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.

翻译：排序与平衡堪称在线匹配文献中最重要的两类算法。针对Karp、Vazirani与Vazirani（STOC 1990）提出的在线二分匹配的整数版本与分数版本，这两类算法分别达到了相同的$1-1/e$最优竞争比。通过引入扰动函数，这两类算法已被推广至加权在线二分匹配问题，包括顶点加权在线二分匹配与AdWords。经典扰动函数选取$f(x)=1-e^{x-1}$，因其在这两种场景下均能达到$1-1/e$的最优竞争比。本文聚焦不同扰动函数的影响，深化了对排序与平衡加权泛化的理解。首先，我们证明经典扰动函数是顶点加权在线二分匹配中唯一的最优扰动函数。与之形成鲜明对比的是，在非加权场景中所有扰动函数均能达到$1-1/e$的最优竞争比。其次，我们证明采用经典扰动函数将排序算法推广至预算未知的AdWords问题时，竞争比至多为$0.624$，从而否定了Vazirani（2021）的猜想。更一般地，作为第一个结果的应用，我们通过建立$1-1/e-0.0003$的上界，证明不存在能达成著名$1-1/e$竞争比的扰动函数。最后，我们提出介于AdWords与预算未知AdWords之间的在线预算可加福利最大化问题，并通过推广平衡算法设计了$1-1/e$的最优竞争比算法。