that finding a solution in this case is easy.) It has been known for over 20 years that this counting problem is already NP-hard for degree-three polynomials over F_2; however, all known reductions increased the number of variables by a considerable amount. We give a subexponential-time reduction from counting solutions to k-CNF formulas to counting solutions to a degree-k^{O(k)} polynomial (over any finite field of O(1) order) which exactly preserves the number of variables. As a corollary, the Strong Exponential Time Hypothesis (even its weak counting variant #SETH) implies that counting solutions to constant-degree polynomials (even over F_2) requires essentially 2^n time. Similar results hold for counting orthogonal pairs of vectors over F_p.
counting complexity
polynomial equations
finite field
derandomization
strong exponential time hypothesis
6:1-6:15
Regular Paper
R. Ryan
Williams
R. Ryan Williams
10.4230/OASIcs.SOSA.2018.6
Thomas Dybdahl Ahle, Rasmus Pagh, Ilya P. Razenshteyn, and Francesco Silvestri. On the complexity of inner product similarity join. In Tova Milo and Wang-Chiew Tan, editors, Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2016, San Francisco, CA, USA, June 26 - July 01, 2016, pages 151-164. ACM, 2016. URL: http://dx.doi.org/10.1145/2902251.2902285.
http://dx.doi.org/10.1145/2902251.2902285
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Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. A duality between clause width and clause density for SAT. In 21st Annual IEEE Conference on Computational Complexity (CCC 2006), 16-20 July 2006, Prague, Czech Republic, pages 252-260. IEEE Computer Society, 2006. URL: http://dx.doi.org/10.1109/CCC.2006.6.
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Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. The complexity of satisfiability of small depth circuits. In Jianer Chen and Fedor V. Fomin, editors, Parameterized and Exact Computation, 4th International Workshop, IWPEC 2009, Copenhagen, Denmark, September 10-11, 2009, Revised Selected Papers, volume 5917 of Lecture Notes in Computer Science, pages 75-85. Springer, 2009. URL: http://dx.doi.org/10.1007/978-3-642-11269-0_6.
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Timothy M. Chan and Ryan Williams. Deterministic apsp, orthogonal vectors, and more: Quickly derandomizing razborov-smolensky. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1246-1255. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch87.
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Radu Curticapean. Parity separation: A scientifically proven method for permanent weight loss. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 47:1-47:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.47.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.47
Holger Dell, Thore Husfeldt, Dániel Marx, Nina Taslaman, and Martin Wahlen. Exponential time complexity of the permanent and the tutte polynomial. ACM Trans. Algorithms, 10(4):21:1-21:32, 2014. URL: http://dx.doi.org/10.1145/2635812.
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Holger Dell and John Lapinskas. Fine-grained reductions from approximate counting to decision. CoRR, abs/1707.04609, 2017. URL: http://arxiv.org/abs/1707.04609.
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Andrej Ehrenfeucht and Marek Karpinski. The computational complexity of (xor, and)-counting problems. Technical Report TR-90-031, International Computer Science Institute, Berkeley, 1990. URL: http://www.icsi.berkeley.edu/pubs/techreports/tr-90-033.pdf.
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Guy Even, Oded Goldreich, Michael Luby, Noam Nisan, and Boban Velickovic. Approximations of general independent distributions. In S. Rao Kosaraju, Mike Fellows, Avi Wigderson, and John A. Ellis, editors, Proceedings of the 24th Annual ACM Symposium on Theory of Computing, May 4-6, 1992, Victoria, British Columbia, Canada, pages 10-16. ACM, 1992. URL: http://dx.doi.org/10.1145/129712.129714.
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Oded Goldreich. In a world of p=bpp. In Oded Goldreich, editor, Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation - In Collaboration with Lidor Avigad, Mihir Bellare, Zvika Brakerski, Shafi Goldwasser, Shai Halevi, Tali Kaufman, Leonid Levin, Noam Nisan, Dana Ron, Madhu Sudan, Luca Trevisan, Salil Vadhan, Avi Wigderson, David Zuckerman, volume 6650 of Lecture Notes in Computer Science, pages 191-232. Springer, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22670-0_20.
http://dx.doi.org/10.1007/978-3-642-22670-0_20
Parikshit Gopalan, Adam R. Klivans, Raghu Meka, Daniel Stefankovic, Santosh Vempala, and Eric Vigoda. An FPTAS for #knapsack and related counting problems. In Rafail Ostrovsky, editor, IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 817-826. IEEE Computer Society, 2011. URL: http://dx.doi.org/10.1109/FOCS.2011.32.
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Parikshit Gopalan, Raghu Meka, and Omer Reingold. DNF sparsification and a faster deterministic counting algorithm. Computational Complexity, 22(2):275-310, 2013. URL: http://dx.doi.org/10.1007/s00037-013-0068-6.
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Ben Gum and Richard J. Lipton. Cheaper by the dozen: Batched algorithms. In Vipin Kumar and Robert L. Grossman, editors, Proceedings of the First SIAM International Conference on Data Mining, SDM 2001, Chicago, IL, USA, April 5-7, 2001, pages 1-11. SIAM, 2001. URL: http://dx.doi.org/10.1137/1.9781611972719.23.
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Edward A. Hirsch. A fast deterministic algorithm for formulas that have many satisfying assignments. Logic Journal of the IGPL, 6(1):59-71, 1998. URL: http://dx.doi.org/10.1093/jigpal/6.1.59.
