Mathematical Methods of Operations Research

, Volume 78, Issue 1, pp 35–59 | Cite as

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

  • Alexander EngauEmail author
  • Miguel F. Anjos
  • Immanuel Bomze
Original Article


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.


Stable set Maximum clique Theta number  Semidefinite programming Interior-point algorithms  Cutting-plane methods Combinatorial optimization 

Mathematical Subject Classifications

90C09 90C20 90C22 90C27 90C35 90C51 90C90 



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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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alexander Engau
    • 1
    Email author
  • Miguel F. Anjos
    • 2
  • Immanuel Bomze
    • 3
  1. 1.University of Colorado DenverDenverUSA
  2. 2.GERAD & École Polytechnique de MontréalMontréalCanada
  3. 3.University of ViennaViennaAustria

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