### Abstract

We consider the Hamiltonian cycle problem (HCP) embedded in a controlled Markov decision process. In this setting, HCP reduces to an optimization problem on a set of Markov chains corresponding to a given graph. We prove that Hamiltonian cycles are minimizers for the trace of the fundamental matrix on a set of all stochastic transition matrices. In case of doubly stochastic matrices with symmetric linear perturbation, we show that Hamiltonian cycles minimize a diagonal element of a fundamental matrix for all admissible values of the perturbation parameter. In contrast to the previous work on this topic, our arguments are primarily based on probabilistic rather than algebraic methods.

Original language | Undefined |
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Article number | 10.1287/moor.1080.0351 |

Pages (from-to) | 71-82 |

Number of pages | 12 |

Journal | Mathematics of operations research |

Volume | 34 |

Issue number | 1 |

DOIs | |

Publication status | Published - 27 Jan 2009 |

### Keywords

- IR-67701
- METIS-263887
- MSC-60J10
- EWI-15437
- Singular perturbation
- Hamiltonian cycle
- Fundamental matrix
- Markov chains

## Cite this

Litvak, N., & Ejov, V. (2009). Markov chains and optimality of the Hamiltonian cycle.

*Mathematics of operations research*,*34*(1), 71-82. [10.1287/moor.1080.0351]. https://doi.org/10.1287/moor.1080.0351