Online Geometric Packing through Online TSP Scheduling

Abstract

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 $\hat{I}\text{c◯}(\sqrt{\log n/\log \log n})$ 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 $x_1,\dots ,x_n$ 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.