Feasible bases for a polytope related to the Hamilton cycle problem
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We study a certain polytope depending on a graph G and a parameter β∈(0,1) which arises from embedding the Hamiltonian cycle problem in a discounted Markov decision process. Eshragh et al. conjectured a lower bound on the proportion of feasible bases corresponding to Hamiltonian cycles in the set of all feasible bases. We make progress towards a proof of the conjecture by proving results about the structure of feasible bases. In particular, we prove three main results: (1) the set of feasible bases is independent of the parameter β when the parameter is close to 1, (2) the polytope can be interpreted as a generalized network flow polytope and (3) we deduce a combinatorial interpretation of the feasible bases. We also provide a full characterization for a special class of feasible bases, and we apply this to provide some computational support for the conjecture.
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