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# Max-Min Greedy Matching

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LIPIcs.APPROX-RANDOM.2019.7.pdf
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## Acknowledgements

A substantial part of this work was conducted in Microsoft Research, Herzlyia. We are grateful to Amos Fiat and Sella Nevo for numerous discussions that contributed significantly to the ideas presented in this paper. We also thank Robert Kleinberg for helpful discussions.

## Cite As

Alon Eden, Uriel Feige, and Michal Feldman. Max-Min Greedy Matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.7

## Abstract

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 left unmatched, if all its neighbors are already matched). The obtained matching is maximal, thus matches at least a 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 a 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 polytime 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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational pricing and auctions
• Mathematics of computing → Matchings and factors
##### Keywords
• Online matching
• Pricing mechanism
• Markets

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