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

Within the context of stochastic probing with commitment, we consider the online stochastic matching 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 study this problem in the generality of a patience (or budget) constraint which limits the number of probes that can be made to edges adjacent to an online node. The patience constraint results in modelling and computational efficiency issues that are not encountered in the special cases of unit patience and full (i.e., unlimited) patience. The stochastic matching problem leads to a variety of settings. Our main contribution is to provide a new LP relaxation and a unified approach for establishing new and improved competitive bounds in three different input model settings (namely, adversarial, random order, and known i.i.d.). In all these settings, the algorithm does not have any control on the ordering of the online nodes. We establish competitive bounds in these settings, all of which generalize the standard non-stochastic setting when edges do not need to be probed (i.e., exist with certainty). All of our competitive ratios hold for arbitrary edge probabilities and patience constraints: (1) A $1-1/e$ ratio when the stochastic graph is known, offline vertices are weighted and online arrivals are adversarial. (2) A $1-1/e$ ratio when the stochastic graph is known, edges are weighted, and online arrivals are given in random order (i.e., in ROM, the random order model). (3) A $1-1/e$ ratio when online arrivals are drawn i.i.d. from a known stochastic type graph and edges are weighted. (4) A (tight) $1/e$ ratio when the stochastic graph is unknown, edges are weighted and online arrivals are given in random order.