On the Perturbation Function of Ranking and Balance for Weighted Online Bipartite Matching

Ranking and Balance are arguably the two most important algorithms in the online matching literature. They achieve the same optimal competitive ratio of $1-1/e$ for the integral version and fractional version of online bipartite matching by Karp, Vazirani, and Vazirani (STOC 1990) respectively. The two algorithms have been generalized to weighted online bipartite matching problems, including vertex-weighted online bipartite matching and AdWords, by utilizing a perturbation function. The canonical choice of the perturbation function is $f(x)=1-e^{x-1}$ as it leads to the optimal competitive ratio of $1-1/e$ in both settings. We advance the understanding of the weighted generalizations of Ranking and Balance in this paper, with a focus on studying the effect of different perturbation functions. First, we prove that the canonical perturbation function is the \emph{unique} optimal perturbation function for vertex-weighted online bipartite matching. In stark contrast, all perturbation functions achieve the optimal competitive ratio of $1-1/e$ in the unweighted setting. Second, we prove that the generalization of Ranking to AdWords with unknown budgets using the canonical perturbation function is at most $0.624$ competitive, refuting a conjecture of Vazirani (2021). More generally, as an application of the first result, we prove that no perturbation function leads to the prominent competitive ratio of $1-1/e$ by establishing an upper bound of $1-1/e-0.0003$. Finally, we propose the online budget-additive welfare maximization problem that is intermediate between AdWords and AdWords with unknown budgets, and we design an optimal $1-1/e$ competitive algorithm by generalizing Balance.