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Christian Hoffmann. Exponential time complexity of weighted counting of independent sets. In Venkatesh Raman and Saket Saurabh, editors, Parameterized and Exact Computation - 5th International Symposium, IPEC 2010, Chennai, India, December 13-15, 2010. Proceedings, volume 6478 of Lecture Notes in Computer Science, pages 180-191. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-17493-3_18.
http://dx.doi.org/10.1007/978-3-642-17493-3_18
Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
http://dx.doi.org/10.1006/jcss.2001.1774
Nathan Linial and Noam Nisan. Approximate inclusion-exclusion. Combinatorica, 10(4):349-365, 1990. URL: http://dx.doi.org/10.1007/BF02128670.
http://dx.doi.org/10.1007/BF02128670
Daniel Lokshtanov, Ramamohan Paturi, Suguru Tamaki, R. Ryan Williams, and Huacheng Yu. Beating brute force for systems of polynomial equations over finite fields. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2190-2202. SIAM, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.143.
http://dx.doi.org/10.1137/1.9781611974782.143
Michael Luby and Boban Velickovic. On deterministic approximation of DNF. Algorithmica, 16(4/5):415-433, 1996. URL: http://dx.doi.org/10.1007/BF01940873.
http://dx.doi.org/10.1007/BF01940873
Michael Luby, Boban Velickovic, and Avi Wigderson. Deterministic approximate counting of depth-2 circuits. In Second Israel Symposium on Theory of Computing Systems, ISTCS 1993, Natanya, Israel, June 7-9, 1993, Proceedings, pages 18-24. IEEE Computer Society, 1993. URL: http://dx.doi.org/10.1109/ISTCS.1993.253488.
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Michael O. Rabin. Probabilistic algorithms in finite fields. SIAM J. Comput., 9(2):273-280, 1980. URL: http://dx.doi.org/10.1137/0209024.
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Emanuele Viola. The sum of D small-bias generators fools polynomials of degree D. Computational Complexity, 18(2):209-217, 2009. URL: http://dx.doi.org/10.1007/s00037-009-0273-5.
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Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005. URL: http://dx.doi.org/10.1016/j.tcs.2005.09.023.
http://dx.doi.org/10.1016/j.tcs.2005.09.023
Ryan Williams and Huacheng Yu. Finding orthogonal vectors in discrete structures. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1867-1877. SIAM, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.135.
http://dx.doi.org/10.1137/1.9781611973402.135
Virginia Vassilevska Williams, Joshua R. Wang, Richard Ryan Williams, and Huacheng Yu. Finding four-node subgraphs in triangle time. In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1671-1680. SIAM, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.111.
http://dx.doi.org/10.1137/1.9781611973730.111
Alan R. Woods. Unsatisfiable systems of equations, over a finite field. In 39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA, pages 202-211. IEEE Computer Society, 1998. URL: http://dx.doi.org/10.1109/SFCS.1998.743444.
http://dx.doi.org/10.1109/SFCS.1998.743444
Creative Commons Attribution 3.0 Unported license
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On Sampling Edges Almost Uniformly
We consider the problem of sampling an edge almost uniformly from an unknown graph, G = (V, E). Access to the graph is provided via queries of the following types: (1) uniform vertex queries, (2) degree queries, and (3) neighbor queries. We describe a new simple algorithm that returns a random edge e in E using \tilde{O}(n/sqrt{eps m}) queries in expectation, such that each edge e is sampled with probability (1 +/- eps)/m. Here, n = |V| is the number of vertices, and m = |E| is the number of edges. Our algorithm is optimal in the sense that any algorithm that samples an edge from an almost-uniform distribution must perform Omega(n/sqrt{m}) queries.
Sublinear Algorithms
Graph Algorithms
Sampling Edges
Query Complexity
7:1-7:9
Regular Paper
Talya
Eden
Talya Eden
Will
Rosenbaum
Will Rosenbaum
10.4230/OASIcs.SOSA.2018.7
Talya Eden, Amit Levi, Dana Ron, and C. Seshadhri. Approximately counting triangles in sublinear time. SIAM J. Comput., 46(5):1603-1646, 2017. URL: http://dx.doi.org/10.1137/15M1054389.
http://dx.doi.org/10.1137/15M1054389
Talya Eden, Dana Ron, and C. Seshadhri. Sublinear time estimation of degree distribution moments: The degeneracy connection. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 7:1-7:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.7.
http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.7
Talya Eden and Will Rosenbaum. On sampling edges almost uniformly. CoRR, abs/1706.09748, 2017. URL: http://arxiv.org/abs/1706.09748.
http://arxiv.org/abs/1706.09748
Uriel Feige. On sums of independent random variables with unbounded variance and estimating the average degree in a graph. SIAM J. Comput., 35(4):964-984, 2006. URL: http://dx.doi.org/10.1137/S0097539704447304.
http://dx.doi.org/10.1137/S0097539704447304
Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. In Frank Thomson Leighton and Peter W. Shor, editors, Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 406-415. ACM, 1997. URL: http://dx.doi.org/10.1145/258533.258627.
http://dx.doi.org/10.1145/258533.258627
Oded Goldreich and Dana Ron. Approximating average parameters of graphs. Random Struct. Algorithms, 32(4):473-493, 2008. URL: http://dx.doi.org/10.1002/rsa.20203.
http://dx.doi.org/10.1002/rsa.20203
Tali Kaufman, Michael Krivelevich, and Dana Ron. Tight bounds for testing bipartiteness in general graphs. SIAM J. Comput., 33(6):1441-1483, 2004. URL: http://dx.doi.org/10.1137/S0097539703436424.
