Hamiltonian cycles and subsets of discounted occupational measures

TL;DR

This paper explores a polytope related to Hamiltonian cycles within a discounted Markov decision process, characterizing feasible bases, analyzing their distribution in random graphs, and proposing an algorithm to find Hamiltonian cycles.

Contribution

It introduces a novel polytope framework linking Hamiltonian cycles to discounted occupational measures and develops an algorithm based on feasible bases analysis.

Findings

Characterized feasible bases of the polytope for general graphs

Analyzed the distribution of feasible bases in random graphs

Developed a random walk algorithm for the reduced polytope

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.