General Graphs are Easier than Bipartite Graphs: Tight Bounds for Secretary Matching

Online algorithms for secretary matching in bipartite weighted graphs have been studied extensively in recent years. We generalize this study to secretary matching in general weighted graphs, for both vertex and edge arrival models. Under vertex arrival, vertices arrive online in a uniformly random order; upon the arrival of a vertex v, the weights of edges from v to all previously arriving vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. We provide a tight 5/12-competitive algorithm for this setting. Interestingly, it outperforms the best possible algorithm for secretary matching in bipartite graphs with 1-sided arrival, which cannot be better than 1/e-competitive. Under edge arrival, edges arrive online in a uniformly random order; upon the arrival of an edge e, its weight is revealed, and the algorithm decides whether to include it in the matching or not. For this setting we provide a 1/4-competitive algorithm, which improves upon the state of the art bound.