  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.
