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Hamiltonian Cycles and Subsets of Discounted Occupational Measures

Author

Listed:
  • Ali Eshragh

    (School of Mathematical and Physical Sciences, University of Newcastle, 2308 Newcastle, New South Wales, Australia)

  • Jerzy A. Filar

    (School of Mathematics and Physics, University of Queensland, 4072 Brisbane, Queensland, Australia)

  • Thomas Kalinowski

    (School of Science and Technology, University of New England, 2351 Armidale, New South Wales, Australia)

  • Sogol Mohammadian

    (School of Mathematical and Physical Sciences, University of Newcastle, 2308 Newcastle, New South Wales, Australia)

Abstract

We study a certain polytope arising from embedding the Hamiltonian cycle problem in a discounted Markov decision process. The Hamiltonian cycle problem can be reduced to finding particular extreme points of a certain polytope associated with the input graph. This polytope is a subset of the space of discounted occupational measures. We characterize the feasible bases of the polytope for a general input graph G and determine the expected numbers of different types of feasible bases when the underlying graph is random. We utilize these results to demonstrate that augmenting certain additional constraints to reduce the polyhedral domain can eliminate a large number of feasible bases that do not correspond to Hamiltonian cycles. Finally, we develop a random walk algorithm on the feasible bases of the reduced polytope and present some numerical results. We conclude with a conjecture on the feasible bases of the reduced polytope.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:ormoor:v:45:y:2020:i:2:p:713-731
    DOI: 10.1287/moor.2019.1009
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    References listed on IDEAS

    as
    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. 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.
    3. Ali Eshragh & Jerzy Filar, 2011. "Hamiltonian Cycles, Random Walks, and Discounted Occupational Measures," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 258-270, May.
    4. Eugene A. Feinberg, 2000. "Constrained Discounted Markov Decision Processes and Hamiltonian Cycles," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 130-140, February.
    5. Jerzy A. Filar & Dmitry Krass, 1994. "Hamiltonian Cycles and Markov Chains," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 223-237, February.
    6. Vladimir Ejov & Jerzy A. Filar & Michael Haythorpe & Giang T. Nguyen, 2009. "Refined MDP-Based Branch-and-Fix Algorithm for the Hamiltonian Cycle Problem," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 758-768, August.
    7. Nelly Litvak & Vladimir Ejov, 2009. "Markov Chains and Optimality of the Hamiltonian Cycle," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 71-82, February.
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