# Greedy Matching: Guarantees and Limitations

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## Abstract

Since Tinhofer proposed the MinGreedy algorithm for maximum cardinality matching in 1984, several experimental studies found the randomized algorithm to perform excellently for various classes of random graphs and benchmark instances. In contrast, only few analytical results are known. We show that MinGreedy cannot improve on the trivial approximation ratio of \(\frac{1}{2}\) whp., even for bipartite graphs. Our hard inputs seem to require a small number of high-degree nodes. This motivates an investigation of greedy algorithms on graphs with maximum degree \(\varDelta \): we show that MinGreedy achieves a \({\frac{{\varDelta }-1}{2{\varDelta }-3}} \)-approximation for graphs with \({\varDelta } {=} 3\) and for \(\varDelta \)-regular graphs, and a guarantee of \({\frac{{\varDelta }-1/2}{2{\varDelta }-2}} \) for graphs with maximum degree \({\varDelta } \). Interestingly, our bounds even hold for the deterministic MinGreedy that breaks all ties arbitrarily. Moreover, we investigate the limitations of the greedy paradigm, using the model of *priority algorithms* introduced by Borodin, Nielsen, and Rackoff. We study deterministic priority algorithms and prove a \({\frac{{\varDelta }-1}{2{\varDelta }-3}}\)-inapproximability result for graphs with maximum degree \({\varDelta } \); thus, these greedy algorithms do not achieve a \(\frac{1}{2} {+} \varepsilon \)-approximation and in particular the \(\frac{2}{3}\)-approximation obtained by the deterministic MinGreedy for \({\varDelta } {=} 3\) is optimal in this class. For *k*-uniform hypergraphs we show a tight \(\frac{1}{k}\)-inapproximability bound. We also study fully randomized priority algorithms and give a \(\frac{5}{6}\)-inapproximability bound. Thus, they cannot compete with matching algorithms of other paradigms.

## Keywords

Maximum matching Greedy algorithms Approximation algorithms Priority algorithms Randomized algorithms## Notes

### Acknowledgments

The authors would like to thank Georg Schnitger for many helpful discussions, and Allan Borodin and Nicolas Peña for their valuable comments on an early draft of the paper.

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