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On transition matrices of Markov chains corresponding to Hamiltonian cycles

Author

Listed:
  • Konstantin Avrachenkov

    (INRIA)

  • Ali Eshragh

    (The University of Newcastle)

  • Jerzy A. Filar

    (Flinders University)

Abstract

In this paper, we present some algebraic properties of a particular class of probability transition matrices, namely, Hamiltonian transition matrices. Each matrix $$P$$ P in this class corresponds to a Hamiltonian cycle in a given graph $$G$$ G on $$n$$ n nodes and to an irreducible, periodic, Markov chain. We show that a number of important matrices traditionally associated with Markov chains, namely, the stationary, fundamental, deviation and the hitting time matrix all have elegant expansions in the first $$n-1$$ n - 1 powers of $$P$$ P , whose coefficients can be explicitly derived. We also consider the resolvent-like matrices associated with any given Hamiltonian cycle and its reverse cycle and prove an identity about the product of these matrices. As an illustration of these analytical results, we exploit them to develop a new heuristic algorithm to determine a non-Hamiltonicity of a given graph.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:annopr:v:243:y:2016:i:1:d:10.1007_s10479-014-1642-2
    DOI: 10.1007/s10479-014-1642-2
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    References listed on IDEAS

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    1. 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.
    2. Eugene A. Feinberg, 2000. "Constrained Discounted Markov Decision Processes and Hamiltonian Cycles," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 130-140, February.
    3. Jerzy A. Filar & Dmitry Krass, 1994. "Hamiltonian Cycles and Markov Chains," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 223-237, February.
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    Cited by:

    1. 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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