When Stochastic Rewards Reduce to Deterministic Rewards in Online Bipartite Matching

TL;DR

The paper addresses online vertex-weighted bipartite matching with stochastic rewards, where each potential match succeeds with probability $p_{it}$. It introduces a meta-reduction using value-preserving distributions to transfer competitive guarantees from deterministic online matching to stochastic rewards in regimes of Identical ($p_{it}=p$) and Decomposable ($p_{it}=p_i p_t$) probabilities, and extends to certain time-correlated scenarios. This approach yields the optimal $1-1/e$ competitive ratio for Perturbed Greedy in those regimes and shows the results hold under specific correlations, while proving a nontrivial upper bound of $0.624$ for general probabilities. The work thus simplifies the analysis of stochastic rewards by reducing to deterministic settings and clarifies when the classical $(1-1/e)$ frontier can be achieved; it also highlights the inherent limitations in the general case and points to directions for tighter bounds and broader correlation models.

Abstract

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.

Theorems & Definitions (13)

• Lemma 2.1: goeldevanur
• Lemma 2.2: albersvazirani
• Theorem 3.1
• Corollary 3.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Theorem 3.2
• Theorem 3.3
• Lemma 3.4