# Online Bipartite Matching with Amortized O(log 2 n) Replacements

@article{Bernstein2019OnlineBM, title={Online Bipartite Matching with Amortized O(log 2 n) Replacements}, author={Aaron Bernstein and Jacob Holm and Eva Rotenberg}, journal={Journal of the ACM (JACM)}, year={2019}, volume={66}, pages={1 - 23} }

In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one-by-one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O(log 2 n…

## 27 Citations

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This paper proposes to consider the general question of how requiring a non-amortized hard budget $k$ on the number of reassignments affects the algorithms' performances, under various models from the literature.

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- Computer ScienceLATIN
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A tight \(O(n \log n)\) upper bound is proved for the total length of shortest augmenting paths for trees improving over \(O(\log ^2 n) bound" in this paper.

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This work shows an O(1)competitive algorithm for online matching on the line with amortized recourse of O(log n), the first non-trivial result for min-cost bipartite matching with recourse, and gives a (1+ε)-competitive algorithm that reassigns any request at most O(ε−1) times.

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This work gives a $(1+\varepsilon)-competitive algorithm that reassigns any request at most $O(n\log n)$ times, the first non-trivial result for min-cost bipartite matching with recourse and obtains a near-optimal result.

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A two-stage model where the trade-off between quality and robustness of solutions is studied, and there exists an algorithm that is $(3,1)$-robust for any metric if one knows the number of arriving nodes in advance.

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This master thesis focuses on randomized composable coresets as building blocks for memoryefficient algorithms in the context of massive graphs, and proposes a new coreset for bipartite graphs with size O(n) which is based on the notion of server flows of [21].

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It is shown that no deterministic algorithm better than $1+1/(k-1)$ exists, and an improvement of the greedy algorithm is presented, which for small values of $k$ outperforms the algorithm of [AMP, 2013].

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