research-article

Inapproximability of Combinatorial Problems via Small LPs and SDPs

Authors:

Sebastian Pokutta,

Daniel ZinkAuthors Info & Claims

STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

Pages 107 - 116

https://doi.org/10.1145/2746539.2746550

Published: 14 June 2015 Publication History

Abstract

Motivated by [12], we provide a framework for studying the size of linear programming formulations as well as semidefinite programming formulations of combinatorial optimization problems without encoding them first as linear programs. This is done via a factorization theorem for the optimization problem itself (and not a specific encoding of such). As a result we define a consistent reduction mechanism that degrades approximation factors in a controlled fashion and which, at the same time, is compatible with approximate linear and semidefinite programming formulations. Moreover, our reduction mechanism is a minor restriction of classical reductions establishing inapproximability in the context of PCP theorems. As a consequence we establish strong linear programming inapproximability (for LPs with a polynomial number of constraints) for several problems that are not 0/1-CSPs: we obtain a 3/2-epsilon inapproximability for Vertex Cover (which is not of the CSP type) answering an open question in [12], we answer a weak version of our sparse graph conjecture posed in [6] showing an inapproximability factor of 1/2+ε for bounded degree IndependentSet, and we establish inapproximability of MaxMULTICUT (a non-binary CSP). In the case of SDPs, we obtain relative inapproximability results for these problems.

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Cited By

Ciardo LŽivný SSaha BServedio R(2023)Approximate Graph Colouring and the Hollow ShadowProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585112(623-631)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585112
Aprile MDrescher MFiorini SHuynh T(2022)A tight approximation algorithm for the cluster vertex deletion problemMathematical Programming10.1007/s10107-021-01744-w197:2(1069-1091)Online publication date: 7-Jan-2022
https://doi.org/10.1007/s10107-021-01744-w
Braun GPokutta SZink D(2019)Affine reductions for LPs and SDPsMathematical Programming: Series A and B10.1007/s10107-017-1221-9173:1-2(281-312)Online publication date: 1-Jan-2019
https://dl.acm.org/doi/10.1007/s10107-017-1221-9
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Index Terms

Inapproximability of Combinatorial Problems via Small LPs and SDPs
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
  2. Mathematical analysis
    1. Mathematical optimization
      1. Continuous optimization
        Linear programming
2. Theory of computation
  1. Computational complexity and cryptography
    1. Problems, reductions and completeness
  2. Design and analysis of algorithms
    1. Mathematical optimization
      1. Continuous optimization
        Linear programming

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cover image ACM Conferences

STOC '15: Proceedings of the forty-seventh annual ACM symposium on Theory of Computing

June 2015

916 pages

ISBN:9781450335362

DOI:10.1145/2746539

General Chair:
Rocco Servedio
Columbia University
,
Program Chair:
Ronitt Rubinfeld
MIT and Tel Aviv University

Copyright © 2015 ACM.

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Published: 14 June 2015

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June 14 - 17, 2015

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STOC '15 Paper Acceptance Rate 93 of 347 submissions, 27%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

Ciardo LŽivný SSaha BServedio R(2023)Approximate Graph Colouring and the Hollow ShadowProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585112(623-631)Online publication date: 2-Jun-2023
https://dl.acm.org/doi/10.1145/3564246.3585112
Aprile MDrescher MFiorini SHuynh T(2022)A tight approximation algorithm for the cluster vertex deletion problemMathematical Programming10.1007/s10107-021-01744-w197:2(1069-1091)Online publication date: 7-Jan-2022
https://doi.org/10.1007/s10107-021-01744-w
Braun GPokutta SZink D(2019)Affine reductions for LPs and SDPsMathematical Programming: Series A and B10.1007/s10107-017-1221-9173:1-2(281-312)Online publication date: 1-Jan-2019
https://dl.acm.org/doi/10.1007/s10107-017-1221-9
Braun GPokutta SRoy A(2018)Strong reductions for extended formulationsMathematical Programming: Series A and B10.5555/3288898.3288958172:1-2(591-620)Online publication date: 1-Nov-2018
https://dl.acm.org/doi/10.5555/3288898.3288958
Braun GPokutta SRoy A(2018)Strong reductions for extended formulationsMathematical Programming10.1007/s10107-018-1316-y172:1-2(591-620)Online publication date: 25-Aug-2018
https://doi.org/10.1007/s10107-018-1316-y
Braun GBrown-Cohen JHuq APokutta SRaghavendra PRoy AWeitz BZink D(2017)The matching problem has no small symmetric SDPMathematical Programming: Series A and B10.1007/s10107-016-1098-z165:2(643-662)Online publication date: 1-Oct-2017
https://dl.acm.org/doi/10.1007/s10107-016-1098-z
Braun GJain RLee TPokutta S(2017)Information-theoretic approximations of the nonnegative rankComputational Complexity10.1007/s00037-016-0125-z26:1(147-197)Online publication date: 1-Mar-2017
https://dl.acm.org/doi/10.1007/s00037-016-0125-z
Chan SLee JRaghavendra PSteurer D(2016)Approximate Constraint Satisfaction Requires Large LP RelaxationsJournal of the ACM10.1145/281125563:4(1-22)Online publication date: 26-Oct-2016
https://dl.acm.org/doi/10.1145/2811255
Braun GFiorini SPokutta S(2016)Average case polyhedral complexity of the maximum stable set problemMathematical Programming: Series A and B10.1007/s10107-016-0989-3160:1-2(407-431)Online publication date: 1-Nov-2016
https://dl.acm.org/doi/10.1007/s10107-016-0989-3
Braun GPokutta SRoy A(2016)Strong Reductions for Extended FormulationsProceedings of the 18th International Conference on Integer Programming and Combinatorial Optimization - Volume 968210.1007/978-3-319-33461-5_29(350-361)Online publication date: 1-Jun-2016
https://dl.acm.org/doi/10.1007/978-3-319-33461-5_29
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