The $(1+(\lambda,\lambda))$ Genetic Algorithm on the Vertex Cover Problem: Crossover Helps Leaving Plateaus

Many discrete optimization problems feature plateaus, which are hard to evolutionary algorithms due to the lack of fitness guidance. While higher mutation rates may assist in making a jump from the plateau to some better search point, an algorithm typically performs random walks on a plateau, possibly with some assistance from diversity mechanisms. The vertex cover problem is one of the important NP-hard problems. We found that the recently proposed
$(1+(\lambda, \lambda))$
genetic algorithm solves certain instances of this problem, including those that are hard to heuristic solvers, much faster than simpler mutation-only evolutionary algorithms. Our theoretical analysis shows that there exists an intricate interplay between the problem structure and the way crossovers are used. It results in a drift towards the points where finding the next improvement is much easier. While this condition is formally proven only on one class of instances and for a subset of search points, experiments show that it is responsible for performance improvements in a much larger range of cases.