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Cybernetics and Systems Analysis

, Volume 45, Issue 1, pp 141–152 | Cite as

LP-oriented upper bounds for the weighted stability number of a graph

  • P. I. StetsyukEmail author
  • A. P. Lykhovyd
Systems Analysis
  • 22 Downloads

Upper bounds for the weighted stability number of a graph are considered that are based on the approximation of its stable set polytope by linear inequalities for odd cycles and p-wheels in the graph. Algorithms are developed for finding upper bounds on the basis of solution of LP problems with a finite number of inequalities produced by the shortest path algorithm for a special graph. The results of test experiments are given for graphs with several hundred or thousand vertices.

Keywords

weighted stability number of a graph polyhedron of stable sets LP-oriented upper bound t-perfect graph p-wheel Wp-perfect graph 

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References

  1. 1.
    M. Gary and D. Johnson, Computers and Hard-to-Solve Problems [Russian translation], Mir, Moscow (1982).Google Scholar
  2. 2.
    M. Grötschel, L. Lóvasz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, Berlin (1988).zbMATHGoogle Scholar
  3. 3.
    R. Wunderling, Paralleler und objektorientierter simplex-algorithmus, Techn. Univ., Berlin (1996), http://www.zib.de/Publications/abstracts/TR-96-09/.zbMATHGoogle Scholar
  4. 4.
    B. V. Cherkassky, A. V. Goldberg, and T. Radzik, “Shortest paths algorithms: Theory and experimental evaluation,” Math. Program., 73, No. 2, 129–174 (1996), http://www.avglab.com/andrew/soft.html.CrossRefMathSciNetGoogle Scholar
  5. 5.
    DIMACS (1995). Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge, http://dimacs.rutgers.edu/Challenges/.
  6. 6.
    N. Z. Shor, Nondifferentiable Optimization and Polynomial Problems, Kluwer, Boston-Dordrecht-London (1998).zbMATHGoogle Scholar
  7. 7.
    E. Cheng and W. H. Cunningham, “Wheel inequalities for stable set polytopes,” Math. Program., 77, No. 3, 389–421 (1997).MathSciNetGoogle Scholar
  8. 8.
    P. I. Stetsyuk and B. M. Chumakov, “Properties of an upper bound of N. Z. Shor for the weighted stability number of a graph,” in: Proc. Intern. Symp. “Computation Optimization Problems (COP-XXXIII), ” V. M. Glushkov Cybernetics Institute of NASU, Kyiv (2007), pp. 271–272.Google Scholar
  9. 9.
    I. V. Sergienko and V. P. Shylo, “Problems of discrete optimization: Challenges and main approaches to solve them,” Cybernetics and Systems Analysis, No. 4, 465–482 (2006).Google Scholar
  10. 10.
    T. Achterberg, Constraint Integer Programming, Techn. Univ., Berlin (2007), http://opus.kobv.de/tuberlin/volltexte/2007/1611/.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Cybernetics InstituteNational Academy of Sciences of UkraineKievUkraine

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