Online Geometric Packing through Online TSP Scheduling

2026-07-24

AI Summary

This work addresses the long-standing open problem of online strip packing of convex polygons under translation, for which no constant-competitive algorithm was previously known. The authors introduce a novel “online TSP scheduling” model that reformulates the packing problem as assigning visit times to arriving points subject to distance constraints. By establishing algorithmic connections between this model, online scheduling, and geometric packing—and leveraging the breakthrough of Azar et al. in online scheduling—they design an online scheduling algorithm with a competitive ratio of $O(\log^2 n)$. This yields the first $O(\log^2 n)$-competitive algorithm for online strip packing of convex polygons, significantly improving upon the prior best bound of $O(n^{0.59})$, and extends to various translational packing settings, including unit hyperspheres in higher dimensions.

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 $Ω(\sqrt{\log n/\log\log n})$ 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.

Problem

• online packing
• convex polygons
• strip packing
• competitive ratio
• online TSP scheduling

Innovation

• online TSP scheduling
• competitive ratio
• online geometric packing
• convex polygons
• online sorting