Bipartite Stochastic Matching: Online, Random Order, and I.I.D. Models

Within the context of stochastic probing with commitment, we consider the stochastic rewards problem, that is, the one sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist, based on known edge probabilities. If a probed edge exists, it must be used in the matching (if possible). We consider the competitiveness of online algorithms in four different models. Namely, we consider the following stochastic reward problems: when the stochastic graph is not known and the order of online arrivals is determined uniformly at random (i.e., the ROM model); when the stochastic graph is known to the algorithm and the order of arrival of online vertices is determined adversarially; when the stochastic graph is known and the order of arrival of online vertices is determined uniformly at random; and finally when the online vertices are generated i.i.d. from a known distribution on the vertices of a known stochastic (type) graph. In all these settings, the algorithm does not have any control on the ordering of the online nodes. We study these models under various assumptions on the patience (number of given probes) of the online vertices, and whether edges or offline vertices are weighted in determining the stochastic reward. Our main contribution is to establish new and improved competitive bounds in these four models. In all settings, there is one ideal benchmark (the optimal offline probing algorithm) relative to which the competitive ratio is defined. For establishing competitive ratios we introduce a new LP relaxation which upper bounds the performance of the ideal benchmark.