An optimal recombination operator for two-parent solutions provides the best solution among those that take the value for each variable from one of the parents (gene transmission property). If the solutions are bit strings, the offspring of an optimal recombination operator is optimal in the smallest hyperplane containing the two parent solutions. Exploring this hyperplane is computationally costly, in general, requiring exponential time in the worst case. However, when the variable interaction graph of the objective function is sparse, exploration can be done in polynomial time. In this article, we present a recombination operator, called Dynastic Potential Crossover (DPX), that runs in polynomial time and behaves like an optimal recombination operator for low-epistasis combinatorial problems. We compare this operator, both theoretically and experimentally, with traditional crossover operators, like uniform crossover and network crossover, and with two recently defined efficient recombination operators: partition crossover and articulation points partition crossover. The empirical comparison uses NKQ Landscapes and MAX-SAT instances. DPX outperforms the other crossover operators in terms of quality of the offspring and provides better results included in a trajectory and a population-based metaheuristic, but it requires more time and memory to compute the offspring.

Generalized Partition Crossover (GPX) is a deterministic recombination operator developed for the Traveling Salesman Problem. Partition crossover operators return the best of 2 k reachable offspring, where k is the number of recombining components. This article introduces a new GPX2 operator, which finds more recombining components than GPX or Iterative Partial Transcription (IPT). We also show that GPX2 has O( n ) runtime complexity, while also introducing new enhancements to reduce the execution time of GPX2. Finally, we experimentally demonstrate the efficiency of GPX2 when it is used to improve solutions found by the multitrial Lin-Kernighan-Helsgaum (LKH) algorithm. Significant improvements in performance are documented on large ( n > 5000 ) and very large ( n = 100 , 000 ) instances of the Traveling Salesman Problem.