Locally Sparsified, Globally Near-Optimal: Matching under Independent Vertex Arrivals

2026-09-22

AI Summary

研究在独立顶点到达情况下，通过限制每个请求的选项数量来实现近似最优匹配的方法，证明了有限局部菜单足以达到全局近似最优。

Abstract

Resource allocation systems often restrict each request to a short list of options before coordinating assignments globally. We study this separation in stochastic bipartite matching under independent vertex arrivals. Each request draws a state from its own known distribution, determining its compatible resources, and independently retains a menu of at most $k$ edges. A maximum matching is then computed on the retained graph. We show that bounded local menus universally suffice for near-optimal matching. For every $\varepsilon>0$, there is a menu size $k_\varepsilon$ depending only on $\varepsilon$ that preserves at least a $(1-\varepsilon)$ fraction of the expected maximum-matching size of the full realized graph. Earlier guarantees required additional assumptions on how matching mass is distributed across edges; our result resolves the unrestricted case. Moreover, the menus are simple to generate from any benchmark matching rule, either by weighted sampling according to the benchmark's edge marginals, or by applying the benchmark to sampled realizations and retaining the resulting partners. Our proof constructs a near-optimal certificate inside the sparsifier by combining a \emph{locally computable} surrogate for the large-marginal edges with a fractional completion from sampled light edges. The surrogate nearly preserves the benchmark's value and endpoint loads while controlling dependencies, which makes the statistical light-edge completion possible.

Problem

• stochastic bipartite matching
• independent vertex arrivals
• resource allocation
• maximum matching
• local menus

Innovation

• Stochastic Bipartite Matching
• Independent Vertex Arrivals
• Local Sparsification
• Near-Optimal Matching
• Weighted Sampling