Shortest Augmenting Paths for Online Matchings on Trees

Shortest Augmenting Paths for Online Matchings on Trees The shortest augmenting path (Sap) algorithm is one of the most classical approaches to the maximum matching and maximum flow problems, e.g., using it Edmonds and Karp (J. ACM 19(2), 248–264 1972) have shown the first strongly polynomial time algorithm for the maximum flow problem. Quite astonishingly, although it has been studied for many years already, this approach is far from being fully understood. This is exemplified by the online bipartite matching problem. In this problem a bipartite graph G = (W ⊎ B, E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. After arrival of this vertex we augment the current matching by using shortest augmenting path. It was conjectured by Chaudhuri et al. (INFOCOM’09) that the total length of all augmenting paths found by Sap is O ( n log n ) $\mathcal {O}(n \log n)$ . However, no better bound than O ( n 2 ) $\mathcal {O}(n^{2})$ is known even for trees. In this paper we prove an O ( n log 2 n ) $\mathcal {O}(n \log ^{2}n)$ upper bound for the total length of augmenting paths for trees. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Theory of Computing Systems Springer Journals

Shortest Augmenting Paths for Online Matchings on Trees

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Publisher
Springer US
Copyright
Copyright © 2018 by The Author(s)
Subject
Computer Science; Theory of Computation
ISSN
1432-4350
eISSN
1433-0490
D.O.I.
10.1007/s00224-017-9838-x
Publisher site
See Article on Publisher Site

Abstract

The shortest augmenting path (Sap) algorithm is one of the most classical approaches to the maximum matching and maximum flow problems, e.g., using it Edmonds and Karp (J. ACM 19(2), 248–264 1972) have shown the first strongly polynomial time algorithm for the maximum flow problem. Quite astonishingly, although it has been studied for many years already, this approach is far from being fully understood. This is exemplified by the online bipartite matching problem. In this problem a bipartite graph G = (W ⊎ B, E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. After arrival of this vertex we augment the current matching by using shortest augmenting path. It was conjectured by Chaudhuri et al. (INFOCOM’09) that the total length of all augmenting paths found by Sap is O ( n log n ) $\mathcal {O}(n \log n)$ . However, no better bound than O ( n 2 ) $\mathcal {O}(n^{2})$ is known even for trees. In this paper we prove an O ( n log 2 n ) $\mathcal {O}(n \log ^{2}n)$ upper bound for the total length of augmenting paths for trees.

Journal

Theory of Computing SystemsSpringer Journals

Published: Jan 24, 2018

References

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