🤖 AI Summary
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.
📝 Abstract
Stochastic dynamic matching problems have recently gained attention in the stochastic-modeling community due to their diverse applications, such as supply-chain management and kidney exchange programs. In this paper, we study a matching problem where items of different classes arrive according to independent Poisson processes. Unmatched items are stored in a queue, and compatibility between items is represented by a simple graph, where items can be matched if their classes are connected. We analyze matching policies in terms of stability, delay, and long-term matching rate optimization. Our approach relies on the conservation equation, which ensures a balance between arrivals and departures in any stable system. Our main contributions are as follows. We establish a link between the existence of stable policies, the dimensionality of the solution set of the conservation equation, and the compatibility graph's structure. We describe the convex polytope formed by non-negative solutions to the conservation equation, and we design policies that can achieve or closely approximate the vertices of this polytope. Lastly, we discuss potential extensions of our results beyond the main assumptions of this paper.