局部稀疏化，全局近最优：独立顶点到达下的匹配

中文摘要 AI 辅助

资源分配系统通常先限制每个请求的选项列表，再全局协调分配。我们研究了独立顶点到达下随机二分匹配中的这种分离。每个请求从其自身已知分布中抽取一个状态，决定其兼容资源，并独立保留至多$k$条边的菜单。然后在保留图上计算最大匹配。我们证明有界局部菜单普遍足以实现近最优匹配。对于每个$\varepsilon>0$，存在仅依赖于$\varepsilon$的菜单大小$k_\varepsilon$，保留完整实现图期望最大匹配大小的至少$(1-\varepsilon)$比例。早期保证需要额外假设匹配质量如何分布在边上；我们的结果解决了无限制情形。此外，菜单可从任何基准匹配规则简单生成，要么根据基准的边边际进行加权采样，要么将基准应用于采样实现并保留所得伙伴。我们的证明通过结合大边际边的局部可计算代理与采样轻边的分数补全，在稀疏化器内构造近最优证书。该代理在控制依赖性的同时几乎保留基准值和端点负载，使得统计轻边补全成为可能。

英文摘要

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.

发表机构

• Google Research（谷歌研究院）
• Tel Aviv University（特拉维夫大学）

机构由 AI 辅助整理，请以论文原文为准。