research-article

The matching polytope has exponential extension complexity

Author:

Thomas RothvossAuthors Info & Claims

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

Pages 263 - 272

https://doi.org/10.1145/2591796.2591834

Published: 31 May 2014 Publication History

Abstract

A popular method in combinatorial optimization is to express polytopes P, which may potentially have exponentially many facets, as solutions of linear programs that use few extra variables to reduce the number of constraints down to a polynomial. After two decades of standstill, recent years have brought amazing progress in showing lower bounds for the so called extension complexity, which for a polytope P denotes the smallest number of inequalities necessary to describe a higher dimensional polytope Q that can be linearly projected on P.

However, the central question in this field remained wide open: can the perfect matching polytope be written as an LP with polynomially many constraints?

We answer this question negatively. In fact, the extension complexity of the perfect matching polytope in a complete n-node graph is 2^Ω(n). By a known reduction this also improves the lower bound on the extension complexity for the TSP polytope from 2^Ω(√n) to 2^Ω(n).

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Aprile M(2022)Extended formulations for matroid polytopes through randomized protocolsOperations Research Letters10.1016/j.orl.2022.01.01150:2(145-149)Online publication date: Mar-2022
https://doi.org/10.1016/j.orl.2022.01.011
Fawzi H(2021)On Polyhedral Approximations of the Positive Semidefinite ConeMathematics of Operations Research10.1287/moor.2020.107746:4(1479-1489)Online publication date: 1-Nov-2021
https://dl.acm.org/doi/10.1287/moor.2020.1077
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Index Terms

The matching polytope has exponential extension complexity
1. Mathematics of computing
  1. Discrete mathematics
2. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

May 2014

984 pages

ISBN:9781450327107

DOI:10.1145/2591796

Program Chair:
David Shmoys
Cornell University

Copyright © 2014 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Published: 31 May 2014

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STOC '14: Symposium on Theory of Computing

May 31 - June 3, 2014

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STOC '14 Paper Acceptance Rate 91 of 319 submissions, 29%;

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Aprile M(2022)Extended formulations for matroid polytopes through randomized protocolsOperations Research Letters10.1016/j.orl.2022.01.01150:2(145-149)Online publication date: Mar-2022
https://doi.org/10.1016/j.orl.2022.01.011
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Wolsey L(2020)ReferencesInteger Programming10.1002/9781119606475.refs(291-309)Online publication date: 19-Sep-2020
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