We study the problem of vertex-weighted online bipartite matching with stochastic rewards where matches may fail with some known probability and the decision maker has to adapt to the sequential realization of these outcomes. Recent works have studied several special cases of this problem and it was known that the (randomized) Perturbed Greedy algorithm due to Aggarwal et al. (SODA, 2011) achieves the best possible competitive ratio guarantee of $(1-1/e)$ in some cases. We give a simple proof of these results by reducing (special cases of) the stochastic rewards problem to the deterministic setting of online bipartite matching (Karp, Vazirani, Vazirani (STOC, 1990)). More broadly, our approach gives conditions under which it suffices to analyze the competitive ratio of an algorithm for the simpler setting of deterministic rewards in order to obtain a competitive ratio guarantee for stochastic rewards. The simplicity of our approach reveals that the Perturbed Greedy algorithm has a competitive ratio of $(1-1/e)$ even in certain settings with correlated rewards, where no results were previously known. Finally, we show that without any special assumptions, the Perturbed Greedy algorithm has a competitive ratio strictly less than $(1-1/e)$ for vertex-weighted online matching with stochastic rewards.

翻译：我们研究带有顶点权重的在线二分图匹配问题，其中匹配可能以已知概率失败，决策者需自适应地应对这些结果的顺序实现。近期工作已研究该问题的若干特例，已知Aggarwal等人（SODA, 2011）提出的（随机化）扰动贪心算法在某些情况下可实现最优竞争比$(1-1/e)$。我们通过将随机奖励问题的（特例）约化为确定性在线二分图匹配问题（Karp, Vazirani, Vazirani, STOC, 1990），给出这些结果的简洁证明。更广泛地，我们的方法给出了充分条件：仅需分析算法在更简单的确定性奖励场景下的竞争比，即可获得随机奖励场景下的竞争比保证。我们的证明简洁性表明，即使在一些先前未获结果的关联奖励设定中，扰动贪心算法仍能达到$(1-1/e)$的竞争比。最后，我们证明不施加任何特殊假设时，对于带有随机奖励的顶点加权在线匹配问题，扰动贪心算法的竞争比严格小于$(1-1/e)$。