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We present a physics inspired heuristic method for solving combinatorialoptimization problems. Our approach is specifically motivated by the desire toavoid trapping in metastable local minima- a common occurrence in hard problemswith multiple extrema. Our method involves (i) coupling otherwise independentsimulations of a system ("replicas") via geometrical distances as well as (ii)probabilistic inference applied to the solutions found by individual replicas.The {\it ensemble} of replicas evolves as to maximize the inter-replicacorrelation while simultaneously minimize the local intra-replica cost function(e.g., the total path length in the Traveling Salesman Problem within eachreplica). We demonstrate how our method improves the performance of rudimentarylocal optimization schemes long applied to the NP hard Traveling SalesmanProblem. In particular, we apply our method to the well-known "$k$-opt"algorithm and examine two particular cases- $k=2$ and $k=3$. With the aid ofgeometrical coupling alone, we are able to determine for the optimum tourlength on systems up to $280$ cities (an order of magnitude larger than thelargest systems typically solved by the bare $k=3$ opt). The probabilisticreplica-based inference approach improves $k-opt$ even further and determinesthe optimal solution of a problem with $318$ cities and find tours whose totallength is close to that of the optimal solutions for other systems with alarger number of cities.

An interacting replica approach applied to the traveling salesman problem