TY - GEN

T1 - Max-min greedy matching

AU - Eden, Alon

AU - Feige, Uriel

AU - Feldman, Michal

N1 - Publisher Copyright:
© Alon Eden, Uriel Feige, and Michal Feldman.

PY - 2019/9

Y1 - 2019/9

N2 - 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.

AB - 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.

KW - Markets

KW - Online matching

KW - Pricing mechanism

UR - http://www.scopus.com/inward/record.url?scp=85072862319&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.APPROX-RANDOM.2019.7

DO - 10.4230/LIPIcs.APPROX-RANDOM.2019.7

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AN - SCOPUS:85072862319

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019

A2 - Achlioptas, Dimitris

A2 - Vegh, Laszlo A.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019

Y2 - 20 September 2019 through 22 September 2019

ER -