Revised: February 17, 2021
Published: April 13, 2022
Abstract: [Plain Text Version]
A bipartite graph $G(U,V;E)$ that admits a perfect matching is given. One player imposes a permutation $\pi$ over $V$, the other player imposes a permutation $\sigma$ over $U$. In the greedy matching algorithm, vertices of $U$ arrive in order $\sigma$ and each vertex is matched to the highest (under $\pi$) 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 $\pi$, and the second (min) player responds with the worst possible $\sigma$ for $\pi$. Does there exist a permutation $\pi$ ensuring to match strictly more than half of the vertices? Can such a permutation 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 $\pi$ for which for every $\sigma$ at least $\rho > 0.51$ fraction of the vertices of $V$ are matched. We provide additional lower and upper bounds for special families of graphs, including regular and Hamiltonian graphs. Our solution solves an open problem regarding the welfare guarantees attainable by pricing in sequential markets with binary unit-demand valuations.
A conference version of this paper appeared in the Proceedings of the 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems, 2019 (APPROX'19).