http://dx.doi.org/10.1137/S0097539703436424
Michal Parnas and Dana Ron. Testing the diameter of graphs. Random Struct. Algorithms, 20(2):165-183, 2002. URL: http://dx.doi.org/10.1002/rsa.10013.
http://dx.doi.org/10.1002/rsa.10013
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A Simple PTAS for the Dual Bin Packing Problem and Advice Complexity of Its Online Version
Recently, Renault (2016) studied the dual bin packing problem in the per-request advice model of online algorithms. He showed that given O(1/eps) advice bits for each input item allows approximating the dual bin packing problem online to within a factor of 1+\eps. Renault asked about the advice complexity of dual bin packing in the tape-advice model of online algorithms. We make progress on this question. Let s be the maximum bit size of an input item weight. We present a conceptually simple online algorithm that with total advice O((s + log n)/eps^2) approximates the dual bin packing to within a 1+eps factor. To this end, we describe and analyze a simple offline PTAS for the dual bin packing problem. Although a PTAS for a more general problem was known prior to our work (Kellerer 1999, Chekuri and Khanna 2006), our PTAS is arguably simpler to state and analyze. As a result, we could easily adapt our PTAS to obtain the advice-complexity result.
We also consider whether the dependence on s is necessary in our algorithm. We show that if s is unrestricted then for small enough eps > 0 obtaining a 1+eps approximation to the dual bin packing requires Omega_eps(n) bits of advice. To establish this lower bound we analyze an online reduction that preserves the advice complexity and approximation ratio from the binary separation problem due to Boyar et al. (2016). We define two natural advice complexity classes that capture the distinction similar to the Turing machine world distinction between pseudo polynomial time algorithms and polynomial time algorithms. Our results on the dual bin packing problem imply the separation of the two classes in the advice complexity world.
dual bin packing
PTAS
tape-advice complexity
8:1-8:12
Regular Paper
Allan
Borodin
Allan Borodin
Denis
Pankratov
Denis Pankratov
Amirali
Salehi-Abari
Amirali Salehi-Abari
10.4230/OASIcs.SOSA.2018.8
Susan F. Assmann, David S. Johnson, Daniel J. Kleitman, and Joseph Y.-T. Leung. On a dual version of the one-dimensional bin packing problem. J. Algorithms, 5(4):502-525, 1984. URL: http://dx.doi.org/10.1016/0196-6774(84)90004-X.
http://dx.doi.org/10.1016/0196-6774(84)90004-X
Laszlo Babai, Peter Frankl, and Janos Simon. Complexity classes in communication complexity theory. In Proc. of the 27th Symp. on Found. of Comput. Sci., SFCS '86, pages 337-347, 1986.
Hans-Joachim Böckenhauer, Dennis Komm, Rastislav Královič, Richard Královič, and Tobias Mömke. On the advice complexity of online problems. Algorithms and Computation, pages 331-340, 2009.
Allan Borodin, Denis Pankratov, and Amirali Salehi-Abari. On conceptually simple algorithms for variants of online bipartite matching. In WAOA'17: The 15th workshop on approximation and online algorithms (To appear), 2017.
Joan Boyar, Lene M Favrholdt, Christian Kudahl, Kim S Larsen, and Jesper W Mikkelsen. Online algorithms with advice: a survey. ACM SIGACT News, 47(3):93-129, 2016.
Joan Boyar, Lene M. Favrholdt, Christian Kudahl, and Jesper W. Mikkelsen. The advice complexity of a class of hard online problems. Theory of Comput. Sys., 2016.
Joan Boyar, Shahin Kamali, Kim S. Larsen, and Alejandro López-Ortiz. Online bin packing with advice. Algorithmica, 74(1):507-527, Jan 2016.
Joan Boyar, Kim S. Larsen, and Morten N. Nielsen. The accommodating function: A generalization of the competitive ratio. SIAM J. on Comput., 31(1):233-258, 2001.
Chandra Chekuri and Sanjeev Khanna. A polynomial time approximation scheme for the multiple knapsack problem. SIAM J. on Comput., 35(3):713-728, 2005.
János Csirik and V. Totik. Online algorithms for a dual version of bin packing. Discrete Applied Mathematics, 21(2):163-167, 1988. URL: http://dx.doi.org/10.1016/0166-218X(88)90052-2.
http://dx.doi.org/10.1016/0166-218X(88)90052-2
Marek Cygan, Lukasz Jez, and Jirí Sgall. Online knapsack revisited. Theory Comput. Sys., 58(1):153-190, 2016.
Christoph Dürr, Christian Konrad, and Marc Renault. On the Power of Advice and Randomization for Online Bipartite Matching. In Proc. of ESA, pages 37:1-37:16, 2016.
Yuval Emek, Pierre Fraigniaud, Amos Korman, and Adi Rosén. Online computation with advice. Theoretical Computer Science, 412(24):2642-2656, 2011.
Edward G. Coffman Jr., Joseph Y.-T. Leung, and D. W. Ting. Bin packing: Maximizing the number of pieces packed. Acta Inf., 9:263-271, 1978.
Hans Kellerer. A polynomial time approximation scheme for the multiple knapsack problem. In Proc. of RANDOM-APPROX, volume 1671, pages 51-62. Springer, 1999.
Marc P Renault. Online algorithms with advice for the dual bin packing problem. Central Eur. J. of Op. Res., pages 1-14, 2016.
Vijay V. Vazirani. Approximation algorithms. Springer, 2001.
