树上 $k$-出租车问题中 DoubleCoverage 双重覆盖算法的势函数分析
A Potential Function Analysis of Double Coverage for the $k$-Taxi Problem on Trees
- University of Toronto（多伦多大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过将最小匹配距离表示为树上的积分，为树上 $k$-出租车问题的 DoubleCoverage 算法提供了纯势函数证明，弥补了此前仅适用于 $k=2$ 的空白。
AI中文摘要：
我们为 HST（层次化分割树）和有界深度的普通加权树上的困难 $k$-出租车问题中 DoubleCoverage 算法的竞争比提供了势函数证明。Buchbinder、Coester 和 Naor（2023）利用时间反向对偶拟合方法获得了这些竞争比，并指出他们不知道在 $k>2$ 情况下存在纯势函数证明。我们观察到，在线和离线出租车配置之间的最小匹配距离可以表示为以根树为基础的积分，即每个点下方出租车数量的绝对差值的积分。这种表示使我们能够在短移动区间上分析 DoubleCoverage 算法，并将所得估计应用于当两辆服务出租车从起点移动到终点时保持不变的势函数。
英文摘要：
We provide potential function proofs of the competitive ratios of DoubleCoverage for the hard $k$-taxi problem on HSTs and general weighted trees of bounded depth. Buchbinder, Coester, and Naor (2023) obtained these ratios using time-reverse dual fitting and noted that they did not know a pure potential proof beyond $k=2$. We observe that the minimum matching distance between the online and offline taxi configurations can be expressed as an integral over the rooted tree of the absolute difference between their taxi counts below each point. This representation lets us analyse DoubleCoverage over short movement intervals and apply the resulting estimates to potentials that remain unchanged when both serving taxis are relocated from pickup to destination.
