Runtime Analysis of RLS and (1+1) EA for the Dynamic Weighted Vertex Cover Problem

Abstract: In this paper, we perform theoretical analyses on the behaviour of an evolutionary algorithm and a randomised search algorithm for the dynamic vertex cover problem based on its dual formulation. The dynamic vertex cover problem has already been theoretically investigated to some extent and it has been shown that using its dual formulation to represent possible solutions can lead to a better approximation behaviour. We improve some of the existing results, i.e. we find a linear expected re-optimization time for a (1+1) EA to re-discover a 2-approximation when edges are dynamically deleted from the graph. Furthermore, we investigate a different setting for applying the dynamism to the problem, in which a dynamic change happens at each step with a probability $P_D$. We also expand these analyses to the weighted vertex cover problem, in which weights are assigned to vertices and the goal is to find a cover set with minimum total weight. Similar to the classical case, the dynamic changes that we consider on the weighted vertex cover problem are adding and removing edges to and from the graph. We aim at finding a maximal solution for the dual problem, which gives a 2-approximate solution for the vertex cover problem. This is equivalent to the maximal matching problem for the classical vertex cover problem.