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Simple and Efficient Leader Election
We provide a simple and efficient population protocol for leader election that uses O(log n) states and elects exactly one leader in O(n (log n)^2) interactions with high probability and in expectation. Our analysis is simple and based on fundamental stochastic arguments. Our protocol combines the tournament based leader elimination by Alistarh and Gelashvili, ICALP'15, with the synthetic coin introduced by Alistarh et al., SODA'17.
population protocols
leader election
distributed
randomized
9:1-9:11
Regular Paper
Petra
Berenbrink
Petra Berenbrink
Dominik
Kaaser
Dominik Kaaser
Peter
Kling
Peter Kling
Lena
Otterbach
Lena Otterbach
10.4230/OASIcs.SOSA.2018.9
Dan Alistarh, James Aspnes, David Eisenstat, Rati Gelashvili, and Ronald L. Rivest. Time-Space Trade-offs in Population Protocols. In Proc. SODA, pages 2560-2579, 2017.
Dan Alistarh, James Aspnes, and Rati Gelashvili. Space-optimal majority in population protocols. CoRR, abs/1704.04947, 2017.
Dan Alistarh and Rati Gelashvili. Polylogarithmic-Time Leader Election in Population Protocols. In Proc. ICALP, pages 479-491, 2015.
Dana Angluin, James Aspnes, Zoë Diamadi, Michael J. Fischer, and René Peralta. Computation in networks of passively mobile finite-state sensors. Distributed Computing, 18(4):235-253, 2006.
Dana Angluin, James Aspnes, and David Eisenstat. Stably Computable Predicates Are Semilinear. In Proc. PODC, pages 292-299, New York, NY, USA, 2006.
Dana Angluin, James Aspnes, David Eisenstat, and Eric Ruppert. The computational power of population protocols. Distributed Computing, 20(4):279-304, 2007.
Andreas Bilke, Colin Cooper, Robert Elsässer, and Tomasz Radzik. Population protocols for leader election and exact majority with O(log² n) states and O(log² n) convergence time. CoRR, abs/1705.01146, 2017.
David Doty and David Soloveichik. Stable leader election in population protocols requires linear time. CoRR, abs/1502.04246, 2015.
Leszek Gasieniec and Grzegorz Stachowiak. Fast Space Optimal Leader Election in Population Protocols. CoRR, abs/1704.07649, 2017.
Richard Karp, Christian Schindelhauer, Scott Shenker, and Berthold Vöcking. Randomized Rumor Spreading. In Proc. FOCS, pages 565-574, 2000.
Michael Mitzenmacher and Eli Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, 2005.
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A Simple Algorithm for Approximating the Text-To-Pattern Hamming Distance
The algorithmic task of computing the Hamming distance between a given pattern of length m and each location in a text of length n, both over a general alphabet \Sigma, is one of the most fundamental algorithmic tasks in string algorithms. The fastest known runtime for exact computation is \tilde O(n\sqrt m). We recently introduced a complicated randomized algorithm for obtaining a (1 +/- eps) approximation for each location in the text in O( (n/eps) log(1/eps) log n log m log |\Sigma|) total time, breaking a barrier that stood for 22 years. In this paper, we introduce an elementary and simple randomized algorithm that takes O((n/eps) log n log m) time.
Pattern Matching
Hamming Distance
Approximation Algorithms
10:1-10:5
Regular Paper
Tsvi
Kopelowitz
Tsvi Kopelowitz
Ely
Porat
Ely Porat
10.4230/OASIcs.SOSA.2018.10
K. Abrahamson. Generalized string matching. In SIAM J. Computing 16 (6), page 1039–1051, 1987.
A. Amir, O. Lipsky, E. Porat, and J. Umanski. Approximate matching in the l1 metric. In CPM, pages 91-103, 2005.
Amihood Amir, Yonatan Aumann, Gary Benson, Avivit Levy, Ohad Lipsky, Ely Porat, Steven Skiena, and Uzi Vishne. Pattern matching with address errors: rearrangement distances. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, pages 1221-1229, 2006.
Amihood Amir, Yonatan Aumann, Piotr Indyk, Avivit Levy, and Ely Porat. Efficient computations of l_1 and l_infinity rearrangement distances. In String Processing and Information Retrieval, 14th International Symposium, SPIRE 2007, Santiago, Chile, October 29-31, 2007, Proceedings, pages 39-49, 2007.
Amihood Amir, Yonatan Aumann, Oren Kapah, Avivit Levy, and Ely Porat. Approximate string matching with address bit errors. In Combinatorial Pattern Matching, 19th Annual Symposium, CPM 2008, Pisa, Italy, June 18-20, 2008, Proceedings, pages 118-129, 2008.
Amihood Amir, Estrella Eisenberg, and Ely Porat. Swap and mismatch edit distance. In Algorithms - ESA 2004, 12th Annual European Symposium, Bergen, Norway, September 14-17, 2004, Proceedings, pages 16-27, 2004.
Amihood Amir, Tzvika Hartman, Oren Kapah, Avivit Levy, and Ely Porat. On the cost of interchange rearrangement in strings. In Algorithms - ESA 2007, 15th Annual European Symposium, Eilat, Israel, October 8-10, 2007, Proceedings, pages 99-110, 2007.
Amihood Amir, Moshe Lewenstein, and Ely Porat. Approximate swapped matching. In Foundations of Software Technology and Theoretical Computer Science, 20th Conference, FST TCS 2000 New Delhi, India, December 13-15, 2000, Proceedings., pages 302-311, 2000.
Alexandr Andoni, Robert Krauthgamer, and Krzysztof Onak. Polylogarithmic approximation for edit distance and the asymmetric query complexity. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 377-386, 2010.
