Hamiltonian cycles and singularly perturbed Markov chains
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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 consist 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, e, 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.