Article

Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut

Authors:

Grant Schoenebeck,

Luca Trevisan,

Madhur TulsianiAuthors Info & Claims

STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing

Pages 302 - 310

https://doi.org/10.1145/1250790.1250836

Published: 11 June 2007 Publication History

Get Access

Abstract

We study linear programming relaxations of Vertex Cover and Max Cutarising from repeated applications of the "lift-and-project" method of Lovasz and Schrijver starting from the standard linear programming relaxation.

For Vertex Cover, Arora, Bollobas, Lovasz and Tourlakis prove thatthe integrality gap remains at least 2-ε after Ω_ε(log n) rounds, where n is the number ofvertices, and Tourlakis proves that integrality gap remains at least 1.5-ε after Ω((log n)²) rounds. Fernandez de laVega and Kenyon prove that the integrality gap of Max Cut is at most 12 + ε after any constant number of rounds. (Theirresult also applies to the more powerful Sherali-Adams method.

We prove that the integrality gap of Vertex Cover remains at least 2-ε after Ω_ε (n) rounds, and that theintegrality gap of Max Cut remains at most 1/2 +ε after Ω_ε(n) rounds.

References

[1]

S. Arora, B. Bollobas, and L. Lovasz. Proving integrality gaps without knowing the linear program. In Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, pages 313--322, 2002.

Digital Library

Google Scholar

[2]

S. Arora, B. Bollobás, L. Lovász, and I. Tourlakis. Proving integrality gaps without knowing the linear program. Theory of Computing, 2(2):19--51, 2006.

Crossref

Google Scholar

[3]

I. Dinur and S. Safra. On the hardness of approximating minimum vertex-cover. Annals of Mathematics, 162(1):439--486, 2005.

Crossref

Google Scholar

[4]

W. Fernandez de la Vega and C. Kenyon-Mathieu. Linear programming relaxations of Maxcut. In Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms, 2007. To appear.

Digital Library

Google Scholar

[5]

M. Goemans and D. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42(6):1115--1145, 1995. Preliminary version in Proc. of STOC'94.

Digital Library

Google Scholar

[6]

S. Khot and O. Regev. Vertex cover might be hard to approximate to within 2-ε. In Proceedings of the 18th IEEE Conference on Computational Complexity, 2003.

Crossref

Google Scholar

[7]

L. Lovasz and A. Schrijver. Cones of matrices and set-functions and 0-1 optimization. SIAM J. on Optimization, 1(12):166--190, 1991.

Crossref

Google Scholar

[8]

G. Schoenebeck, L. Trevisan, and M. Tulsiani. Tight integrality gaps for lovasz-schrijver lp relaxations of vertex cover and max cut. Technical Report TR06-132, Electronic Colloquium on Computational Complexity, 2006.

Google Scholar

[9]

H.D. Sherali and W.P. Adams. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics, 3:411--430, 1990.

Digital Library

Google Scholar

[10]

I. Tourlakis. New lower bounds for vertex cover in the Lovasz-Schrijver hierarchy. In Proceedings of the 21st IEEE Conference on Computational Complexity, 2006.

Digital Library

Google Scholar

Cited By

View all

Hopkins MLin T(2022)Explicit Lower Bounds Against Ω(n)-Rounds of Sum-of-Squares2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00069(662-673)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00069
Kothari PManohar P(2021)A stress-free sum-of-squares lower bound for coloringProceedings of the 36th Computational Complexity Conference10.4230/LIPIcs.CCC.2021.23Online publication date: 20-Jul-2021
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2021.23
Hopkins SSchramm TTrevisan L(2020)Subexponential LPs Approximate Max-Cut2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00092(943-953)Online publication date: Nov-2020
https://doi.org/10.1109/FOCS46700.2020.00092
Show More Cited By

Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut
1. Theory of computation

Recommendations

Integrality Gaps of $2-o(1)$ for Vertex Cover SDPs in the Lovász-Schrijver Hierarchy

Linear and semidefinite programming are highly successful approaches for obtaining good approximations for NP-hard optimization problems. For example, breakthrough approximation algorithms for Max Cut and Sparsest Cut use semidefinite programming. ...
Semidefinite Programming Relaxations of the Traveling Salesman Problem and Their Integrality Gaps
The traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization. Several semidefinite programming relaxations have been proposed recently that exploit a variety of mathematical structures including, for example, algebraic ...
Integrality gap of the vertex cover linear programming relaxation
Abstract
We give a characterization result for the integrality gap of the natural linear programming relaxation for the vertex cover problem. We show that integrality gap of the standard linear programming relaxation for any graph G equals 2 − 2 χ f ( G ) ...

