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Strengthened semidefinite relaxations via a second lifting for the Max-Cut problem

Strengthened semidefinite relaxations via a second lifting for the Max-Cut problem
Strengthened semidefinite relaxations via a second lifting for the Max-Cut problem
In this paper we study two strengthened semidefinite programming relaxations for the Max-Cut problem. Our results hold for every instance of Max-Cut; in particular, we make no assumptions about the edge weights. We prove that the first relaxation provides a strengthening of the Goemans-Williamson relaxation. The second relaxation is a further tightening of the first one and we prove that its feasible set corresponds to a convex set that is larger than the cut polytope but nonetheless is strictly contained in the intersection of the elliptope and the metric polytope. Both relaxations are obtained using Lagrangian relaxation. Hence our results also exemplify the strength and flexibility of Lagrangian relaxation for obtaining a variety of SDP relaxations with different properties. We also address some practical issues in the solution of these SDP relaxations. Because Slater's constraint qualification fails for both of them, we project their feasible sets onto a lower dimensional space in a way that does not affect the sparsity of these relaxations but guarantees Slater's condition. Some preliminary numerical results are included.
Max-Cut problem, semidefinite programming relaxations, Lagrangian relaxation, Cut polytope metric polytope
0166-218X
79-106
Anjos, Miguel F.
0103b24d-cab5-4109-9924-dc9eebd04257
Wolkowicz, Henry
3ef6bc24-0956-4eb2-a542-802d82454cf7
Anjos, Miguel F.
0103b24d-cab5-4109-9924-dc9eebd04257
Wolkowicz, Henry
3ef6bc24-0956-4eb2-a542-802d82454cf7

Anjos, Miguel F. and Wolkowicz, Henry (2002) Strengthened semidefinite relaxations via a second lifting for the Max-Cut problem. Discrete Applied Mathematics, 119 (1-2), 79-106. (doi:10.1016/S0166-218X(01)00266-9).

Record type: Article

Abstract

In this paper we study two strengthened semidefinite programming relaxations for the Max-Cut problem. Our results hold for every instance of Max-Cut; in particular, we make no assumptions about the edge weights. We prove that the first relaxation provides a strengthening of the Goemans-Williamson relaxation. The second relaxation is a further tightening of the first one and we prove that its feasible set corresponds to a convex set that is larger than the cut polytope but nonetheless is strictly contained in the intersection of the elliptope and the metric polytope. Both relaxations are obtained using Lagrangian relaxation. Hence our results also exemplify the strength and flexibility of Lagrangian relaxation for obtaining a variety of SDP relaxations with different properties. We also address some practical issues in the solution of these SDP relaxations. Because Slater's constraint qualification fails for both of them, we project their feasible sets onto a lower dimensional space in a way that does not affect the sparsity of these relaxations but guarantees Slater's condition. Some preliminary numerical results are included.

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More information

Published date: 2002
Keywords: Max-Cut problem, semidefinite programming relaxations, Lagrangian relaxation, Cut polytope metric polytope
Organisations: Operational Research

Identifiers

Local EPrints ID: 29672
URI: http://eprints.soton.ac.uk/id/eprint/29672
ISSN: 0166-218X
PURE UUID: 917ec2d4-20c7-4616-ada6-a8b4ff3944cf

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Date deposited: 12 May 2006
Last modified: 15 Jul 2019 19:08

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