research-article

Matroid matching: the power of local search

Authors:

Maxim Sviridenko,

Jan VondrakAuthors Info & Claims

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

Pages 369 - 378

https://doi.org/10.1145/1806689.1806741

Published: 05 June 2010 Publication History

Abstract

We consider the classical matroid matching problem. Unweighted matroid matching for linear matroids was solved by Lovasz, and the problem is known to be intractable for general matroids. We present a PTAS for unweighted matroid matching for general matroids. In contrast, we show that natural LP relaxations have an Ω(n) integrality gap and moreover, Ω(n) rounds of the Sherali-Adams hierarchy are necessary to bring the gap down to a constant. More generally, for any fixed k>=2 and ε>0, we obtain a (k/2+ε)-approximation for matroid matching in k-uniform hypergraphs, also known as the matroid k-parity problem. As a consequence, we obtain a (k/2+ε)-approximation for the problem of finding the maximum-cardinality set in the intersection of k matroids. We have also designed a 3/2-approximation for the weighted version of a special case of matroid matching, the matchoid problem.

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Cited By

Adamczyk MSviridenko MWard J(2016)Submodular Stochastic Probing on MatroidsMathematics of Operations Research10.1287/moor.2015.076641:3(1022-1038)Online publication date: 1-Aug-2016
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Badanidiyuru AVondrák JChekuri C(2014)Fast algorithms for maximizing submodular functionsProceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms10.5555/2634074.2634184(1497-1514)Online publication date: 5-Jan-2014
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Soto J(2014)A simple PTAS for weighted matroid matching on strongly base orderable matroidsDiscrete Applied Mathematics10.1016/j.dam.2012.10.019164(406-412)Online publication date: 1-Feb-2014
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Index Terms

Matroid matching: the power of local search
1. Theory of computation
  1. Randomness, geometry and discrete structures

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cover image ACM Conferences

STOC '10: Proceedings of the forty-second ACM symposium on Theory of computing

June 2010

812 pages

ISBN:9781450300506

DOI:10.1145/1806689

General Chair:
Michael Mitzenmacher
Harvard
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Program Chair:
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Caltech

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Published: 05 June 2010

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Cited By

Adamczyk MSviridenko MWard J(2016)Submodular Stochastic Probing on MatroidsMathematics of Operations Research10.1287/moor.2015.076641:3(1022-1038)Online publication date: 1-Aug-2016
https://dl.acm.org/doi/10.1287/moor.2015.0766
Badanidiyuru AVondrák JChekuri C(2014)Fast algorithms for maximizing submodular functionsProceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms10.5555/2634074.2634184(1497-1514)Online publication date: 5-Jan-2014
https://dl.acm.org/doi/10.5555/2634074.2634184
Soto J(2014)A simple PTAS for weighted matroid matching on strongly base orderable matroidsDiscrete Applied Mathematics10.1016/j.dam.2012.10.019164(406-412)Online publication date: 1-Feb-2014
https://dl.acm.org/doi/10.1016/j.dam.2012.10.019
Filmus Y(2013)Inequalities on submodular functions via term rewritingInformation Processing Letters10.1016/j.ipl.2013.03.018113:13(457-464)Online publication date: 1-Jul-2013
https://dl.acm.org/doi/10.1016/j.ipl.2013.03.018
Kakimura NTakamatsu M(2012)Matching problems with delta-matroid constraintsProceedings of the Eighteenth Computing: The Australasian Theory Symposium - Volume 12810.5555/2523693.2523704(83-92)Online publication date: 31-Jan-2012
https://dl.acm.org/doi/10.5555/2523693.2523704
Cheung HLau LLeung KRandall D(2011)Algebraic algorithms for linear matroid parity problemsProceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms10.5555/2133036.2133141(1366-1382)Online publication date: 23-Jan-2011
https://dl.acm.org/doi/10.5555/2133036.2133141
Feldman MNaor JSchwartz RWard J(2011)Improved approximations for k-exchange systemsProceedings of the 19th European conference on Algorithms10.5555/2040572.2040658(784-798)Online publication date: 5-Sep-2011
https://dl.acm.org/doi/10.5555/2040572.2040658
Vondrák JChekuri CZenklusen RFortnow LVadhan S(2011)Submodular function maximization via the multilinear relaxation and contention resolution schemesProceedings of the forty-third annual ACM symposium on Theory of computing10.1145/1993636.1993740(783-792)Online publication date: 6-Jun-2011
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Soto J(2011)A simple PTAS for Weighted Matroid Matching on Strongly Base Orderable MatroidsElectronic Notes in Discrete Mathematics10.1016/j.endm.2011.05.01437(75-80)Online publication date: Aug-2011
https://doi.org/10.1016/j.endm.2011.05.014
Feldman MNaor JSchwartz RWard J(2011)Improved Approximations for k-Exchange SystemsAlgorithms – ESA 201110.1007/978-3-642-23719-5_66(784-798)Online publication date: 2011
https://doi.org/10.1007/978-3-642-23719-5_66

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