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Mathematical Programming

, Volume 94, Issue 1, pp 137–166 | Cite as

Maximum stable set formulations and heuristics based on continuous optimization

  • Samuel Burer
  • Renato D.C. Monteiro
  • Yin Zhang

Abstract.

 The stability number α(G) for a given graph G is the size of a maximum stable set in G. The Lovász theta number provides an upper bound on α(G) and can be computed in polynomial time as the optimal value of the Lovász semidefinite program. In this paper, we show that restricting the matrix variable in the Lovász semidefinite program to be rank-one and rank-two, respectively, yields a pair of continuous, nonlinear optimization problems each having the global optimal value α(G). We propose heuristics for obtaining large stable sets in G based on these new formulations and present computational results indicating the effectiveness of the heuristics.

Keywords

Computational Result Polynomial Time Nonlinear Optimization Matrix Variable Continuous Optimization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Samuel Burer
    • 1
  • Renato D.C. Monteiro
    • 2
  • Yin Zhang
    • 3
  1. 1.Department of Management Sciences, University of Iowa, Iowa City, Iowa 52242, USA, e-mail: samuel-burer@uiowa.edu. This author was supported in part by NSF Grants CCR-9902010, CCR-0203426 and INT-9910084.US
  2. 2.School of ISyE, Georgia Institute of Technology, Atlanta, Georgia 30332, USA, e-mail: monteiro@isye.gatech.edu. This author was supported in part by NSF Grants CCR-9902010, CCR-0203113 and INT-9910084.US
  3. 3.Department of Computational and Applied Mathematics, Rice University, Houston, Texas 77005, USA, e-mail: zhang@caam.rice.edu. This author was supported in part by DOE Grant DE-FG03-97ER25331, DOE/LANL Contract 03891-99-23 and NSF Grant DMS-9973339.US

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