Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities

The rich literature on online Bayesian selection problems has long focused on so-called prophet inequalities, which compare the gain of an online algorithm to that of a “prophet” who knows the future. An equally natural, though significantly less well-studied, benchmark is the optimum online algorithm, which may be omnipotent (i.e., computationally unbounded), but not omniscient. What is the computational complexity of the optimum online? How well can a polynomial-time algorithm approximate it? Motivated by applications in ride hailing, we study the above questions for the online stochastic maximum-weight matching problem under vertex arrivals. For this problem, a number of $1/2$-competitive algorithms are known. This is the best possible ratio for this problem, as it generalizes the original single-item prophet inequality problem. We present a polynomial-time algorithm, which approximates the optimal online algorithm within a factor of 0.51—strictly more than the best-possible constant against a prophet. In contrast, we show that it is PSPACE-hard to approximate this problem within some universal constant $\alpha <1$.

Funding: Financial support from the National Science Foundation [Grants CCF1763970, CCF1812919, and CCF191070], the Office of Naval Research [Grant N000141912550], and Cisco Research is gratefully acknowledged.