Greedy Online Matching on Random Graphs∗

In the online bipartite matching problem, vertices in one partition are given up front and vertices in the other partition arrive sequentially. When a vertex arrives, it reveals its neighboring edges and it must be immediately matched or left unmatched. We determine asymptotic matching sizes given by a variety of greedy online matching algorithms on random graphs. On G(n, n, p), we show that an entire class of greedy algorithms has a performance ratio of at least 0.837 for all functions p = p(n), where the performance ratio is defined as the asymptotic expected ratio of matching size given by the algorithm to the maximum matching size. This class includes the Ranking algorithm of Karp et al. [12]. On the random regular graph G(n, n, r) with r = 2, we show that the performance ratios of a random greedy algorithm and a degree greedy algorithm are 0.877 and 0.917, respectively. Analogous results are shown for a non-bipartite online matching problem.