The invisible hand algorithm: Solving the assignment problem with statistical physics

We propose a novel method for solving the assignment problem using techniques adapted from statistical physics. We derive a convex effective energy function whose unique minimum corresponds to the optimal assignment. Steepest descent results in a continuous-time dynamical system that is guaranteed to converge arbitrarily close to the optimal solution. Our algorithm has an appealing economic interpretation and has very interesting connections to the discrete auction algorithm proposed by Bertsekas. We also derive an alternative discrete algorithm for minimizing the effective energy based on a theorem by Sinkhorn. Keywords--Assignment problem, Statistical physics, Auction algorithm, Interior point method, Optimization, Neural networks.