Prophet Matching Meets Probing with Commitment

Abstract: 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. Our algorithms respect commitment, in that if a probed edge exists, it must be used in the matching. We study this matching problem subject to a downward-closed constraint on each online node's allowable edge probes. Our setting generalizes the commonly studied patience (or time-out) constraint which limits the number of probes that can be made to an online node's adjacent edges. We introduce a new LP that we prove is a relaxation of an optimal offline probing algorithm (the adaptive benchmark) and which overcomes the limitations of previous LP relaxations. (1) A tight $\frac{1}{2}$ ratio when the stochastic graph is generated from a known stochastic type graph where the $t^{th}$ online node is drawn independently from a known distribution $\scr{D}{\pi(t)}$ 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 $t^{th}$ online node is drawn independently from a known distribution $\scr{D}{\pi(t)}$ and $\pi$ is a random permutation. We refer to this setting as the known i.d. stochastic matching problem with random order arrivals. Our results improve upon the previous best competitive ratio of $0.46$ in the known i.i.d. setting against the standard adaptive benchmark. Moreover, we are the first to study the prophet secretary matching problem in the context of probing, where we match the best known classical result.