Certifying Capacity and Gaps of the Rank-Dependent Gain-Sharing Framework for Ranking

The randomized primal-dual framework of Devanur, Jain, and Kleinberg (SODA 2013) splits each unit matching gain between endpoint dual variables according to the matched offline rank. The gain-sharing function affects the certificate but not Ranking's decisions. Keeping the splitting rule and expected edgewise feasibility criterion fixed, we define certifying capacity as the strongest uniform guarantee obtainable by functional optimization over a prescribed function class. Its shortfall from Ranking's actual competitiveness is the certifying gap. We study unweighted online bipartite matching under adversarial arrivals on graphs with online degree at most $d$ and on $d$-regular graphs. The optimized protocol is tight on both degree-one classes and in the unrestricted setting, but has a strictly positive gap on both degree-two classes, even over all measurable nonnegative complementary splits. In both degree-two settings, we determine Ranking's exact competitiveness and the exact certifying capacity over nondecreasing gain-sharing functions. For general $d$, we optimize over nondecreasing, right-continuous functions $f:[0,1]\to[0,1]$ with $f(1)=1$, using their probability-measure representation. Both the certifying capacity and actual competitiveness converge to $1-1/e$ in both instance families as $d\to\infty$. The resulting gap is at most $1/(2ed)+O(d^{-2})$ for online degree at most $d$; on $d$-regular graphs, it vanishes but is at least $(1/(2e)-o(1))/d$. These results distinguish function-class restrictions from limitations of the fixed protocol and motivate tighter randomized primal-dual frameworks.