Max-Min Greedy Matching

A bipartite graph G(U,V; E) that admits a perfect matching is given. One player imposes a linear order π over V, the other player imposes a linear order σ over U. In the greedy matching algorithm, vertices of U arrive in order σ and each vertex is matched to the highest (under π) yet unmatched neighbor in V (or is left unmatched, if all its neighbors are already matched). The matching obtained is maximal, thus matches at least half of the vertices. The max-min greedy matching problem asks: Suppose the first (max) player reveals π, and the second (min) player responds with the worst possible σ for π. Does there exist a linear order π ensuring to match strictly more than half of the vertices? Can such a linear order be computed in polynomial time? The main result of this paper is an affirmative answer for these questions: we show that there exists a polynomial-time algorithm to compute π for which for every σ at least ρ > 0.51 fraction of the vertices of V are matched. We provide additional lower and upper bounds for.