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A hybrid simulation-optimization algorithm for the Hamiltonian cycle problem

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  • Ali Eshragh
  • Jerzy Filar
  • Michael Haythorpe

Abstract

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). Copyright Springer Science+Business Media, LLC 2011

Suggested Citation

  • Ali Eshragh & Jerzy Filar & Michael Haythorpe, 2011. "A hybrid simulation-optimization algorithm for the Hamiltonian cycle problem," Annals of Operations Research, Springer, vol. 189(1), pages 103-125, September.
    • Handle: RePEc:spr:annopr:v:189:y:2011:i:1:p:103-125:10.1007/s10479-009-0565-9
    DOI: 10.1007/s10479-009-0565-9
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    References listed on IDEAS

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    1. Reuven Rubinstein, 1999. "The Cross-Entropy Method for Combinatorial and Continuous Optimization," Methodology and Computing in Applied Probability, Springer, vol. 1(2), pages 127-190, September.
    2. Rubinstein, Reuven Y., 1997. "Optimization of computer simulation models with rare events," European Journal of Operational Research, Elsevier, vol. 99(1), pages 89-112, May.
    3. Eugene A. Feinberg, 2000. "Constrained Discounted Markov Decision Processes and Hamiltonian Cycles," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 130-140, February.
    4. Jerzy A. Filar & Dmitry Krass, 1994. "Hamiltonian Cycles and Markov Chains," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 223-237, February.
    5. L. Margolin, 2005. "On the Convergence of the Cross-Entropy Method," Annals of Operations Research, Springer, vol. 134(1), pages 201-214, February.
    6. Zdravko I. Botev & Dirk P. Kroese, 2008. "An Efficient Algorithm for Rare-event Probability Estimation, Combinatorial Optimization, and Counting," Methodology and Computing in Applied Probability, Springer, vol. 10(4), pages 471-505, December.
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    Cited by:

    1. Konstantin Avrachenkov & Ali Eshragh & Jerzy A. Filar, 2016. "On transition matrices of Markov chains corresponding to Hamiltonian cycles," Annals of Operations Research, Springer, vol. 243(1), pages 19-35, August.
    2. Michael Haythorpe & Walter Murray, 2022. "Finding a Hamiltonian cycle by finding the global minimizer of a linearly constrained problem," Computational Optimization and Applications, Springer, vol. 81(1), pages 309-336, January.
    3. Ali Eshragh & Jerzy A. Filar & Thomas Kalinowski & Sogol Mohammadian, 2020. "Hamiltonian Cycles and Subsets of Discounted Occupational Measures," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 713-731, May.

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