Online Stochastic Matching: A Polytope Perspective
This paper studies online stochastic matching under graph-theoretic compatibility constraints, where items of distinct classes arrive according to independent Poisson processes, compatibility is encoded by an undirected graph, and unmatched items are queued. Targeting the joint optimization of stability, matching delay, and long-run matching rate, we establish—for the first time—the fundamental connection between the existence of stable policies, the dimension of the convex polyhedron formed by nonnegative solutions to conservation equations, and the structural properties of the compatibility graph. We propose a novel policy design paradigm wherein performance bounds are characterized by the vertices of this polyhedron. Leveraging stochastic process modeling, graph theory, and convex analysis, we construct stable policies that either achieve or approximate these vertex bounds. These policies maximize the long-run matching rate while ensuring system stability and yield tight theoretical bounds on matching delay in terms of graph structure.