Toward a Better Understanding of Randomized Greedy Matching

There has been a long history of studying randomized greedy matching algorithms since the work by Dyer and Frieze [9]. We follow this trend and consider the problem formulated in the oblivious setting, in which the vertex set of a graph is known to the algorithm but not the edge set. The algorithm can make queries for the existence of the edge between any pair of vertices but must include the edge into the matching if it exists, i.e., as in the query-commit model by Gamlath et al. [12]. We revisit the Modified Randomized Greedy (MRG) algorithm by Aronson et al. [1] that is proved to achieve a (0.5+ε)-approximation. In each step of the algorithm, an unmatched vertex is chosen uniformly at random and matched to a randomly chosen neighbor (if exists). We study a weaker version of the algorithm named Random Decision Order (RDO) that, in each step, randomly picks an unmatched vertex and matches it to an arbitrary neighbor (if exists). We prove that the RDO algorithm provides a 0.639-approximation for bipartite graphs and 0.531-approximation for general graphs. As a corollary, we substantially improve the approximation ratio of MRG.

Furthermore, we generalize the RDO algorithm to the edge-weighted case and prove that it achieves a 0.501-approximation ratio. This result solves the open question by Chan et al. [4] and Gamlath et al. [12] about the existence of an algorithm that beats greedy in edge-weighted general graphs, where the greedy algorithm probes the edges in descending order of edge-weights. We also present a variant of the algorithm that achieves a (1-1/e)-approximation for edge-weighted bipartite graphs, which generalizes the (1-1/e)-approximation ratio of Gamlath et al. [12] for the stochastic setting to the case when the realizations of edges are arbitrarily correlated, where in the stochastic setting, there is a known probability associated with each pair of vertices that indicates the probability that an edge exists between the two vertices, when the pair is probed.