  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.
