Improved Analysis of RANKING for Online Vertex-Weighted Bipartite Matching in the Random Order Model

In this paper, we consider the online vertex-weighted bipartite matching problem in the random arrival model. We consider the generalization of the RANKING algorithm for this problem introduced by Huang, Tang, Wu, and Zhang (TALG 2019), who show that their algorithm has a competitive ratio of 0.6534. We show that assumptions in their analysis can be weakened, allowing us to replace their derivation of a crucial function $g$ on the unit square with a linear program that computes the values of a best possible $g$ under these assumptions on a discretized unit square. We show that the discretization does not incur much error, and show computationally that we can obtain a competitive ratio of 0.6629. To compute the bound over our discretized unit square we use parallelization, and still needed two days of computing on a 64-core machine. Furthermore, by modifying our linear program somewhat, we can show computationally an upper bound on our approach of 0.6688; any further progress beyond this bound will require either further weakening in the assumptions of $g$ or a stronger analysis than that of Huang et al.