A. Backurs and P. Indyk. Edit distance cannot be computed in strongly subquadratic time (unless SETH is false). In Accepted to 56th IEEE Symposium on Foundations of Computer Science (FOCS), 2015.
Ziv Bar-Yossef, T. S. Jayram, Robert Krauthgamer, and Ravi Kumar. Approximating edit distance efficiently. In 45th Symposium on Foundations of Computer Science (FOCS 2004), 17-19 October 2004, Rome, Italy, Proceedings, pages 550-559, 2004.
Ayelet Butman, Noa Lewenstein, Benny Porat, and Ely Porat. Jump-matching with errors. In String Processing and Information Retrieval, 14th International Symposium, SPIRE 2007, Santiago, Chile, October 29-31, 2007, Proceedings, pages 98-106, 2007.
Amit Chakrabarti and Oded Regev. An optimal lower bound on the communication complexity of gap-hamming-distance. SIAM J. Comput., 41(5):1299-1317, 2012.
Raphael Clifford. Matrix multiplication and pattern matching under hamming norm. http://www.cs.bris.ac.uk/Research/Algorithms/events/BAD09/BAD09/Talks/BAD09-Hammingnotes.pdf. Retrieved August 2015.
http://www.cs.bris.ac.uk/Research/Algorithms/events/BAD09/BAD09/Talks/BAD09-Hammingnotes.pdf
Raphaël Clifford, Klim Efremenko, Benny Porat, Ely Porat, and Amir Rothschild. Mismatch sampling. Information and Computation, 214:112-118, 2012.
Raphaël Clifford, Klim Efremenko, Ely Porat, and Amir Rothschild. From coding theory to efficient pattern matching. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009, pages 778-784, 2009. URL: http://dl.acm.org/citation.cfm?id=1496770.1496855.
http://dl.acm.org/citation.cfm?id=1496770.1496855
Raphaël Clifford, Klim Efremenko, Ely Porat, and Amir Rothschild. Pattern matching with don't cares and few errors. Journal of Computer System Science, 76(2):115-124, 2010.
Raphaël Clifford and Ely Porat. A filtering algorithm for k-mismatch with don't cares. Inf. Process. Lett., 110(22):1021-1025, 2010. URL: http://dx.doi.org/10.1016/j.ipl.2010.08.012.
http://dx.doi.org/10.1016/j.ipl.2010.08.012
Graham Cormode and S. Muthukrishnan. The string edit distance matching problem with moves. In Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 6-8, 2002, San Francisco, CA, USA., pages 667-676, 2002.
M.J. Fischer and M.S. Paterson. String matching and other products. r.m. karp (ed.), complexity of computation. In SIAM–AMS Proceedings, vol. 7,, page 113–125, 1974.
T. S. Jayram, Ravi Kumar, and D. Sivakumar. The one-way communication complexity of hamming distance. Theory of Computing, 4(1):129-135, 2008.
H. Karloff. Fast algorithms for approximately counting mismatches. In Inf. Process. Lett. 48 (2), pages 53-60, 1993.
Tsvi Kopelowitz and Ely Porat. Breaking the variance: Approximating the hamming distance in 1/ε time per alignment. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS, pages 601-613, 2015.
Vladimir Levenshtein. Binary codes capable of correcting spurious insertions and deletions of ones. In Probl. Inf. Transmission 1, page 8–17, 1965.
O. Lipsky and E. Porat. Approximated pattern matching with the l1, l2 and linfinit metrics. In SPIRE, pages 212-223, 2008.
R. Lowrance and R. A. Wagner. An extension of the string-to-string correction problem. J. of the ACM, pages 177-183, 1975.
Benny Porat and Ely Porat. Exact and approximate pattern matching in the streaming model. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA, pages 315-323, 2009.
Benny Porat, Ely Porat, and Asaf Zur. Pattern matching with pair correlation distance. In String Processing and Information Retrieval, 15th International Symposium, SPIRE 2008, Melbourne, Australia, November 10-12, 2008. Proceedings, pages 249-256, 2008.
Ely Porat and Klim Efremenko. Approximating general metric distances between a pattern and a text. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20-22, 2008, pages 419-427, 2008.
Ely Porat and Ohad Lipsky. Improved sketching of hamming distance with error correcting. In Combinatorial Pattern Matching, 18th Annual Symposium, CPM 2007, London, Canada, July 9-11, 2007, Proceedings, pages 173-182, 2007.
Ariel Shiftan and Ely Porat. Set intersection and sequence matching. In String Processing and Information Retrieval, 16th International Symposium, SPIRE 2009, Saariselkä, Finland, August 25-27, 2009, Proceedings, pages 285-294, 2009.
David P. Woodruff. Optimal space lower bounds for all frequency moments. In Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, New Orleans, Louisiana, USA, January 11-14, 2004, pages 167-175, 2004.
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Compact LP Relaxations for Allocation Problems
We consider the restricted versions of Scheduling on Unrelated Machines and the Santa Claus problem. In these problems we are given a set of jobs and a set of machines. Every job j has a size p_j and a set of allowed machines \Gamma(j), i.e., it can only be assigned to those machines. In the first problem, the objective is to minimize the maximum load among all machines; in the latter problem it is to maximize the minimum load. For these problems, the strongest LP relaxation known is the configuration LP. The configuration LP has an exponential number of variables and it cannot be solved exactly unless P = NP.
Our main result is a new LP relaxation for these problems. This LP has only O(n^3) variables and constraints. It is a further relaxation of the configuration LP, but it obeys the best bounds known for its integrality gap (11/6 and 4).
