Solutions to recursive distributional equations for the mean-field TSP and related problems

Abstract: For several combinatorial optimization problems over random structures, the theory of local weak convergence from probability and the cavity method from statistical physics can be used to deduce a recursive equation for the distribution of a quantity of interest. We show that there is a unique solution to such a recursive distributional equation (RDE) when the optimization problem is the traveling salesman problem (TSP) or from a related family of minimum weight d-factor problems (which includes minimum weight matching) on a complete graph (or complete bipartite graph) with independent and identically distributed edge-weights from the exponential distribution. We analyze the dynamics of the operator induced by the RDE on the space of distributions, and prove that the iterates of the operator, starting from any arbitrary distribution, converges to the fixed-point solution, modulo shifts. The existence of a solution to the RDE in such a problem helps in proving results about the limit of the optimal solution of the combinatorial problem. The convergence of the iterates of the operator is important in proving results about the validity of belief propagation for iteratively finding the optimal solution.