Hybrid evolutionary algorithms on minimum vertex cover for random graphs

This paper analyzes the hierarchical Bayesian optimization algorithm (hBOA) on minimum vertex cover for standard classes of random graphs and transformed SAT instances. The performance of hBOA is compared with that of the branch-and-bound problem solver (BB), the simple genetic algorithm (GA) and the parallel simulated annealing (PSA). The results indicate that BB is significantly outperformed by all the other tested methods, which is expected as BB is a complete search algorithm and minimum vertex cover is an NP-complete problem. The best performance is achieved with hBOA; nonetheless, the performance differences between hBOA and other evolutionary algorithms are relatively small, indicating that mutation-based search and recombination-based search lead to similar performance on the tested problem instances.

[1]  Thomas H. Cormen,et al.  Introduction to algorithms [2nd ed.] , 2001 .

[2]  Reinhard Lüling,et al.  Load balancing for distributed branch & bound algorithms , 1992, Proceedings Sixth International Parallel Processing Symposium.

[3]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[4]  Peng-Jun Wan,et al.  Optimal placement of wavelength converters in trees and trees of rings , 1999, Proceedings Eight International Conference on Computer Communications and Networks (Cat. No.99EX370).

[5]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[6]  Martin Pelikan,et al.  Searching for Ground States of Ising Spin Glasses with Hierarchical BOA and Cluster Exact Approximation , 2006, Scalable Optimization via Probabilistic Modeling.

[7]  A. Rbnyi.  ON THE EVOLUTION OF RANDOM GRAPHS , 2001 .

[8]  David Maxwell Chickering,et al.  A Bayesian Approach to Learning Bayesian Networks with Local Structure , 1997, UAI.

[9]  M. Mézard,et al.  Analytic and Algorithmic Solution of Random Satisfiability Problems , 2002, Science.

[10]  Dan Gusfield,et al.  Optimal, Efficient Reconstruction of Phylogenetic Networks with Constrained Recombination , 2004, J. Bioinform. Comput. Biol..

[11]  Shumeet Baluja,et al.  A Method for Integrating Genetic Search Based Function Optimization and Competitive Learning , 1994 .

[12]  Xiuzhen Huang,et al.  Maximum common subgraph: some upper bound and lower bound results , 2006, BMC Bioinform..

[13]  Martin Pelikan,et al.  Hierarchical Bayesian optimization algorithm: toward a new generation of evolutionary algorithms , 2010, SICE 2003 Annual Conference (IEEE Cat. No.03TH8734).

[14]  A. Hartmann.  Phase Transitions in Combinatorial Optimization Problems - Basics, Algorithms and Statistical Mechanics , 2005 .

[15]  Nir Friedman,et al.  Learning Bayesian Networks with Local Structure , 1996, UAI.

[16]  Béla Bollobás,et al.  Random Graphs , 1985 .

[17]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[18]  J. A. Lozano,et al.  Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation , 2001 .

[19]  Rémi Monasson,et al.  Determining computational complexity from characteristic ‘phase transitions’ , 1999, Nature.

[20]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[21]  Wei Li,et al.  Many hard examples in exact phase transitions , 2003, Theor. Comput. Sci..

[22]  Franz Rothlauf,et al.  Evaluation-Relaxation Schemes for Genetic and Evolutionary Algorithms , 2004 .

[23]  Georges R. Harik,et al.  Finding Multimodal Solutions Using Restricted Tournament Selection , 1995, ICGA.

[24]  Alexander K. Hartmann,et al.  The number of guards needed by a museum: A phase transition in vertex covering of random graphs , 2000, Physical review letters.

[25]  Michael R. Fellows,et al.  Kernelization Algorithms for the Vertex Cover Problem: Theory and Experiments , 2004, ALENEX/ANALC.

[26]  H. Rieger,et al.  New Optimization Algorithms in Physics , 2004 .

[27]  E. L. Lawler,et al.  Branch-and-Bound Methods: A Survey , 1966, Oper. Res..

[28]  Ke Xu,et al.  A Simple Model to Generate Hard Satisfiable Instances , 2005, IJCAI.

[29]  H. Mühlenbein,et al.  From Recombination of Genes to the Estimation of Distributions I. Binary Parameters , 1996, PPSN.

[30]  Edward P. K. Tsang,et al.  Foundations of constraint satisfaction , 1993, Computation in cognitive science.

[31]  D. Goldberg,et al.  Escaping hierarchical traps with competent genetic algorithms , 2001 .

[32]  M. Bauer,et al.  Core percolation in random graphs: a critical phenomena analysis , 2001, cond-mat/0102011.

[33]  B. Bollobás.  The evolution of random graphs , 1984 .

[34]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[35]  B. Chakrabarti,et al.  Quantum Annealing and Related Optimization Methods , 2008 .

[36]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[37]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[38]  Wei Li,et al.  Exact Phase Transitions in Random Constraint Satisfaction Problems , 2000, J. Artif. Intell. Res..

[39]  Alexander K. Hartmann,et al.  Typical solution time for a vertex-covering algorithm on finite-connectivity random graphs , 2001, Physical review letters.

[40]  S Cocco,et al.  Trajectories in phase diagrams, growth processes, and computational complexity: how search algorithms solve the 3-satisfiability problem. , 2001, Physical review letters.

[41]  David E. Goldberg,et al.  A Survey of Optimization by Building and Using Probabilistic Models , 2002, Comput. Optim. Appl..

[42]  Ketan Kotecha,et al.  A Hybrid Genetic Algorithm for Minimum Vertex Cover Problem , 2003, IICAI.

[43]  Martin Pelikan,et al.  Scalable Optimization via Probabilistic Modeling: From Algorithms to Applications (Studies in Computational Intelligence) , 2006 .

[44]  Dan Gusfield,et al.  Efficient reconstruction of phylogenetic networks with constrained recombination , 2003, Computational Systems Bioinformatics. CSB2003. Proceedings of the 2003 IEEE Bioinformatics Conference. CSB2003.

[45]  D. E. Goldberg,et al.  Genetic Algorithms in Search , 1989 .

[46]  Alexander K. Hartmann,et al.  The typical-case complexity of a vertex-covering algorithm on finite-connectivity random graphs , 2000 .

[47]  Bao Zheng,et al.  Solving vertex covering problems using hybrid genetic algorithms , 2000, WCC 2000 - ICSP 2000. 2000 5th International Conference on Signal Processing Proceedings. 16th World Computer Congress 2000.

[48]  M. Weigt,et al.  Minimal vertex covers on finite-connectivity random graphs: a hard-sphere lattice-gas picture. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.