For the configuration LP these bounds were obtained using two local search algorithm. These algorithms, however, differ significantly in presentation. In this paper, we give a meta algorithm based on the local search ideas. With an instantiation for each objective function, we prove the bounds for the new compact LP relaxation (in particular, for the configuration LP). This way, we bring out many analogies between the two proofs, which were not apparent before.
Linear programming
unrelated machines
makespan
max-min
restricted assignment
santa claus
11:1-11:19
Regular Paper
Klaus
Jansen
Klaus Jansen
Lars
Rohwedder
Lars Rohwedder
10.4230/OASIcs.SOSA.2018.11
Chidambaram Annamalai. Lazy local search meets machine scheduling. CoRR, abs/1611.07371, 2016. URL: http://arxiv.org/abs/1611.07371.
http://arxiv.org/abs/1611.07371
Chidambaram Annamalai, Christos Kalaitzis, and Ola Svensson. Combinatorial algorithm for restricted max-min fair allocation. In Piotr Indyk, editor, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1357-1372. SIAM, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.90.
http://dx.doi.org/10.1137/1.9781611973730.90
Arash Asadpour, Uriel Feige, and Amin Saberi. Santa claus meets hypergraph matchings. ACM Trans. Algorithms, 8(3):24:1-24:9, 2012. URL: http://dx.doi.org/10.1145/2229163.2229168.
http://dx.doi.org/10.1145/2229163.2229168
Nikhil Bansal and Maxim Sviridenko. The santa claus problem. In Jon M. Kleinberg, editor, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 31-40. ACM, 2006. URL: http://dx.doi.org/10.1145/1132516.1132522.
http://dx.doi.org/10.1145/1132516.1132522
Ivona Bezáková and Varsha Dani. Allocating indivisible goods. SIGecom Exchanges, 5(3):11-18, 2005. URL: http://dx.doi.org/10.1145/1120680.1120683.
http://dx.doi.org/10.1145/1120680.1120683
Tomás Ebenlendr, Marek Krcál, and Jirí Sgall. Graph balancing: A special case of scheduling unrelated parallel machines. Algorithmica, 68(1):62-80, 2014. URL: http://dx.doi.org/10.1007/s00453-012-9668-9.
http://dx.doi.org/10.1007/s00453-012-9668-9
Klaus Jansen and Lars Rohwedder. On the configuration-lp of the restricted assignment problem. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2670-2678. SIAM, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.176.
http://dx.doi.org/10.1137/1.9781611974782.176
Klaus Jansen and Lars Rohwedder. A quasi-polynomial approximation for the restricted assignment problem. In Friedrich Eisenbrand and Jochen Könemann, editors, Integer Programming and Combinatorial Optimization - 19th International Conference, IPCO 2017, Waterloo, ON, Canada, June 26-28, 2017, Proceedings, volume 10328 of Lecture Notes in Computer Science, pages 305-316. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-59250-3_25.
http://dx.doi.org/10.1007/978-3-319-59250-3_25
Jan Karel Lenstra, David B. Shmoys, and Éva Tardos. Approximation algorithms for scheduling unrelated parallel machines. Math. Program., 46:259-271, 1990. URL: http://dx.doi.org/10.1007/BF01585745.
http://dx.doi.org/10.1007/BF01585745
Lukás Polácek and Ola Svensson. Quasi-polynomial local search for restricted max-min fair allocation. ACM Trans. Algorithms, 12(2):13:1-13:13, 2016. URL: http://dx.doi.org/10.1145/2818695.
http://dx.doi.org/10.1145/2818695
Ola Svensson. Santa claus schedules jobs on unrelated machines. SIAM J. Comput., 41(5):1318-1341, 2012. URL: http://dx.doi.org/10.1137/110851201.
http://dx.doi.org/10.1137/110851201
José Verschae and Andreas Wiese. On the configuration-lp for scheduling on unrelated machines. J. Scheduling, 17(4):371-383, 2014. URL: http://dx.doi.org/10.1007/s10951-013-0359-4.
http://dx.doi.org/10.1007/s10951-013-0359-4
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Just Take the Average! An Embarrassingly Simple 2^n-Time Algorithm for SVP (and CVP)
We show a 2^{n+o(n)}-time (and space) algorithm for the Shortest Vector Problem on lattices (SVP) that works by repeatedly running an embarrassingly simple "pair and average" sieving-like procedure on a list of lattice vectors. This matches the running time (and space) of the current fastest known algorithm, due to Aggarwal, Dadush, Regev, and Stephens-Davidowitz (ADRS, in STOC, 2015), with a far simpler algorithm. Our algorithm is in fact a modification of the ADRS algorithm, with a certain careful rejection sampling step removed.
The correctness of our algorithm follows from a more general "meta-theorem," showing that such rejection sampling steps are unnecessary for a certain class of algorithms and use cases. In particular, this also applies to the related 2^{n + o(n)}-time algorithm for the Closest Vector Problem (CVP), due to Aggarwal, Dadush, and Stephens-Davidowitz (ADS, in FOCS, 2015), yielding a similar embarrassingly simple algorithm for gamma-approximate CVP for any gamma = 1+2^{-o(n/log n)}. (We can also remove the rejection sampling procedure from the 2^{n+o(n)}-time ADS algorithm for exact CVP, but the resulting algorithm is still quite complicated.)
Lattices
SVP
CVP
12:1-12:19
Regular Paper
Divesh
Aggarwal
Divesh Aggarwal
Noah
Stephens-Davidowitz
Noah Stephens-Davidowitz
10.4230/OASIcs.SOSA.2018.12
Divesh Aggarwal, Daniel Dadush, Oded Regev, and Noah Stephens-Davidowitz. Solving the Shortest Vector Problem in 2ⁿ time via discrete Gaussian sampling. In STOC, 2015.
