A Tight Analysis of the Maximal Matching Heuristic

We study the algorithm that iteratively removes adjacent vertices from a simple, undirected graph until no edge remains. This algorithm is a well-known 2-approximation to three classical NP-hard optimization problems: MINIMUM VERTEX COVER, MINIMUM MAXIMAL MATCHING and MINIMUM EDGE DOMINATING SET. We show that the worst-case approximation factor of this simple method can be expressed in a finer way when assumptions on the density of the graph is made. For graphs with an average degree at least n, called weakly -dense graphs, we show that the asymptotic approximation factor is min{2, 1/(1 − √ 1 − )}. For graphs with a minimum degree at least n – strongly -dense graphs – we show that the asymptotic approximation factor is min{2, 1/ }. These bounds are obtained through a careful analysis of the tight examples.