通过在线旅行商问题调度进行在线几何包装
Online Geometric Packing through Online TSP Scheduling
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中文总结 AI 辅助
研究凸多边形在线平移包装问题，当前最佳算法竞争比为\(O(n^{0.59})\)。引入在线旅行商问题调度，给出\(O(\log^2 n)\)竞争算法，由此得到凸多边形在线平移条带包装等问题的同竞争比算法，揭示包装与在线排序联系在算法设计上的应用。
中文摘要 AI 辅助
我们考虑通过平移将凸多边形在线包装到一个条带中的问题。几十年来，矩形的在线算法具有常数竞争比已为人所知，但凸多边形的当前最佳算法竞争比为\(O(n^{\log_2 3 - 1}\log n)=O(n^{0.59})\)，其中\(n\)是多边形数量。我们引入了一个新的在线问题——在线旅行商问题调度，其中点\(x_1,\ldots,x_n\)从度量空间\((M,d)\)在线到达，每个\(x_i\)到达时必须分配一个访问时间\(p_i\in[0,\infty)\)，满足\(|p_i - p_j|\geq d(x_i,x_j)\)。我们给出了在线旅行商问题调度的\(O(\log^2 n)\)竞争算法，并展示了这如何暗示凸多边形在线平移条带包装的\(O(\log^2 n)\)竞争算法。还证明了其他平移包装问题也能达到相同竞争比。我们的在线旅行商问题调度算法基于阿扎尔等人在线排序的突破。表明包装与在线排序的联系不仅可用于下限，也可用于算法。
英文摘要
We consider the problem of online packing of convex polygons into a strip by translations. While online algorithms with a constant competitive ratio have been known for rectangles for decades [Baker and Schwarz, SICOMP 1983], the current best algorithm for convex polygons has competitive ratio $O(n^{\log_2 3-1}\log n) = O(n^{0.59})$, where $n$ is the number of polygons. This algorithm was described by Aamand, Abrahamsen, Beretta, and Kleist [SODA 2023], who also proved a lower bound of $Ω(\sqrt{\log n/\log\log n})$ on the competitive ratio of any algorithm. Their lower bound is obtained via a reduction from \emph{online sorting}, a problem introduced in the same paper, for which they established a lower bound on the competitive ratio. We introduce a new, natural online problem that we call online TSP scheduling. Here, points $x_1,\ldots,x_n$ arrive online from a metric space $(M,d)$, and upon arrival each $x_i$ must be assigned a visit time $p_i\in[0,\infty)$ satisfying $|p_i-p_j|\ge d(x_i,x_j)$ for all $j<i$. The cost of the schedule is $\max_i p_i$. We present an $O(\log^2 n)$-competitive algorithm for online TSP scheduling, and show how this implies an $O(\log^2 n)$-competitive algorithm for online translational strip packing of convex polygons. We also prove that the same competitive ratio is achievable for other translational packing problems, including online packing of $d$-dimensional unit hyperdisks in $\mathbb R^{d+1}$, whose offline version was studied by Alt, Cabello, Cheong, Park, and Seiferth [Comp. Geom. 2026]. Our algorithm for online TSP scheduling builds on a recent breakthrough for online sorting by Azar, Panigrahi, and Vardi [SODA 2026]. We thus show that the connection between packing and online sorting can be used not only for lower bounds, but also for algorithms.