Divesh Aggarwal, Daniel Dadush, and Noah Stephens-Davidowitz. Solving the Closest Vector Problem in 2ⁿ time - The discrete Gaussian strikes again! In FOCS, 2015.
Miklós Ajtai. The shortest vector problem in L_2 is NP-hard for randomized reductions (extended abstract). In Jeffrey Scott Vitter, editor, Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 10-19. ACM, 1998. URL: http://dx.doi.org/10.1145/276698.276705.
http://dx.doi.org/10.1145/276698.276705
Miklós Ajtai. Generating hard instances of lattice problems. In Complexity of computations and proofs, volume 13 of Quad. Mat., pages 1-32. Dept. Math., Seconda Univ. Napoli, Caserta, 2004. Preliminary version in STOC'96.
Miklós Ajtai, Ravi Kumar, and D. Sivakumar. A sieve algorithm for the shortest lattice vector problem. In Jeffrey Scott Vitter, Paul G. Spirakis, and Mihalis Yannakakis, editors, Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, pages 601-610. ACM, 2001. URL: http://dx.doi.org/10.1145/380752.380857.
http://dx.doi.org/10.1145/380752.380857
Miklós Ajtai, Ravi Kumar, and D. Sivakumar. Sampling short lattice vectors and the closest lattice vector problem. In CCC, pages 41-45, 2002.
Erdem Alkim, Léo Ducas, Thomas Pöppelmann, and Peter Schwabe. Post-quantum key exchange - A new hope. In USENIX Security Symposium, 2016.
László Babai. On lovász' lattice reduction and the nearest lattice point problem. Combinatorica, 6(1):1-13, 1986. URL: http://dx.doi.org/10.1007/BF02579403.
http://dx.doi.org/10.1007/BF02579403
Wojciech Banaszczyk. New bounds in some transference theorems in the geometry of numbers. Mathematische Annalen, 296(4):625-635, 1993. URL: http://dx.doi.org/10.1007/BF01445125.
http://dx.doi.org/10.1007/BF01445125
Anja Becker, Léo Ducas, Nicolas Gama, and Thijs Laarhoven. New directions in nearest neighbor searching with applications to lattice sieving. In SODA, 2016.
Joppe W. Bos, Craig Costello, Léo Ducas, Ilya Mironov, Michael Naehrig, Valeria Nikolaenko, Ananth Raghunathan, and Douglas Stebila. Frodo: Take off the ring! Practical, quantum-secure key exchange from LWE. In CCS, 2016.
Zvika Brakerski, Adeline Langlois, Chris Peikert, Oded Regev, and Damien Stehlé. Classical hardness of learning with errors. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 575-584. ACM, 2013. URL: http://dx.doi.org/10.1145/2488608.2488680.
http://dx.doi.org/10.1145/2488608.2488680
R. de Buda. Some optimal codes have structure. Selected Areas in Communications, IEEE Journal on, 7(6):893-899, Aug 1989.
Irit Dinur, Guy Kindler, Ran Raz, and Shmuel Safra. Approximating CVP to within almost-polynomial factors is NP-hard. Combinatorica, 23(2):205-243, 2003.
Nicolas Gama and Phong Q. Nguyen. Finding short lattice vectors within Mordell’s inequality. In STOC, 2008.
Craig Gentry, Chris Peikert, and Vinod Vaikuntanathan. Trapdoors for hard lattices and new cryptographic constructions. In STOC, pages 197-206, 2008.
Oded Goldreich, Daniele Micciancio, Shmuel Safra, and Jean-Pierre Seifert. Approximating shortest lattice vectors is not harder than approximating closest lattice vectors. Inf. Process. Lett., 71(2):55-61, 1999. URL: http://dx.doi.org/10.1016/S0020-0190(99)00083-6.
http://dx.doi.org/10.1016/S0020-0190(99)00083-6
Ishay Haviv and Oded Regev. Tensor-based hardness of the Shortest Vector Problem to within almost polynomial factors. Theory of Computing, 8(23):513-531, 2012. Preliminary version in STOC'07.
Hendrik W. Lenstra Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538-548, 1983. URL: http://dx.doi.org/10.1287/moor.8.4.538.
http://dx.doi.org/10.1287/moor.8.4.538
Ravi Kannan. Minkowski’s convex body theorem and integer programming. Math. Oper. Res., 12(3):415-440, 1987. URL: http://dx.doi.org/10.1287/moor.12.3.415.
http://dx.doi.org/10.1287/moor.12.3.415
Subhash Khot. Hardness of approximating the Shortest Vector Problem in lattices. Journal of the ACM, 52(5):789-808, 2005. Preliminary version in FOCS'04.
Philip Klein. Finding the closest lattice vector when it’s unusually close. In SODA, 2000.
A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Math. Ann., 261(4):515-534, 1982. URL: http://dx.doi.org/10.1007/BF01457454.
http://dx.doi.org/10.1007/BF01457454
Daniele Micciancio. The Shortest Vector Problem is NP-hard to approximate to within some constant. SIAM Journal on Computing, 30(6):2008-2035, 2001. Preliminary version in FOCS 1998.
Daniele Micciancio and Panagiotis Voulgaris. Faster exponential time algorithms for the Shortest Vector Problem. In SODA, 2010.
Daniele Micciancio and Panagiotis Voulgaris. A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. SIAM Journal on Computing, 42(3):1364-1391, 2013.
