Degree-Parameterized Analysis of Sampling-Based Online Matching

2026-09-13

AI Summary

研究了基于采样的在线二部图匹配问题，通过确定性贪婪采样和黑盒采样-匹配两种方法，分析了最大离线度d和采样比例θ对匹配效果的影响。

Abstract

We study edge-weighted online bipartite matching under random arrival order, parameterized by the maximum offline degree $d$ and sampling fraction $θ$. We analyze two sampling-based frameworks. For \emph{Deterministic Greedy Sampling}, which computes prices from a fixed-size initial sample and then applies a local threshold rule, we derive an explicit worst-case competitive ratio and prove it tight within this policy family for every fixed $d\ge2$ and $θ\in[0,1]$. The optimal sampling choice interpolates between no sampling for $d=1,2$ and a dense-limit guarantee of approximately $0.2562$, improving on the classical $1/8$ analysis while retaining linear per-arrival time. We also derive worst-case bounds on the variance of the number of matched offline agents, including order-tight behavior as $θ\to1_-$ and an $O(θ)$ bound as $θ\to0_+$ for fixed $m,d$. For \emph{Black-Box Sampling--Matching}, we introduce prefix-dependent reweighting followed by an arbitrary approximate offline matching solver and prove a transfer theorem whose guarantee is the offline approximation ratio times an explicit function of $d$ and $θ$. With exact matching, the framework recovers the classical $1/e$ guarantee in the unbounded-degree limit.

Problem

• online bipartite matching
• sampling-based methods
• competitive ratio
• variance
• offline degree

Innovation

• Deterministic Greedy Sampling
• Black-Box Sampling--Matching
• competitive ratio
• sampling fraction
• offline degree