发表机构
Nanyang Technological University; Shanghai University of Finance and Economics（南洋理工大学; 上海财经大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明经典Ranking算法在可重用资源在线二分图匹配中可实现约0.512的竞争比，超越1/2，并给出从无权重到权重匹配的黑盒归约，解决了开放问题。
AI 中文摘要
我们研究具有单位库存可重用资源的在线二分图匹配问题，其中请求以对抗性固定的顺序到达，每次使用资源后，该资源将在由资源相关分布决定的独立时长内不可用。基准算法预先知道所有请求，但在选择相应使用之前无法观察时长。Karp、Vazirani和Vazirani（STOC 1990）的经典Ranking算法为资源固定一个均匀随机的优先级顺序，并将每个到达的请求匹配到其最高优先级的可用邻居。对于无权重的不可重用资源，它实现了最优竞争比$1-1/e$，但对于可重用资源，它是否能超越$1/2$仍然是一个开放问题。我们证明，对于具有资源相关随机时长的无权重资源，Ranking算法实现了$(5-2\sqrt3)/3\approx0.511966$的竞争比。我们还给出了从无权重Ranking到资源权重匹配的黑盒归约：任何无权重竞争比$\alpha>1/2$都能产生严格高于$1/2$的权重比。通过独立采样时长分布，该归约给出了$0.500034$的权重比。这些结果解决了Delong等人（MOR 2024）留下的两个问题：Ranking是否超越$1/2$，以及在随机时长下是否能超越$1/2$。我们逐个资源地分析Ranking，而不是逐个请求地分析。对于确定性时长，这给出了对覆盖函数的随机顺序贪心算法的归约。然后，我们通过比较Ranking和贪心算法的剩余调度，将分析扩展到随机时长，并对随机排名进行更精细的分析以获得所述的$0.511$界限。对于权重归约，我们在相似权重组内应用Ranking，并使用加权贪心算法来控制组间的损失。
英文摘要
We study online bipartite matching with unit-inventory reusable resources, where requests arrive in an adversarially fixed order, and each use of a resource makes it unavailable for an independent duration drawn from a resource-dependent distribution. The benchmark knows all requests in advance but cannot observe a duration before choosing the corresponding use. The classical Ranking algorithm of Karp, Vazirani, and Vazirani (STOC 1990) fixes a uniformly random priority order of the resources and matches each arriving request to its highest-priority available neighbor. It achieves the optimal competitive ratio $1-1/e$ for unweighted nonreusable resources, but whether it beats $1/2$ for reusable resources has remained open. We prove that, for unweighted resources with resource-dependent stochastic durations, Ranking achieves a competitive ratio of $(5-2\sqrt3)/3\approx0.511966$. We also give a black-box reduction from unweighted Ranking to resource-weighted matching: any unweighted competitive ratio $α>1/2$ yields a weighted ratio strictly above $1/2$. With independent sampling access to the duration distributions, the reduction gives a weighted ratio of $0.500034$. These results resolve two questions left open by Delong et al. (MOR 2024): whether Ranking beats $1/2$, and whether one can beat $1/2$ under stochastic durations. We analyze Ranking resource by resource, rather than request by request. For deterministic durations, this gives a reduction to random-order greedy for a coverage function. We then extend the analysis to stochastic durations by comparing the residual schedules of Ranking and a greedy algorithm, and apply a finer analysis of the random ranks to obtain the stated $0.511$ bound. For the weighted reduction, we apply Ranking within groups of similar weights and uses weighted greedy to control the loss between groups.
