On the Complexity of Weighted Greedy Matchings

Motivated by the fact that in several cases a matching in a graph is stable if and only if it is produced by a greedy algorithm, we study the problem of computing a maximum weight greedy matching on edge-weighted graphs, denoted GreedyMatching. This problem is to compute a matching with the maximum weight when every edge that is being added to the matching is chosen among the available edges with the currently largest weight. We prove that GreedyMatching is hard even to approximate; in particular, it is APX-complete, even on bipartite graphs of maximum degree 3 and only 5 different weight values. Our results imply that, unless P=NP, there is no polynomial time approximation algorithm with ratio better than 0.9474 for graphs with integer weights and 0.889 for graphs with rational weights. By slightly modifying our reduction we prove that also the two decision variations of the problem, where a specific edge or vertex needs to be matched, are strongly NP-complete, even on bipartite graphs with 7 different weight values. On the positive side, we consider a simple randomized greedy matching algorithm, which we call Rgma, and we study its performance on the special class of bush graphs, i.e. on edge-weighted graphs where all edges with equal weight form a star (bush). We prove that Rgma achieves a 2 3 -approximation on bush graphs, where the each bush has at most two leafs, and we conjecture that the same approximation ratio is achieved even when applied to general bush graphs. The importance of the performance of Rgma on general bush graphs is highlighted by the fact that, as our results show, a ρ-approximation ratio of Rgma for GreedyMatching on general bush graphs implies that, for every ǫ < 1, the randomized algorithm MRG (first defined in [4]) achieves a (ρ − ǫ)-approximation of the maximum cardinality matching in unweighted graphs. Thus, an affirmative answer to our conjecture that ρ = 2 3 would solve a longstanding open problem [4, 17].