Daniele Micciancio and Michael Walter. Practical, predictable lattice basis reduction. In Eurocrypt, 2016.
NIST post-quantum standardization call for proposals. http://csrc.nist.gov/groups/ST/post-quantum-crypto/cfp-announce-dec2016.html, 2016. Accessed: 2017-04-02.
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Chris Peikert. A decade of lattice cryptography. Foundations and Trends in Theoretical Computer Science, 10(4):283-424, 2016.
Xavier Pujol and Damien Stehlé. Solving the Shortest Lattice Vector Problem in time 2^2.465 n. IACR Cryptology ePrint Archive, 2009:605, 2009.
Oded Regev. On lattices, learning with errors, random linear codes, and cryptography. J. ACM, 56(6):34:1-34:40, 2009. URL: http://dx.doi.org/10.1145/1568318.1568324.
http://dx.doi.org/10.1145/1568318.1568324
Oded Regev and Noah Stephens-Davidowitz. An inequality for Gaussians on lattices. SIDMA, 2017.
Claus-Peter Schnorr. A hierarchy of polynomial time lattice basis reduction algorithms. Theor. Comput. Sci., 53:201-224, 1987. URL: http://dx.doi.org/10.1016/0304-3975(87)90064-8.
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Adi Shamir. A polynomial-time algorithm for breaking the basic merkle-hellman cryptosystem. IEEE Trans. Information Theory, 30(5):699-704, 1984. URL: http://dx.doi.org/10.1109/TIT.1984.1056964.
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Noah Stephens-Davidowitz. On the Gaussian Measure Over Lattices. PhD thesis, New York University, 2017.
Peter van Emde Boas. Another NP-complete problem and the complexity of computing short vectors in a lattice. Technical report, University of Amsterdam, Department of Mathematics, Netherlands, 1981. Technical Report 8104.
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Complex Semidefinite Programming and Max-k-Cut
In a second seminal paper on the application of semidefinite
programming to graph partitioning problems, Goemans and Williamson
showed in 2004 how to formulate and round a complex semidefinite program to give what is to date still the best-known approximation guarantee of .836008 for Max-3-Cut. (This approximation ratio was also achieved independently around the same time by De Klerk et
al..) Goemans and Williamson left open the problem of how to apply their techniques to Max-k-Cut for general k. They point out that it does not seem straightforward or even possible to formulate a good quality complex semidefinite program for the general Max-k-Cut problem, which presents a barrier for the further application of their techniques.
We present a simple rounding algorithm for the standard semidefinite
programmming relaxation of Max-k-Cut and show that it is equivalent to the rounding of Goemans and Williamson in the case of Max-3-Cut. This allows us to transfer the elegant analysis of Goemans and Williamson for Max-3-Cut to Max-k-Cut. For k > 3, the resulting approximation ratios are about .01 worse than the best known guarantees. Finally, we present a generalization of our rounding algorithm and conjecture (based on computational observations) that it matches the best-known guarantees of De Klerk et al.
Graph Partitioning
Max-k-Cut
Semidefinite Programming
13:1-13:11
Regular Paper
Alantha
Newman
Alantha Newman
10.4230/OASIcs.SOSA.2018.13
Gunnar Andersson, Lars Engebretsen, and Johan Håstad. A new way of using semidefinite programming with applications to linear equations mod p. J. Algorithms, 39(2):162-204, 2001. URL: http://dx.doi.org/10.1006/jagm.2000.1154.
http://dx.doi.org/10.1006/jagm.2000.1154
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http://dx.doi.org/10.1023/B:JOCO.0000038911.67280.3f
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http://dx.doi.org/10.1007/BF02523688
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http://dx.doi.org/10.1145/227683.227684
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http://dx.doi.org/10.1145/301250.301431
Creative Commons Attribution 3.0 Unported license
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A Simple, Space-Efficient, Streaming Algorithm for Matchings in Low Arboricity Graphs
We present a simple single-pass data stream algorithm using O((log n)/eps^2) space that returns an (alpha + 2)(1 + eps) approximation to the size of the maximum matching in a graph of arboricity alpha.
data streams
matching
planar graphs
arboricity
14:1-14:4
Regular Paper
Andrew
McGregor
Andrew McGregor
Sofya
Vorotnikova
Sofya Vorotnikova
10.4230/OASIcs.SOSA.2018.14
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Simple Analyses of the Sparse Johnson-Lindenstrauss Transform
For every n-point subset X of Euclidean space and target distortion 1+eps for 0<eps<1, the Sparse Johnson Lindenstrauss Transform (SJLT) of (Kane, Nelson, J. ACM 2014) provides a linear dimensionality-reducing map f:X-->l_2^m where f(x) = Ax for A a matrix with m rows where (1) m = O((log n)/eps^2), and (2) each column of A is sparse, having only O(eps m) non-zero entries. Though the constructions given for such A in (Kane, Nelson, J. ACM 2014) are simple, the analyses are not, employing intricate combinatorial arguments. We here give two simple alternative proofs of their main result, involving no delicate combinatorics. One of these proofs has already been tested pedagogically, requiring slightly under forty minutes by the third author at a casual pace to cover all details in a blackboard course lecture.
dimensionality reduction
Johnson-Lindenstrauss
Sparse Johnson-Lindenstrauss Transform
15:1-15:9
Regular Paper
Michael B.
Cohen
Michael B. Cohen
T.S.
Jayram
T.S. Jayram
Jelani
Nelson
Jelani Nelson
10.4230/OASIcs.SOSA.2018.15
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode