Mehta and Panigrahi (2012) proposed Online Matching with Stochastic Rewards, which generalizes the Online Bipartite Matching problem of Karp, Vazirani, and Vazirani (1990) by associating the edges with success probabilities. This new feature captures the pay-per-click model in online advertising. Recently, Huang and Zhang (2020) studied this problem under the online primal dual framework using the Configuration Linear Program (LP), and got the best known competitive ratios of the Stochastic Balance algorithm. Their work suggests that the more expressive Configuration LP is more suitable for this problem than the Matching LP. This paper advances the theory of Configuration LP in two directions. Our technical contribution includes a characterization of the joint matching outcome of an offline vertex and \emph{all its neighbors}. This characterization may be of independent interest, and is aligned with the spirit of Configuration LP. By contrast, previous analyses of Ranking generally focus on only one neighbor. Second, we designed a Stochastic Configuration LP that captures a stochastic benchmark proposed by Goyal and Udwani (2020), who used a Path-based LP. The Stochastic Configuration LP is smaller and simpler than the Path-based LP. Moreover, using the new LP we improved the competitive ratio of Stochastic Balance from $0.596$ to $0.611$ when the success probabilities are infinitesimal, and to $0.613$ when the success probabilities are further equal.

翻译：Mehta和Panigrahi（2012）提出了带有随机奖励的在线匹配问题，该问题通过为边关联成功概率，推广了Karp、Vazirani和Vazirani（1990）的在线二分匹配问题。这一新特性捕捉了在线广告中的按点击付费模型。近期，Huang和Zhang（2020）在在线原始对偶框架下利用配置线性规划研究了该问题，并得到了随机平衡算法已知最优的竞争比。他们的工作表明，相比匹配线性规划，表达能力更强的配置线性规划更适合该问题。本文在配置线性规划理论的两个方向取得进展。我们的技术贡献包括对离线顶点及其所有邻居的联合匹配结果的特征刻画。这一特征刻画可能具有独立研究价值，且与配置线性规划的思想相契合。相比之下，先前对Ranking算法的分析通常仅聚焦于单个邻居。其次，我们设计了随机配置线性规划，以刻画Goyal和Udwani（2020）提出的随机基准（他们使用了基于路径的线性规划）。随机配置线性规划比基于路径的线性规划更简洁且更易处理。此外，利用新线性规划，我们将随机平衡算法在成功概率无穷小时竞争比从$0.596$提升至$0.611$，当成功概率进一步相等时提升至$0.613$。