Greedy Local Search and Vertex Cover in Sparse Random Graphs

Recently, various randomized search heuristics have been studied for the solution of the minimum vertex cover problem, in particular for sparse random instances according to the G (n ,c /n ) model, where c > 0 is a constant. Methods from statistical physics suggest that the problem is easy if c < e . This work starts with a rigorous explanation for this claim based on the refined analysis of the Karp-Sipser algorithm by Aronson et al. Subsequently, theoretical supplements are given to experimental studies of search heuristics on random graphs. For c < 1, a greedy and randomized local-search heuristic finds an optimal cover in polynomial time with a probability arbitrarily close to 1. This behavior relies on the absence of a giant component. As an additional insight into the randomized search, it is shown that the heuristic fails badly also on graphs consisting of a single tree component of maximum degree 3.