## Abstract

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.

Original language | American English |
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Article number | 6 |

Journal | Theory of Computing |

Volume | 18 |

DOIs | |

State | Published - 2022 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2022 Alon Eden, Uriel Feige, and Michal Feldman.

## Keywords

- Online matching
- markets
- pricing mechanism