TY - GEN

T1 - The demand query model for bipartite matching

AU - Nisan, Noam

N1 - Publisher Copyright: Copyright © 2021 by SIAM

PY - 2021

Y1 - 2021

N2 - We introduce a “concrete complexity” model for studying algorithms for matching in bipartite graphs. The model is based on the “demand query” model used for combinatorial auctions. Most (but not all) known algorithms for bipartite matching seem to be translatable into this model including exact, approximate, sequential, parallel, and online ones. A perfect matching in a bipartite graph can be found in this model with O(n3/2) demand queries (in a bipartite graph with n vertices on each side) and our main open problem is to either improve the upper bound or prove a lower bound. An improved upper bound could yield “normal” algorithms whose running time is better than the fastest ones known, while a lower bound would rule out a faster algorithm for bipartite matching from within a large class of algorithms. Our main result is a lower bound for finding an approximately maximum size matching in parallel: A deterministic algorithm that runs in no(1) rounds, where each round can make at most n1.99 demand queries cannot find a matching whose size is within no(1) factor of the maximum. This is in contrast to randomized algorithms that can find a matching whose size is 99% of the maximum in O(log n) rounds, each making n demand queries.

AB - We introduce a “concrete complexity” model for studying algorithms for matching in bipartite graphs. The model is based on the “demand query” model used for combinatorial auctions. Most (but not all) known algorithms for bipartite matching seem to be translatable into this model including exact, approximate, sequential, parallel, and online ones. A perfect matching in a bipartite graph can be found in this model with O(n3/2) demand queries (in a bipartite graph with n vertices on each side) and our main open problem is to either improve the upper bound or prove a lower bound. An improved upper bound could yield “normal” algorithms whose running time is better than the fastest ones known, while a lower bound would rule out a faster algorithm for bipartite matching from within a large class of algorithms. Our main result is a lower bound for finding an approximately maximum size matching in parallel: A deterministic algorithm that runs in no(1) rounds, where each round can make at most n1.99 demand queries cannot find a matching whose size is within no(1) factor of the maximum. This is in contrast to randomized algorithms that can find a matching whose size is 99% of the maximum in O(log n) rounds, each making n demand queries.

UR - http://www.scopus.com/inward/record.url?scp=85105347268&partnerID=8YFLogxK

M3 - Conference contribution

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 592

EP - 599

BT - ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

A2 - Marx, Daniel

T2 - 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

Y2 - 10 January 2021 through 13 January 2021

ER -