License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2019.48
URN: urn:nbn:de:0030-drops-104528
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10452/
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### A Weighted Approach to the Maximum Cardinality Bipartite Matching Problem with Applications in Geometric Settings

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### Abstract

We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph G(A cup B,E) with edge weights of 0 or 1. Let w <= n be an upper bound on the weight of any matching in G. Consider the subgraph induced by all the edges of G with a weight 0. Suppose every connected component in this subgraph has O(r) vertices and O(mr/n) edges. We present an algorithm to compute a maximum cardinality matching in G in O~(m(sqrt{w} + sqrt{r} + wr/n)) time.
When all the edge weights are 1 (symmetrically when all weights are 0), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in O(m sqrt{n}) time. However, if we can carefully assign weights of 0 and 1 on its edges such that both w and r are sub-linear in n and wr=O(n^{gamma}) for gamma < 3/2, then we can compute maximum cardinality matching in G in o(m sqrt{n}) time. Using our algorithm, we obtain a new O~(n^{4/3}/epsilon^4) time algorithm to compute an epsilon-approximate bottleneck matching of A,B subsetR^2 and an 1/(epsilon^{O(d)}}n^{1+(d-1)/(2d-1)}) poly log n time algorithm for computing epsilon-approximate bottleneck matching in d-dimensions. All previous algorithms take Omega(n^{3/2}) time. Given any graph G(A cup B,E) that has an easily computable balanced vertex separator for every subgraph G'(V',E') of size |V'|^{delta}, for delta in [1/2,1), we can apply our algorithm to compute a maximum matching in O~(mn^{delta/1+delta}) time improving upon the O(m sqrt{n}) time taken by the HK-Algorithm.

### BibTeX - Entry

```@InProceedings{lahn_et_al:LIPIcs:2019:10452,
author =	{Nathaniel Lahn and Sharath Raghvendra},
title =	{{A Weighted Approach to the Maximum Cardinality Bipartite Matching Problem with Applications in Geometric Settings}},
booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
pages =	{48:1--48:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-104-7},
ISSN =	{1868-8969},
year =	{2019},
volume =	{129},
editor =	{Gill Barequet and Yusu Wang},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10452},
URN =		{urn:nbn:de:0030-drops-104528},
doi =		{10.4230/LIPIcs.SoCG.2019.48},
annote =	{Keywords: Bipartite matching, Bottleneck matching}
}
```

 Keywords: Bipartite matching, Bottleneck matching Collection: 35th International Symposium on Computational Geometry (SoCG 2019) Issue Date: 2019 Date of publication: 11.06.2019

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