Journal of Combinatorial Optimization

, Volume 21, Issue 4, pp 434–457 | Cite as

A Branch and Cut solver for the maximum stable set problem

  • Steffen Rebennack
  • Marcus Oswald
  • Dirk Oliver Theis
  • Hanna Seitz
  • Gerhard Reinelt
  • Panos M. Pardalos
Article

Abstract

This paper deals with the cutting-plane approach to the maximum stable set problem. We provide theoretical results regarding the facet-defining property of inequalities obtained by a known project-and-lift-style separation method called edge-projection, and its variants. An implementation of a Branch and Cut algorithm is described, which uses edge-projection and two other separation tools which have been discussed for other problems: local cuts (pioneered by Applegate, Bixby, Chvátal and Cook) and mod-k cuts. We compare the performance of this approach to another one by Rossi and Smiriglio (Oper. Res. Lett. 28:63–74, 2001) and discuss the value of the tools we have tested.

Keywords

Maximum stable set problem Cutting-plane algorithm Branch and Cut Separation algorithm Edge-projection 

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References

  1. ABACUS (2006) A branch and cut solver. Version 2.3.0. http://www.informatik.uni-koeln.de/abacus/
  2. Applegate D, Bixby R, Chvátal V, Cook W (2001) TSP cuts which do not conform to the template paradigm. In: Computational combinatorial optimization. LNCS, vol 2241. Springer, Berlin, pp 157–222 CrossRefGoogle Scholar
  3. Arora S, Safra S (1992) Probabilistic checking of proofs; a new characterization of NP. In: Proceedings 33rd IEEE symposium on foundations of computer science. IEEE Computer Society, Los Angeles, pp 2–13 CrossRefGoogle Scholar
  4. Avenali A (2007) Resolution branch and bound and an application: the maximum weighted stable set problem. Oper Res 55(5):932–948 CrossRefMATHMathSciNetGoogle Scholar
  5. Balas E, Yu CS (1986) Finding a maximum clique in an arbitrary graph. SIAM J Comput 14(4):1054–1068 CrossRefMathSciNetGoogle Scholar
  6. Bomze IM, Budinich M, Pardalos PM, Pelillo M (1999) The maximum clique problem. Handbook of combinatorial optimization. Kluwer Academic, Boston Google Scholar
  7. Butenko S (2003) Maximum independent set and related problems, with applications. PhD thesis, University of Florida, USA Google Scholar
  8. Campelo M, Correa RC (2009) A Lagrangian relaxation for the maximum stable set problem. http://arxiv.org/PS_cache/arxiv/pdf/0903/0903.1407v1.pdf
  9. Caprara A, Fiscetti M, Letchford AN (2000) On the separation of maximally violated mod-k cuts. Math Program 87(1):37–56 MATHMathSciNetGoogle Scholar
  10. de Vries S, Vohra RV (2003) Combinatorial auctions: a survey. INFORMS J Comput 15(3):284–309 CrossRefMathSciNetGoogle Scholar
  11. Dukanovich I, Rendl F (2007) Semidefinite programming relaxations for graph coloring and maximal clique problems. Math Prog B 109:345–365 CrossRefGoogle Scholar
  12. Elf M, Gutwenger C, Jünger M, Rinaldi G (2001) Branch-and-cut algorithms for combinatorial optimization and their implementation in ABACUS. In: Computational combinatorial optimization. LNCS, vol 2241. Springer, Berlin, pp 157–222 CrossRefGoogle Scholar
  13. Fricke L (2007) Google Scholar
  14. Fujisawa K, Morito S, Kubo M (1995) Experimental analyses of the life span method for the maximum stable set problem. Inst Stat Math Coop Res Rep 75:135–165 Google Scholar
  15. Garey MR, Johnson DS (1979) In: Klee V (ed) Computers and intractability, a guide to the theory of NP-completeness. A series of books in the mathematical sciences. Freeman, New York Google Scholar
  16. Gerards AMH, Schrijver A (1986) Matrices with the Edmonds-Johnson property. Combinatorica 6:365–379 CrossRefMATHMathSciNetGoogle Scholar
  17. Grötschel M, Lovasz L, Schrijver A (1988) Geometric algorithms and combiantorial optimization. Springer, Berlin Google Scholar
  18. Gruber G, Rendl F (2003) Computational experience with stable set relaxations. SIAM J Opt 13:1014–1028 CrossRefMATHMathSciNetGoogle Scholar
  19. ILOG CPLEX, Version 8.100. http://www.ilog.com/products/cplex/
  20. Mannino C, Sassano A (1996) Edge projection and the maximum cardinality stable set problem. In: DIMACS series in discrete mathematics and theoretical computer science, vol 26. AMS, New York, pp 249–261 Google Scholar
  21. Nemhauser GL, Sigismondi G (1992) A strong cutting plane/branch-and-bound algorithm for node packing. J Oper Res Soc 43:443–457 MATHGoogle Scholar
  22. Nemhauser GL, Trotter LE Jr (1975) Vertex packings: structural properties and algorithms. Math Program 8:232–248 CrossRefMATHMathSciNetGoogle Scholar
  23. Padberg MW (1973) On the facial structure of set packing polyhedra. Math Program 5:199–215 CrossRefMATHMathSciNetGoogle Scholar
  24. Rebennack S (2006) Maximum stable set problem: a branch & cut solver. Diplomarbeit, Ruprecht–Karls Universität Heidelberg, Heidelberg, Germany Google Scholar
  25. Rebennack S (2008) Stable set problem: branch & cut algorithms. In: Floudas CA, Pardalos PM (eds) Encyclopedia of optimization, 2nd edn. Springer, Berlin, pp 3676–3688 Google Scholar
  26. Rossi F, Smriglio S (2001) A branch-and-cut algorithm for the maximum cardinality stable set problem. Oper Res Lett 28:63–74 CrossRefMATHMathSciNetGoogle Scholar
  27. Second DIMACS Challenge (1992/1993). http://mat.gsia.cmu.edu/challenge.html
  28. Sewell EC (1998) A branch and bound algorithm for the stability number of a sparse graph. INFORMS J Comput 10(4):438–447 CrossRefMathSciNetGoogle Scholar
  29. Strickland DM, Barnes E, Sokol JS (2005) Optimal protein structure alignment using maximum cliques. Oper Res 53:389–402 CrossRefMATHMathSciNetGoogle Scholar
  30. Warren JS, Hicks IV (2006) Combinatorial branch-and-bound for the maximum weight independent set problem. http://ie.tamu.edu/people/faculty/Hicks/jeff.rev.pdf
  31. Warrier D (2007) A branch, price, and cut appraoch to solving the maximum weighted independent set problem. PhD thesis, Texas A&M University Google Scholar
  32. Warrier D, Wilhelm WE, Warren JS, Hicks IV (2005) A branch-and-price approach for the maximum weight independent set proble. Networks 46(4):198–209 CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Steffen Rebennack
    • 1
  • Marcus Oswald
    • 2
  • Dirk Oliver Theis
    • 3
  • Hanna Seitz
    • 2
  • Gerhard Reinelt
    • 2
  • Panos M. Pardalos
    • 1
  1. 1.Department of Industrial & Systems EngineeringUniversity of FloridaGainesvilleUSA
  2. 2.Discrete Optimization Research GroupRuprecht-Karls Universität HeidelbergHeidelbergGermany
  3. 3.Fakultät für Mathematik (IMO)OvG-Universität MagdeburgMagdeburgGermany

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