### Abstract

Original language | Undefined |
---|---|

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

*Mathematics of operations research*,

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

}

*Mathematics of operations research*, vol. 34, no. 1, 10.1287/moor.1080.0351, pp. 71-82. https://doi.org/10.1287/moor.1080.0351

**Markov chains and optimality of the Hamiltonian cycle.** / Litvak, Nelli; Ejov, Vladimir.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Markov chains and optimality of the Hamiltonian cycle

AU - Litvak, Nelli

AU - Ejov, Vladimir

N1 - 10.1287/moor.1080.0351

PY - 2009/1/27

Y1 - 2009/1/27

N2 - 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.

AB - 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.

KW - IR-67701

KW - METIS-263887

KW - MSC-60J10

KW - EWI-15437

KW - Singular perturbation

KW - Hamiltonian cycle

KW - Fundamental matrix

KW - Markov chains

U2 - 10.1287/moor.1080.0351

DO - 10.1287/moor.1080.0351

M3 - Article

VL - 34

SP - 71

EP - 82

JO - Mathematics of operations research

JF - Mathematics of operations research

SN - 0364-765X

IS - 1

M1 - 10.1287/moor.1080.0351

ER -