AI 中文总结
本文针对随机到达的边加权在线二分匹配，提出基于度参数和采样比例的两种框架，给出紧竞争比与方差界，并证明黑盒采样-匹配的转移定理，改进经典结果。
AI 中文摘要
我们研究了在随机到达顺序下的边加权在线二分匹配问题，该问题以最大离线度$d$和采样比例$\theta$为参数。我们分析了两种基于采样的框架。对于\textit{确定性贪心采样}，该框架从固定大小的初始样本计算价格，然后应用局部阈值规则，我们推导出其显式的最坏情况竞争比，并证明该比率在此策略族内对每个固定的$d\ne 2$和$\theta\in[0,1]$都是紧的。最优采样选择在$d=1,2$时不采样与稠密极限下约$0.2562$的保证之间插值，改进了经典的$1/8$分析，同时保持了每次到达的线性时间。我们还推导了匹配离线代理数量的方差的最坏情况界，包括当$\theta\to 1_-$时的阶紧行为和固定$m,d$时$\theta\to 0_+$的$O(\theta)$界。对于\textit{黑盒采样-匹配}，我们引入了前缀相关的重新加权，随后使用任意近似离线匹配求解器，并证明了一个转移定理，其保证是离线近似比乘以$d$和$\theta$的显式函数。使用精确匹配时，该框架在无界度极限下恢复了经典的$1/e$保证。
英文摘要
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.
CommentsA preliminary version of this work has been accepted to WINE 2026
