A Markov Chain Monte Carlo Algorithm for the Quadratic Assignment Problem Based on Replicator Equations

This paper proposes an optimization algorithm for the Quadratic Assignment Problem (QAP) based on replicator equations. If the growth rate of a replicator equation is composed of the performance index and the constraints of the QAP suitably, by increasing the value of the control parameter in the growth rate, the equilibrium solutions which correspond to the feasible solutions of the QAP become stable in order from the one with smaller value of the performance index. Based on the characteristics of the system, the following optimization algorithm is constructed; the control parameter is set so that the equilibrium solutions corresponding to the feasible solutions with smaller values of the performance index become stable, and then in the solution space of the replicator equations, a Markov chain Monte Carlo algorithm is carried out. The proposed algorithm is applied to many problem instances in the QAPLIB. It is revealed that the algorithm can obtain the solutions equivalent to the best known solutions in short time. Especially, for some large scale instances, the new solutions with the same cost as the best known solutions are obtained.