An Efficient Hybrid Genetic Algorithm for the Quadratic Traveling Salesman Problem

Abstract

The traveling salesman problem (TSP) is the most well-known problem in combinatorial optimization which has been studied for many decades. This paper focuses on dealing with one of the most difficult TSP variants named the quadratic traveling salesman problem (QTSP) that has numerous planning applications in robotics and bioinformatics. The goal of QTSP is similar to TSP which finds a cycle visiting all nodes exactly once with minimum total costs. However, the costs in QTSP are associated with three vertices traversed in succession (instead of two like in TSP). This leads to a quadratic objective function that is much harder to solve. To efficiently solve the problem, we propose a hybrid genetic algorithm including a local search procedure for intensification and a new mutation operator for diversification. The local search is composed of a restricted double-bridge move (a variant of 4-Opt); and we show the neighborhood can be evaluated in O(n2), the same complexity as for the classical TSP. The mutation phase is inspired by a ruin-and-recreate scheme. Experimental results conducted on benchmark instances show that our method significantly outperforms state-of-the-art algorithms in terms of solution quality. Out of the 800 instances tested, it finds 437 new best-known solutions.