Comments

Information & Contributors

Information

Published In

STOC '07: Proceedings of the thirty-ninth annual ACM symposium on Theory of computing

June 2007

734 pages

ISBN:9781595936318

DOI:10.1145/1250790

General Chair:
David Johnson
AT&T Labs - Research
,
Program Chair:
Uriel Feige
Microsoft Research and Weizmann Institute

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 ACM 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]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 11 June 2007

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Article

Conference

STOC07

Sponsor:

STOC07: Symposium on Theory of Computing

June 11 - 13, 2007

California, San Diego, USA

Acceptance Rates

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

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

36
Total Citations
View Citations
444
Total Downloads

Downloads (Last 12 months)7
Downloads (Last 6 weeks)1

Reflects downloads up to 10 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Hopkins MLin T(2022)Explicit Lower Bounds Against Ω(n)-Rounds of Sum-of-Squares2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00069(662-673)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00069
Kothari PManohar P(2021)A stress-free sum-of-squares lower bound for coloringProceedings of the 36th Computational Complexity Conference10.4230/LIPIcs.CCC.2021.23Online publication date: 20-Jul-2021
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2021.23
Hopkins SSchramm TTrevisan L(2020)Subexponential LPs Approximate Max-Cut2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00092(943-953)Online publication date: Nov-2020
https://doi.org/10.1109/FOCS46700.2020.00092
Georgiou KJiang ALee EOlave ASeong IUpadhyaya T(2020)Lift & project systems performing on the partial-vertex-cover polytopeTheoretical Computer Science10.1016/j.tcs.2020.03.004Online publication date: Mar-2020
https://doi.org/10.1016/j.tcs.2020.03.004
O'Donnell RSchramm TAceto L(2019)Sherali-adams strikes backProceedings of the 34th Computational Complexity Conference10.4230/LIPIcs.CCC.2019.8(1-30)Online publication date: 17-Jul-2019
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2019.8
Kurpisz AMastrolilli MMathieu CMömke TVerdugo VWiese A(2018)Semidefinite and linear programming integrality gaps for scheduling identical machinesMathematical Programming: Series A and B10.5555/3288898.3288949172:1-2(231-248)Online publication date: 1-Nov-2018
https://dl.acm.org/doi/10.5555/3288898.3288949
Razborov A(2017)On the Width of Semialgebraic Proofs and AlgorithmsMathematics of Operations Research10.1287/moor.2016.084042:4(1106-1134)Online publication date: 1-Nov-2017
https://dl.acm.org/doi/10.1287/moor.2016.0840
Kurpisz AMastrolilli MMathieu CMömke TVerdugo VWiese A(2017)Semidefinite and linear programming integrality gaps for scheduling identical machinesMathematical Programming10.1007/s10107-017-1152-5172:1-2(231-248)Online publication date: 10-May-2017
https://doi.org/10.1007/s10107-017-1152-5
Kolliopoulos SMoysoglou Y(2016)Integrality gaps for strengthened linear relaxations of capacitated facility locationMathematical Programming: Series A and B10.1007/s10107-015-0916-z158:1-2(99-141)Online publication date: 1-Jul-2016
https://dl.acm.org/doi/10.1007/s10107-015-0916-z
Kurpisz AMastrolilli MMathieu CMömke TVerdugo VWiese A(2016)Semidefinite and Linear Programming Integrality Gaps for Scheduling Identical MachinesProceedings of the 18th International Conference on Integer Programming and Combinatorial Optimization - Volume 968210.1007/978-3-319-33461-5_13(152-163)Online publication date: 1-Jun-2016
https://dl.acm.org/doi/10.1007/978-3-319-33461-5_13
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Cited By

Recommendations

Integrality Gaps of $2-o(1)$ for Vertex Cover SDPs in the Lovász-Schrijver Hierarchy

Semidefinite Programming Relaxations of the Traveling Salesman Problem and Their Integrality Gaps

Integrality gap of the vertex cover linear programming relaxation