How Many Needles are in a Haystack, or How to Solve #P-Complete Counting Problems Fast
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We present two randomized entropy-based algorithms for approximating quite general #P-complete counting problems, like the number of Hamiltonian cycles in a graph, the permanent, the number of self-avoiding walks and the satisfiability problem. In our algorithms we first cast the underlying counting problem into an associate rare-event probability estimation, and then apply dynamic importance sampling (IS) to estimate efficiently the desired counting quantity. We construct the IS distribution by using two different approaches: one based on the cross-entropy (CE) method and the other one on the stochastic version of the well known minimum entropy (MinxEnt) method. We also establish convergence of our algorithms and confidence intervals for some special settings and present supportive numerical results, which strongly suggest that both ones (CE and MinxEnt) have polynomial running time in the size of the problem.
KeywordsCross-entropy Rare-event probability estimation Hamilton cycles Self-avoiding walks #P-complete problems Stochastic and simulation-based optimization
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- T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, inc, 1991.Google Scholar
- P. T. de Boer, D. P. Kroese, S. Mannor, and R. Y. Rubinstein, “A tutorial on the cross-entropy method,” Annals of Operations Research, 2005.Google Scholar
- Jun Gu et al., “Algorithms for the satisfiability (SAT) problem: A survey,” DIMACS Series in Discrete Mathematics, 1991.Google Scholar
- J. N. Kapur and Kesavan H. K., Entropy Optimization with Applications, Academic Press, Inc., 1992.Google Scholar
- R. Motwani and R. Raghavan, Randomized Algorithms, Cambridge University Press, 1997.Google Scholar
- R. Y. Rubinstein, “A stochastic minimum cross-entropy method for combinatorial optimization and rare-event estimation,” Methodology and Computing in Applied Probability (1) pp. 1–46, 2005.Google Scholar
- R. Y. Rubinstein and B. Melamed, Modern Simulation and Modeling, John Wiley & Sons, Inc., 1998.Google Scholar
- R. Y. Rubinstein and D. P. Kroese, The Cross-Entropy Method: a Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning, Springer, 2004.Google Scholar
- D. J. A. Welsh, Complexity: Knots, Colouring and Counting, Cambridge University Press, 1993.Google Scholar