AI 中文总结
该研究针对每批最大延迟收费的在线服务问题，证明两种服务语义的离线最优值相等，设计多项式时间算法得到不同度量下的竞争比，还给出相关下界并分析请求位置隐藏时的竞争比。
AI 中文摘要
我们研究每服务批次收取一次最大等待时间费用的在线服务问题。请求在有限度量空间的点到达，移动服务器需支付移动费用，且每次服务行程需支付该行程所服务请求的最大等待时间。我们区分两种服务语义：可选服务（遇到的请求可暂不处理）和自动服务（遇到的请求必须服务）。尽管两种语义的最优调度结构不同，但我们证明它们的离线最优值相等。在有限直线和显式表示的加权树上，共同的离线值可通过多项式时间动态规划计算，而任意有限度量上的精确优化是NP难的。对于在线问题，我们证明了与度量无关的组轨迹引理，该引理将空间分离的请求组分配给时间窗口的两个奇偶类，由此在两种服务语义下，直线上的确定性多项式时间竞争比为10，加权树上为12，任意有限度量上为20；若使用精确的度量斯坦纳树（metric-Steiner-tree）预言机，一般度量的竞争比可提升至12。该多项式算法使用终端最小生成树（terminal-MST）权重的半尺度运行最大值，运行最大值是必要的，因为终端MST权重在新请求到达时不具有单调性。固定两点直线为所有上述度量类提供了确定性可见服务的下界3。最后，当请求位置在被访问前隐藏时，二进探索（dyadic exploration）在已知有限直线上的竞争比为84，该现象是直线特有的：一个隐藏请求在直线上给出确定性和随机下界3和2，而d叶单位星型图给出下界2d-1和d。
英文摘要
We study online service with one maximum-waiting-time charge per service batch. The persistent server endpoint prevents a phase-by-phase comparison with the offline optimum: an offline schedule may merge many online phases, share movement globally, and finish at unrelated endpoints. Our main contribution is a metric-independent \emph{group--trajectory certificate framework} that restores such a comparison. For ordered request groups in disjoint time windows, a certificate value is bounded both by the window length and by the metric Steiner cost of the group. After normalizing the offline schedule into consecutive arrival blocks, strictly interior groups are charged to offline delay, while boundary groups induce connectors of congestion at most two along the offline trajectory. One color class therefore has certificate sum at most $2\OPT$; a parity decomposition yields $\sum_h C_h\le4\OPT$. Consequently, any phase rule whose cost is at most $αC_h$ is $4α$-competitive. For visible service, this theorem yields deterministic ratios $10$ on a line, $12$ on a weighted tree, and $20$ on an arbitrary finite metric; the last algorithm is polynomial and uses a phase-local terminal-MST envelope, while an exact metric-Steiner oracle gives ratio $12$. Structurally, elective and automatic schedules can have different event structures but equal offline optimal values. The common value is computable exactly in polynomial time on lines and explicitly represented weighted trees, whereas exact optimization on arbitrary finite metrics is NP-hard. Finally, we use spatial blindness---announced requests whose locations are revealed only when visited---as a stress test: dyadic exploration preserves a constant ratio on a known finite line, while a single hidden request on a star forces a loss linear in its degree.
