Estimating attraction basin sizes of combinatorial optimization problems

Abstract

Given a particular instance of a combinatorial optimization problem, the knowledge about the attraction basin sizes can help to analyze the difficulty encountered by local search algorithms while solving it. As calculating these sizes exhaustively is computationally intractable, we focus on methods for their estimation. The accuracy of some of these estimation methods depends on the way in which the sample of solutions of the search space is chosen. In this paper, we propose a novel sampling method, which incorporates the knowledge obtained by the already explored solutions into the sampling strategy. So, in contrast to those that already exist, our method can adapt its behavior to the characteristics of the particular attraction basin. We apply our proposal to a number of instances of three famous problems: the quadratic assignment problem, the linear ordering problem and the permutation flow shop scheduling problem. We consider permutation sizes n= 10 and n= 12 and three different neighborhoods: adjacent swap, 2-exchange and insert, and observe that the new method generally outperforms those that already exist.