Max-min greedy matching

Alon Eden, Uriel Feige, Michal Feldman

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations

Abstract

A bipartite graph G(U, V ; E) that admits a perfect matching is given. One player imposes a permutation π over V , the other player imposes a permutation 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 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 π, and the second (min) player responds with the worst possible σ for π, does there exist a permutation π 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 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 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.

Original languageAmerican English
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019
EditorsDimitris Achlioptas, Laszlo A. Vegh
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771252
DOIs
StatePublished - Sep 2019
Externally publishedYes
Event22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019 - Cambridge, United States
Duration: 20 Sep 201922 Sep 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume145
ISSN (Print)1868-8969

Conference

Conference22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019
Country/TerritoryUnited States
CityCambridge
Period20/09/1922/09/19

Bibliographical note

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

Keywords

  • Markets
  • Online matching
  • Pricing mechanism

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