Exploration and Exploitation Bias of Crossover and Path Relinking for Permutation Problems

For most combinatorial optimization problems the computational complexity of evaluating a single solution is much higher than the cost of evaluating an incremental change made by a local search operator. To benefit from this computational gain crossover can be implemented as a path–following algorithm. As a result crossover becomes more similar to path relinking. In this paper we compare the search bias of crossover and path relinking for permutation problems where the absolute position of the elements is decisive. Calculations show that uniform permutation crossover (UPX) can reach many more permutations from a given parent couple than path relinking. UPX is therefore more exploratory than random path relinking, which is itself more exploratory than greedy path relinking. It is important for users to understand the differences in search bias of the operators so they can choose the exploration operator which they deem most fit for their problem. We conclude with a small experiment on an instance of the quadratic assignment problem.