Hamiltonian Cycles and Markov Chains

In this paper we derive new characterizations of the Hamiltonian cycles of a directed graph, and a new LP-relaxation of the Traveling Salesman Problem. Our results are obtained via an embedding of these combinatorial optimization problems in suitably perturbed controlled Markov chains. This embedding lends probabilistic interpretation to many of the quantities of interest, which in turn lead naturally to the introduction of a quadratic entropy-like function.