On the Approximation Performance of Degree Heuristics for Matching

TL;DR

This paper analyzes the approximation performance of degree-sensitive greedy algorithms for maximum matching, establishing tight bounds and identifying optimal heuristics like Karp-Sipser.

Contribution

It introduces the class of degree sensitive greedy algorithms and proves their optimal approximation guarantees, including the Karp-Sipser algorithm.

Findings

Karp-Sipser achieves optimal approximation guarantee D/(2D-2)

Degree-sensitive heuristics have tight worst-case bounds

The analysis applies to bipartite graphs

Abstract

In the design of greedy algorithms for the maximum cardinality matching problem the utilization of degree information when selecting the next edge is a well established and successful approach. We define the class of "degree sensitive" greedy matching algorithms, which allows us to analyze many well-known heuristics, and provide tight approximation guarantees under worst case tie breaking. We exhibit algorithms in this class with optimal approximation guarantee for bipartite graphs. In particular the Karp-Sipser algorithm, which picks an edge incident with a degree-1 node if possible and otherwise an arbitrary edge, turns out to be optimal with approximation guarantee D/(2D-2), where D is the maximum degree.