Advertisement

An Improved Simulated Annealing Algorithm for the Maximum Independent Set Problem

  • Xinshun Xu
  • Jun Ma
  • Hua Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4113)

Abstract

The maximum independent set problem is a classic graph optimization problem. It is well known that it is an NP-Complete problem. In this paper, an improved simulated annealing algorithm is presented for the maximum independent set problem. In this algorithm, an acceptance function is defined for every vertex. This can help the algorithm find a near optimal solution to a problem. Simulations are performed on benchmark graphs and random graphs. The simulation results show that the proposed algorithm provides a high probability of finding optimal solutions.

Keywords

Random Graph Simulated Annealing Algorithm Maximum Clique Acceptance Function Circle Graph 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Stinson, D.R.: An Introduction to the Design and Analysis of Algorithms, 2nd edn. The Charles Babbage Research Center, Winnipeg, Manitoba, Canada (1987)Google Scholar
  2. 2.
    Karp, R.M.: Reducibility among Combinatorial Problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computation, pp. 85–103. Plenum Press, New York (1972)Google Scholar
  3. 3.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, San Francisco (1979)MATHGoogle Scholar
  4. 4.
    Johnson, D.: Approximate Algorithms for Combinatorial Problems. Journal of Computer and System Sciences 9, 256–278 (1974)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Abello, J., Pardalos, P.M., Resende, M.G.C.: On the Maximum Clique Problems in very Large Graphs. In: Abello, J., Vitter, J.S. (eds.) External Memory Algorithms and Visualization. DIMACS series on discrete mathematics and theoretical Computer Science, vol. 50, pp. 119–130. American Mathematical Society, Providence (1999)Google Scholar
  6. 6.
    Avondo-Bodeno, G.: Economic Applications of the Theory of Graphs. Gordon and Breach Science, New York (1962)Google Scholar
  7. 7.
    Berge, C.: The Theory of Graphs and Its Applications. John Wiley and Sons, New York (1962)MATHGoogle Scholar
  8. 8.
    Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillot, M.: The maximum clique problem. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, pp. 1–74. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  9. 9.
    Karp, R.M., Wigderson, A.: A Fast Parallel Algorithm for the Maximal Independent Set Problem. Journal of the ACM 32(4), 762–773 (1985)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Alon, N., Babai, L., Itai, A.: A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem. Journal of Algorithms 7, 567–583 (1986)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Goldberg, M., Spencer, T.: A New Parallel Algorithm for the Maximal Independent Set Problem. SIAM Journal on Computing 18, 419–427 (1989)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Goldberg, M., Spencer, T.: Constructing a Maximal Independent Set in Parallel. SIAM Journal on Discrete Mathematics 2, 322–328 (1989)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Goldberg, M., Spencer, T.: An Efficient Parallel Algorithm that Finds Independent Sets of Guaranteed Size. SIAM Journal on Discrete Mathematics 6, 443–459 (1993)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Takefuji, Y., Chen, L., Lee, K., Huffman, J.: Parallel Algorithm for Finding a Near Maximum Independent Set of the Circle Graph. IEEE Transaction on Neural Networks 1, 263–267 (1990)CrossRefGoogle Scholar
  15. 15.
    Bäck, T., Khuri, S.: An Evolutionary Heuristic for the Maximum Independent Set Problem. In: Michalewicz, Z., Schaffer, J.D., Schwefel, H.P., Fogel, D.B., Kitano, H. (eds.) Proceeding of the First IEEE Conference on Evolutionary Computation, pp. 531–535. IEEE Press, New York (1994)CrossRefGoogle Scholar
  16. 16.
    Busygin, S., Butenko, S., Pardalos, P.: A Heuristic for the Maximum Independent Set Problem Based on Optimization of a Quadratic over a Sphere. Journal of Combinatorial Optimization 6, 287–297 (2002)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Berman, P., Fujito, T.: On Approximation Properties of the Independent Set Problem for Low Degree Graphs. Theory of Computing Systems 32, 115–132 (1999)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Dahlhaus, E., Karpinski, M., Kelsen, P.: An Efficient Parallel Algorithm for Computing a Maximal Independent Set in a Hypergraph of Dimension 3. Information Processing Letters 42, 309–313 (1992)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Pham, D.T., Karaboga, D.: Intelligent Optimisation Techniques: Genetic Algorithm, Tabu Search, Simulated Annealing, and Neural Networks. Springer, Heidelberg (2000)Google Scholar
  20. 20.
    Halldórsson, M.M., Radhakrishnan, J.: Greed is good: Approximating Independent Sets in Sparse and Bounded-degree Graphs. Algorithmica 18, 145–163 (1997)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Yuan, S.-Y., Kuo, S.-Y.: A New Technique for Optimization Problems in Graph Theory. IEEE Transactions on Computers 47, 190–196 (1998)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Matual, D.W.: On the Complete Subgraph of a Random Graph”. In: Bose, R., Dowling, T. (eds.) Proceeding of the Second Chapel Hill Conference on Combinatory Mathematics and its Applications, pp. 356–369. Chapel Hill, North Carolina (1970)Google Scholar
  23. 23.
    Bertoni, A., Campadelli, P., Grossi, G.: A Neural Algorithm for the Maximum Clique Problem: Analysis, Experiments, and Circuit Implementation. Algorithmica 33, 71–88 (2002)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Jagota, S.: Approximating Maximum Clique with a Hopfield Network. IEEE Transactions on Neural Networks 6, 724–735 (1995)CrossRefGoogle Scholar
  25. 25.
    Funabiki, N., Nishikawa, S.: Comparisons of Energy-descent Optimization Algorithms for Maximum Clique Problems. IEICE Transactions on Fundamentals E79-A(4), 452–460 (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Xinshun Xu
    • 1
  • Jun Ma
    • 1
  • Hua Wang
    • 1
  1. 1.School of Computer Science and TechnologyShandong UniversityJinanChina

Personalised recommendations