Prophet Inequality Matching Meets Probing with Commitment.

Within the context of stochastic probing with commitment, we consider the online stochastic matching problem for bipartite graphs 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). In addition to improving upon existing stochastic bipartite matching results, our results can also be seen as extensions to multi-item prophet inequalities. We study this matching problem for given constraints on the allowable sequences of probes adjacent to an online node. Our setting generalizes the patience (or time-out) constraint which limits the number of probes that can be made to edges. The generality of our setting leads to some modelling and computational efficiency issues that are not encountered in previous works. We establish new competitive bounds all of which generalize the standard non-stochastic setting when edges do not need to be probed (i.e., exist with certainty). Specifically, we establish the following competitive ratio results for a general formulation of edge constraints, arbitrary edge weights, and arbitrary edge probabilities: (1) A tight $\frac{1}{2}$ ratio when the stochastic graph is generated from a known stochastic type graph where the $\pi(i)^{th}$ online node is drawn independently from a known distribution ${\cal D}_{\pi(i)}$ and $\pi$ is chosen adversarially. We refer to this setting as the known i.d. stochastic matching problem with adversarial arrivals. (2) A $1-1/e$ ratio when the stochastic graph is generated from a known stochastic type graph where the $\pi(i)^{th}$ online node is drawn independently from a known distribution ${\cal D}_{\pi(i)}$ and $\pi$ is a random permutation. This is referred to as the known i.d. stochastic matching problem with random order arrivals.