Given an undirected graph G, an edge dominating set is a subset F of edges such that each edge not in F is adjacent to some edge in F, and computing the minimum size of an edge dominating set is known to be NP-hard. Since the size of any edge dominating set is at least half of the maximum size µ(G) of a matching in G, we study the problem of testing whether a given graph G has an edge dominating set of size ⌈µ(G)/2⌉ or not. In this paper, we prove that the problem is NP-complete, whereas we design an O*(2.0801µ(G)/2)-time and polynomial-space algorithm to the problem.