Prophet Inequality for Bipartite Matching: Merits of Being Simple and Nonadaptive

Abstract

We consider the Bayesian online selection problem of a matching in bipartite graphs, that is, the weighted online matching problem where the edges arrive online and edge weights are generated from a known distribution. This setting corresponds to the intersection of two matroids in the work of Kleinberg and Weinberg [ 40 ] and Feldman et al. [ 27 ]. We study a simple class of nonadaptive policies that we call vertex-additive policies. A vertex-additive policy assigns static prices to every vertex in the graph and accepts only those edges whose weight exceeds the sum of the prices on the edge endpoints. We show that there exists a vertex-additive policy with the expected payoff of at least one-third of the prophet’s payoff and present a gradient descent algorithm that quickly converges to the desired vector of vertex prices. Our results improve on the adaptive online policies of Kleinberg and Weinberg and Feldman et al. for the intersection of two matroids in two ways: our policy is nonadaptive and has a better approximation guarantee of 3 instead of the previous guarantees of 5.82 in Kleinberg and Weinberg and 5.43 in Feldman et al. We give a complementary lower bound of 2.25 for any online algorithm in the bipartite matching setting. Funding: This work was supported by Fundamental Research Funds of the Central Universities in China, a Science and Technology Innovation 2030 major project [“New Generation of Artificial Intelligence,” Project 2018AAA0100903], the Shanghai Municipal Education Commission Innovation Program, and the Program for Innovative Research Team of the Shanghai University of Finance and Economics (IRTSHUFE).