Analysis of population-based evolutionary algorithms for the vertex cover problem

Recently it has been proved that the (1+1)-EA produces poor worst-case approximations for the vertex cover problem. In this paper the result is extended to the (1+lambda)-EA by proving that, given a polynomial time, the algorithm can only find poor covers for an instance class of bipartite graphs. Although the generalisation of the result to the (mu+1)-EA is more difficult, hints are given in this paper to show that this algorithm may get stuck on the local optimum of bipartite graphs as well because of premature convergence. However a simple diversity maintenance mechanism can be introduced into the EA for optimising the bipartite instance class effectively. It is proved that the diversity mechanism combined with one point crossover can change the runtime for some instance classes from exponential to polynomial in the number of nodes of the graph.