Fast EAX Algorithm Considering Population Diversity for Traveling Salesman Problems

Abstract

This paper proposes an evolutionary algorithm (EA) that is applied to the traveling salesman problem (TSP). Existing approximation methods to address the TSP known to be state-of-the-art heuristics almost exclusively utilize Lin-Kernighan local search (LKLS) and its variants. We propose an EA that does not use LKLS, and demonstrate that it is comparable with these heuristics even though it does not use them. The proposed EA uses edge assembly crossover (EAX) that is known to be an efficient and effective crossover for solving TSPs. We first propose a modified EAX algorithm that can be executed more efficiently than the original, which is 2–7 times faster. We then propose a selection model that can efficiently maintain population diversity at negligible computational cost. The edge entropy measure is used as an indicator of population diversity.

The proposed method called EAX-1AB(ENT) is applied to TSP benchmarks up to instances of 13509 cities. Experimental results reveal that EAX-1AB(ENT) with a population of 200 can almost always find optimal solutions effectively in most TSP benchmarks up to instances of 5915 cities. In the experiments, a previously proposed EAs using EAX can find an optimal solution of usa13509 with reasonable computational cost due to the fast EAX algorithm proposed in this paper. We also demonstrate that EAX-1AB(ENT) is comparable to well-known LKLS methods when relatively small populations such as 30 are used.