Hamiltonian Cycles and Singularly Perturbed Markov Chains

We consider the Hamiltonian cycle problem embedded in a singularly perturbed Markov decision process. We also consider a functional on the space of deterministic policies of the process that consists of the (1,1)-entry of the fundamental matrices of the Markov chains induced by the same policies. We show that when the perturbation parameter, ε, is less than or equal to 1/N², the Hamiltonian cycles of the directed graph are precisely the minimizers of our functional over the space of deterministic policies. In the process, we derive analytical expressions for the possible N distinct values of the functional over the, typically, much larger space of deterministic policies.