Quasi-Optimal Recombination Operator

The output of an optimal recombination operator for two parent solutions is a solution with the best possible value for the objective function among all the solutions fulfilling the gene transmission property: the value of any variable in the offspring must be inherited from one of the parents. This set of solutions coincides with the largest dynastic potential for the two parent solutions of any recombination operator with the gene transmission property. In general, exploring the full dynastic potential is computationally costly, but if the variables of the objective function have a low number of non-linear interactions among them, the exploration can be done in \(O(4^{\beta }(n+m)+n^2)\) time, for problems with n variables, m subfunctions and \(\beta \) a constant. In this paper, we propose a quasi-optimal recombination operator, called Dynastic Potential Crossover (DPX), that runs in \(O(4^{\beta }(n+m)+n^2)\) time in any case and is able to explore the full dynastic potential for low-epistasis combinatorial problems. We compare this operator, both theoretically and experimentally, with two recently defined efficient recombination operators: Partition Crossover (PX) and Articulation Points Partition Crossover (APX). The empirical comparison uses NKQ Landscapes and MAX-SAT instances.