, Volume 65, Issue 4, pp 754-771,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 26 May 2012

Fixed-Parameter Evolutionary Algorithms and the Vertex Cover Problem

Abstract

In this paper, we consider multi-objective evolutionary algorithms for the Vertex Cover problem in the context of parameterized complexity. We consider two different measures for the problem. The first measure is a very natural multi-objective one for the use of evolutionary algorithms and takes into account the number of chosen vertices and the number of edges that remain uncovered. The second fitness function is based on a linear programming formulation and proves to give better results. We point out that both approaches lead to a kernelization for the Vertex Cover problem. Based on this, we show that evolutionary algorithms solve the vertex cover problem efficiently if the size of a minimum vertex cover is not too large, i.e., the expected runtime is bounded by O(f(OPT)⋅n c ), where c is a constant and f a function that only depends on OPT. This shows that evolutionary algorithms are randomized fixed-parameter tractable algorithms for the vertex cover problem.

A conference version appeared in the Proceedings of the Genetic and Evolutionary Computation Conference 2009 [15].