在 $\Delta$-叉 HST 上的近最优在线度量匹配
Near-Optimal Online Metric Matching on $Δ$-ary HST
- University of Iowa（爱荷华大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文针对在线度量匹配问题，在 HST 上提出一种期望竞争比为 $O((\log\log \Delta) \cdot \log \Delta)$ 的算法，该比率与服务器数无关，且近最优。研究问题为在线匹配，核心方法为基于受限重分配模型的框架，主要贡献为开发新算法并证明其近最优性。
中文摘要 AI 辅助
在在线度量匹配问题中，我们有 $n$ 个服务器，它们位于某个度量空间中的已知位置。请求按顺序到达某些位置，并且到达后，请求必须匹配到一个未匹配给先前请求的服务器。目标是使匹配成本最小化。对于具有不知情对手的随机算法，已知的最佳竞争比是通过将度量空间嵌入到 HST 中，然后在度量空间由 HST 定义的环境中解决问题来获得的。Bansal 等人（Algorithmica, 2014）引入了在线度量匹配的一个框架，其中在受限重分配模型中开发算法，然后将其转换为真正的在线算法。使用此框架，他们为 HST 获得了 $O(\log n)$ 的期望竞争比；这也为一般度量空间提供了 $O(\log^2 n)$ 的最佳已知竞争比。在本文中，我们重新审视该框架，旨在开发新算法。对于每个节点最多有 $\Delta$ 个子节点的 HST，我们通过此框架开发了一种算法，其期望竞争比为 $O((\log\log \Delta) \cdot \log \Delta)$。特别是，该比率与 $n$（服务器/请求的数量）无关。它是近最优的，因为任何算法的期望竞争比为 $\Omega(\log \Delta)$。
英文摘要
In the online metric matching problem, we have $n$ servers with known locations in some metric space. Requests arrive one-by-one at certain locations, and upon arrival a request must be matched to a server that was not matched to a previous request. The goal is to minimize the matching cost. For randomized algorithms with an oblivious adversary, the best known competitive ratio is obtained by embedding the metric space into an HST, and then solving the problem in the setting where the metric space is defined by the HST. Bansal et al. (Algorithmica, 2014) introduced a framework for online metric matching where one develops an algorithm in a restricted reassignment model, and then transforms this into a true online algorithm. Using this framework, they obtained an expected competitive ratio of $O(\log n)$ for HSTs; this also gives the best known competitive ratio of $O(\log^2 n)$ for general metrics. In this paper, we revisit this framework with the aim of developing new algorithms. For HSTs where each node has at most $Δ$ children, we develop an algorithm via this framework with an expected competitive ratio of $O((\log\log Δ) \cdot \log Δ)$. In particular, this ratio is independent of $n$, the number of servers/requests. It is near-optimal, as the expected competitive ratio of any algorithm is $Ω(\log Δ)$.
