A Fusion Mechanism for the Generalized Asymmetric Partition Crossover

Partition crossover operators use information about the interaction between decision variables to recombine solutions. The Generalized Asymmetric Partition Crossover (GAPX) was recently proposed for the asymmetric Traveling Salesman Problem (TSP). Unlike former partition crossover operators, GAPX is capable of finding crossover points by splitting vertices of degree 4. GAPX also finds recombining components with more than two crossover points. The first step of GAPX is to define candidate components for recombination by finding connected components in the union graph formed by two parents. Some of the candidate components are infeasible for recombination. However, candidate components can be fused in order to create new recombining components. We introduce a fusion mechanism that allows GAPX to find more recombining components. When $k$ recombining components are found, GAPX generates the best of 2k offspring at cost O(n). When two local optima are recombined by partition crossover, the offspring is very often a local optimum. Fusion can be used to increase k, allowing GAPX to exploit many more offspring. Experimental results show that GAPX with fusion is capable of improving solutions generated by the LKH heuristic. Very good results are also obtained by a hybrid Genetic Algorithm that uses GAPX with fusion.