Quasi-Optimal Recombination Operator

Abstract

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 β (n + m) + n 2) time, for problems with n variables, m subfunctions and β a constant. In this paper, we propose a quasi-optimal recombination operator, called Dynastic Potential Crossover (DPX), that runs in O(4 β (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.

Dynastic Potential Crossover Operator

“…This work extends a conference paper (Chicano et al, 2019) where DPX was first presented. The new contributions of this journal version are as follows:…”

Local optima network analysis for MAX-SAT

“…We used benchmark instances having a low number of variables in order to be able to compute the global optimum in all of them. In particular, we used instances with 40 variables, which is the minimum number of variables found in instances of the MAX-SAT Evaluation 2017 1 . The ratio of the number of clauses to the number of variables is known to have an impact on the search difficulty [13].…”

“…For all the instances we computed the global optimum using an exact method, in order to be certain that the sampling processes reached it. Instead of using an exhaustive enumeration (which could take a long time) we applied the recently proposed Dynastic Potential Crossover Operator (DPX) [1]. This operator is able to explore the full dynastic potential of two parent solutions providing the best among the solutions in that space.…”