Greedy Approaches to Online Stochastic Matching

TL;DR

This paper studies online stochastic matching with probing constraints, analyzing the competitiveness of algorithms in adversarial and random order models, and extends classical bipartite matching results to this stochastic setting.

Contribution

It advances understanding of how classical bipartite matching results extend to stochastic probing models with online arrivals and constraints.

Findings

Analyzes competitiveness in adversarial and random order models.

Extends classical bipartite matching results to stochastic probing.

Provides insights into algorithm performance under probing constraints.

Abstract

Within the context of stochastic probing with commitment, we consider the online stochastic matching problem; that is, the one-sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist based on edge probabilities that become known when an online vertex arrives. If a probed edge exists, it must be used in the matching (if possible). We consider the competitiveness of online algorithms in both the adversarial order model (AOM) and the random order model (ROM). More specifically, we consider a bipartite stochastic graph $G = (U,V,E)$ where $U$ is the set of offline vertices, $V$ is the set of online vertices and $G$ has edge probabilities $(p_{e})_{e \in E}$ and edge weights $(w_{e})_{e \in E}$. Additionally, $G$ has probing constraints $(\scr{C}_{v})_{v \in V}$, where $\scr{C}_v$ indicates which sequences of edges adjacent to an online vertex $v$ can be probed. We assume that $U$ is known in advance, and that $\scr{C}_v$, together with the edge probabilities and weights adjacent to an online vertex are only revealed when the online vertex arrives. This model generalizes the various settings of the classical bipartite matching problem, and so our main contribution is in making progress towards understanding which classical results extend to the stochastic probing model.