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 -