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Adaptive, Restart, Randomized Greedy Heuristics for Maximum Clique

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Abstract

This paper presents some adaptive restart randomized greedy heuristics for MAXIMUM CLIQUE. The algorithms are based on improvements and variations of previously-studied algorithms by the authors. Three kinds of adaptation are studied: adaptation of the initial state (AI) given to the greedy heuristic, adaptation of vertex weights (AW) on the graph, and no adaptation (NA). Two kinds of initialization of the vertex-weights are investigated: unweighted initialization (w i := 1) and degree-based initialization (w i := d i where d i is the degree of vertex i in the graph). Experiments are conducted on several kinds of graphs (random, structured) with six combinations: {NA, AI, and AW} × {unweighted initialization, degree-based initialization. A seventh state of the art semi-greedy algorithm, DMclique, is evaluated as a benchmark algorithm. We concentrate on the problem of finding large cliques in large, dense graphs in a relatively short amount of time. We find that the different strategies produce different effects, and that different algorithms work best on different kinds of graphs.

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References

  • Arora, S., C. Lund, R. Motwani, M. Sudan, and M. Szegedy. (1992). “Proof Verification and Hardness of Approximation Problems.” In The Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science. pp.14–23.

  • Balas, E. and C.S. Yu. (1986). “Finding a Maximum Clique in an Arbitrary Graph.” SIAM Journal on Computing 15(4).

  • Baluja, S. and S. Davies. (1997). “Combining Multiple Optimization Runs with Optimal Dependency Trees.” Technical Report, Department of Computer Science, Carnegie-Mellon University.

  • Boese, K.D., A.B. Kahng, and S. Muddu. (1994). “A NewAdaptive Multistart Technique for Combinatorial Global Optimizations.” Operations Research Letters16,101–113.

    Google Scholar 

  • Bomze, I.M., M. Budinich, P.M. Pardalos, and M. Pelillo. (1999). “The Maximum Clique Problem.” In D.-Z. Du and P.M. Pardalos (eds.), Handbook of Combinatorial Optimization (Supplement Volume A). Dordrecht: Kluwer Academic Publishers, pp.1–74.

    Google Scholar 

  • Boyan, J. and W. Buntine. (To appear). “Statistical Machine Learning Methods for Large-Scale Optimization Problems.” Neural Computing Surveys 3, http://www.icsi.berkeley.edu/jagota/NCS.

  • Feo, T.A., M.G.C. Resende, and S.H. Smith. (1994). “A Greedy Randomized Adaptive Search Procedure for Maximum Independent Set.”Operations Research 42, 860–878.

    Google Scholar 

  • Garey, M.R. and D.S. Johnson. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: Freeman.

    Google Scholar 

  • Grossman, T. (1996) “Applying the INN Model to the MaxClique Problem.” In D.S. Johnson and M.A. Trick (eds.), DIMACS Series: Second DIMACS Challenge. American Mathematical Society, pp. 125–145. Proceedings of the Second DIMACS Challenge: Cliques, Coloring, and Satisfiability.

  • Jagota, A. (1996). “An Adaptive, Multiple Restarts Neural Network Algorithm for Graph Coloring.” European Journal of Operational Research93,257–270.

    Google Scholar 

  • Jagota, A. and K.W. Regan. (1997). “Performance of Neural Net Heuristics for Maximum Clique on Diverse Highly Compressible Graphs.” Journal of Global Optimization10, 439–465.

    Google Scholar 

  • Jagota, A., L. Sanchis, and R. Ganesan. (1996). “Approximating Maximum Clique Using Neural Network and Related Heuristics.” In D.S. Johnson and M.A. Trick (eds.), DIMACS Series: Second DIMACS Challenge. American Mathematical Society, pp.169–204.Proceedings of the Second DIMACS Challenge: Cliques, Coloring, and Satisfiability.

  • Johnson, D.S., C.R. Aragon, L.A. McGeoch, and C. Schevon. (1991). “Optimization by Simulated Annealing: An Experimental Evaluation, Part II (Graph Coloring and Number Partitioning).” Operations Research39, 378–406.

    Google Scholar 

  • Pardalos, P.M. and J. Xue. (1994). “The Maximum Clique Problem.” Journal of Global Optimization4,301–328.

    Google Scholar 

  • Pardalos, P.M., J. Rappe, and M.G.C. Resende. (1998). “An Exact Parallel Algorithm for the Maximum Clique Problem.” In R. De Leone, A. Murli, P.M. Pardalos, and G. Toraldo (eds.), High Performance Algorithms and Software in Nonlinear Optimization.Dordrecht: Kluwer Academic Publishers, pp. 279–300.

    Google Scholar 

  • Sanchis, L. (1994). “Test Case Construction for the Vertex Cover Problem.” In N. Dean and G.E. Shannon (eds.), Computational Support for Discrete Mathematics. American Mathematical Society, pp. 315–326.

  • Sanchis, L. and A. Jagota. (1996). “Some Experimental and Theoretical Results on Test Case Generators for the Maximum Clique Problem.” INFORMS Journal on Computing8(2),87–102.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Engineering, Baskin School of Engineering, University of California, Santa Cruz, CA, 95064

    Arun Jagota

  2. Department of Computer Science, Colgate University, Hamilton, NY, 13346

    Laura A. Sanchis

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  1. Arun Jagota
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  2. Laura A. Sanchis
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Jagota, A., Sanchis, L.A. Adaptive, Restart, Randomized Greedy Heuristics for Maximum Clique. Journal of Heuristics 7, 565–585 (2001). https://doi.org/10.1023/A:1011925109392

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  • DOI: https://doi.org/10.1023/A:1011925109392

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