A hybrid simulation-optimization algorithm for the Hamiltonian cycle problem
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In this paper, we propose a new hybrid algorithm for the Hamiltonian cycle problem by synthesizing the Cross Entropy method and Markov decision processes. In particular, this new algorithm assigns a random length to each arc and alters the Hamiltonian cycle problem to the travelling salesman problem. Thus, there is now a probability corresponding to each arc that denotes the probability of the event “this arc is located on the shortest tour.” Those probabilities are then updated as in cross entropy method and used to set a suitable linear programming model. If the solution of the latter yields any tour, the graph is Hamiltonian. Numerical results reveal that when the size of graph is small, say less than 50 nodes, there is a high chance the algorithm will be terminated in its cross entropy component by simply generating a Hamiltonian cycle, randomly. However, for larger graphs, in most of the tests the algorithm terminated in its optimization component (by solving the proposed linear program).
KeywordsHamiltonian cycle problem Markov decision process Cross-entropy method
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- Ali, S. M., & Silvey, S. D. (1966). A general class of coefficients of divergence of one distribution from another. Journal of the Royal Statistical Society, B, 28, 131–142. Google Scholar
- Applegate, D. L., Bixby, R. E., Chvatal, V., & Cook, W. J. (2007). The traveling salesman problem: a computational study. Princeton: Princeton University Press. Google Scholar
- The Cross-Entropy Method (2009). http://www.cemethod.org/. Accessed 11 Feb 2009.
- Ejov, V., Filar, J., Haythorpe, M., & Nguyen, G. (2008b). Refined MDP-based branch-and-bound algorithm for the Hamiltonian cycles problem. Mathematics of Operations Research (to appear). Google Scholar
- Eshragh, A., & Modarres, M. (2009). A new approach to distribution fitting: decision on beliefs. Journal of Industrial and Systems Engineering (to appear). Google Scholar
- Eshragh Jahromi, A., & Akhavan Niaki, S. T. (2003). Application of decision on beliefs in response surface methodology. In Proceeding of the 54th session of international statistical institute, Berlin, Germany, 2003. Google Scholar
- Filar, J. A., & Vrieze, K. (1996). Competitive Markov decision processes (1st edn.). Berlin: Springer. Google Scholar
- Gentle, J. E. (2004). Random number generation and Monte Carlo methods (2nd edn.). Berlin: Springer. Google Scholar
- Luenberger, D. G. (2003). Linear and nonlinear programming (2nd edn.). Dordrecht: Kluwer Academic Publishers. Google Scholar
- Puterman, M. L. (2005). Markov decision processes: discrete stochastic dynamic programming (1st edn.). New York: Wiley-Interscience. Google Scholar
- Rubinstein, R. Y., & Kroese, D. P. (2004). The cross-entropy method: a unified approach to combinatorial optimization, Monte-Carlo simulation, and machine learning. Berlin: Springer. Google Scholar
- Vaisman, R. (2009). TSP Random tour generation algorithm. http://iew3.technion.ac.il/CE/pubs.php. Accessed 11 Feb 2009.