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Constraint selection in a build-up interior-point cutting-plane method for solving relaxations of the stable-set problem

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Abstract

The stable-set problem is an NP-hard problem that arises in numerous areas such as social networking, electrical engineering, environmental forest planning, bioinformatics clustering and prediction, and computational chemistry. While some relaxations provide high-quality bounds, they result in very large and expensive conic optimization problems. We describe and test an integrated interior-point cutting-plane method that efficiently handles the large number of nonnegativity constraints in the popular doubly-nonnegative relaxation. This algorithm identifies relevant inequalities dynamically and selectively adds new constraints in a build-up fashion. We present computational results showing the significant benefits of this approach in comparison to a standard interior-point cutting-plane method.

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Acknowledgments

We gratefully acknowledge the helpful comments by the associate editor and two anonymous referees that have allowed us to improve this paper.

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Authors and Affiliations

  1. University of Colorado Denver, Denver, CO, USA

    Alexander Engau

  2. GERAD & École Polytechnique de Montréal, Montréal, QC, Canada

    Miguel F. Anjos

  3. University of Vienna, Vienna, Austria

    Immanuel Bomze

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  1. Alexander Engau
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  2. Miguel F. Anjos
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Correspondence to Alexander Engau.

Additional information

Alexander Engau: Research partially supported by the DFG Emmy Noether project “Combinatorial Optimization in Physics (COPhy)” at the University of Cologne, Germany and by MITACS, a Network of Centres of Excellence for the Mathematical Sciences in Canada.

Miguel F. Anjos: Research partially supported by the Natural Sciences and Engineering Research Council of Canada, and by a Humboldt Research Fellowship.

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Engau, A., Anjos, M.F. & Bomze, I. Constraint selection in a build-up interior-point cutting-plane method for solving relaxations of the stable-set problem. Math Meth Oper Res 78, 35–59 (2013). https://doi.org/10.1007/s00186-013-0431-z

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