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Optimizing over the closure of rank inequalities with a small right-hand side for the maximum stable set problem via bilevel programming

Optimizing over the closure of rank inequalities with a small right-hand side for the maximum stable set problem via bilevel programming
Optimizing over the closure of rank inequalities with a small right-hand side for the maximum stable set problem via bilevel programming
In the context of the maximum stable set problem, rank inequalities impose that the cardinality of any set of vertices contained in a stable set be, at most, as large as the stability number of the subgraph induced by such a set. Rank inequalities are very general, as they subsume many classical inequalities such as clique, hole, antihole, web, and antiweb inequalities. In spite of their generality, the exact separation of rank inequalities has never been addressed without the introduction of topological restrictions on the induced subgraph and the tightness of their closure has never been investigated systematically. In this work, we propose a methodology for optimizing over the closure of all rank inequalities with a right-hand side no larger than a small constant without imposing any restrictions on the topology of the induced subgraph. Our method relies on the exact separation of a relaxation of rank inequalities, which we call relaxed k-rank inequalities, whose closure is as tight. We investigate the corresponding separation problem, a bilevel programming problem asking for a subgraph of maximum weight with a bound on its stability number, whose study could be of independent interest. We first prove that the problem is SP2 -hard and provide some insights on its polyhedral structure. We then propose two exact methods for its solution: a branch-and-cut algorithm (which relies on a family of faced-defining inequalities which we introduce in this paper) and a purely combinatorial branch-and-bound algorithm. Our computational results show that the closure of rank inequalities with a right-hand side no larger than a small constant can yield a bound that is stronger, in some cases, than Lovasz’s Theta function, and substantially stronger than bounds obtained with standard inequalities that are valid for the stable set problem, including odd-cycle inequalities and wheel inequalities.
0899-1499
Coniglio, Stefano
03838248-2ce4-4dbc-a6f4-e010d6fdac67
Gualandi, Stefano
8eaa785d-388a-45e5-bba2-6bda706ab2ae
Coniglio, Stefano
03838248-2ce4-4dbc-a6f4-e010d6fdac67
Gualandi, Stefano
8eaa785d-388a-45e5-bba2-6bda706ab2ae

Coniglio, Stefano and Gualandi, Stefano (2021) Optimizing over the closure of rank inequalities with a small right-hand side for the maximum stable set problem via bilevel programming. INFORMS Journal on Computing. (In Press)

Record type: Article

Abstract

In the context of the maximum stable set problem, rank inequalities impose that the cardinality of any set of vertices contained in a stable set be, at most, as large as the stability number of the subgraph induced by such a set. Rank inequalities are very general, as they subsume many classical inequalities such as clique, hole, antihole, web, and antiweb inequalities. In spite of their generality, the exact separation of rank inequalities has never been addressed without the introduction of topological restrictions on the induced subgraph and the tightness of their closure has never been investigated systematically. In this work, we propose a methodology for optimizing over the closure of all rank inequalities with a right-hand side no larger than a small constant without imposing any restrictions on the topology of the induced subgraph. Our method relies on the exact separation of a relaxation of rank inequalities, which we call relaxed k-rank inequalities, whose closure is as tight. We investigate the corresponding separation problem, a bilevel programming problem asking for a subgraph of maximum weight with a bound on its stability number, whose study could be of independent interest. We first prove that the problem is SP2 -hard and provide some insights on its polyhedral structure. We then propose two exact methods for its solution: a branch-and-cut algorithm (which relies on a family of faced-defining inequalities which we introduce in this paper) and a purely combinatorial branch-and-bound algorithm. Our computational results show that the closure of rank inequalities with a right-hand side no larger than a small constant can yield a bound that is stronger, in some cases, than Lovasz’s Theta function, and substantially stronger than bounds obtained with standard inequalities that are valid for the stable set problem, including odd-cycle inequalities and wheel inequalities.

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Accepted/In Press date: 21 June 2021

Identifiers

Local EPrints ID: 450458
URI: http://eprints.soton.ac.uk/id/eprint/450458
ISSN: 0899-1499
PURE UUID: a03bf443-793d-4f1c-b7f7-75c0e056c442
ORCID for Stefano Coniglio: ORCID iD orcid.org/0000-0001-9568-4385

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Date deposited: 28 Jul 2021 16:32
Last modified: 29 Jul 2021 01:45

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Contributors

Author: Stefano Gualandi

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