Online Packing of Orthogonal Polygons

2026-03-23

AI Summary

This study addresses the online bin packing, strip packing, and minimum bounding box area problems for axis-aligned regular hexagons and octagons under translation. Employing competitive analysis, computational geometry, and combinatorial optimization, it establishes the first systematic lower bounds on competitive ratios: Ω(n/log n) for general hexagons, yet constant-competitive algorithms exist for symmetric or small-sized instances. In contrast, for octagons and their skeletal structures, any algorithm incurs a competitive ratio of at least n, rendering only trivial strategies optimal. The work also derives tight upper and lower bounds for variants optimizing strip height, perimeter, and area, thereby revealing the critical role of geometric symmetry and degenerate configurations in determining the competitiveness of online packing algorithms.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Online Learning & Bandits

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

While rectangular and box-shaped objects dominate the classic discourse of theoretic investigations, a fascinating frontier lies in packing more complex shapes. Given recent insights that convex polygons do not allow for constant competitive online algorithms for diverse variants under translation, we study orthogonal polygons, in particular of small complexity. For translational packings of orthogonal 6-gons, we show that the competitive ratio of any online algorithm that aims to pack the items into a minimal number of unit bins is in $Ω(n / \log n)$, where $n$ denotes the number of objects. In contrast, we show that constant competitive algorithms exist when the orthogonal 6-gons are symmetric or small. For (orthogonally convex) orthogonal 8-gons, we show that the trivial $n$-competitive algorithm, which places each item in its own bin, is best-possible, i.e., every online algorithm has an asymptotic competitive ratio of at least $n$. This implies that for general orthogonal polygons, the trivial algorithm is best possible. Interestingly, for packing degenerate orthogonal polygons (with thickness $0$), called skeletons, the change in complexity is even more drastic. While constant competitive algorithms for 6-skeletons exist, no online algorithm for 8-skeletons achieves a competitive ratio better than $n$. For other packing variants of orthogonal 6-gons under translation, our insights imply the following consequences. The asymptotic competitive ratio of any online algorithm is in $Ω(n / \log n)$ for strip packing, and there exist online algorithms with competitive ratios in $O(1)$ for perimeter packing, or in $O(\sqrt{n})$ for minimizing the area of the bounding box. Moreover, the critical packing density is positive (if every object individually fits into the interior of a unit bin).

Problem

• online packing
• orthogonal polygons
• competitive ratio
• bin packing
• computational geometry

Innovation

• online packing
• orthogonal polygons
• competitive ratio
• skeletons
• bin packing