On the Approximation Performance of Degree Heuristics for Matching

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.