Online Sorting and Translational Packing of Convex Polygons
Anders Aamand, Mikkel Abrahamsen, Lorenzo Beretta, Linda Kleist

TL;DR
This paper proves the non-existence of competitive online algorithms for packing convex polygons and related problems, by establishing a lower bound through a novel connection to an online sorting problem, contrasting with known results for rectangles.
Contribution
It introduces a new lower bound for online convex polygon packing problems by linking them to an online sorting problem, showing no competitive algorithms exist for these tasks.
Findings
No competitive algorithms for online convex polygon packing
Lower bound established via online sorting connection
Offline convex packing admits constant factor approximations
Abstract
We investigate several online packing problems in which convex polygons arrive one by one and have to be placed irrevocably into a container, while the aim is to minimize the used space. Among other variants, we consider strip packing and bin packing, where the container is the infinite horizontal strip or a collection of bins, respectively. We draw interesting connections to the following online sorting problem OnlineSorting: We receive a stream of real numbers , , one by one. Each real must be placed in an array with initially empty cells without knowing the subsequent reals. The goal is to minimize the sum of differences of consecutive reals in . The offline optimum is to place the reals in sorted order so the cost is at most . We show that for any -competitive online…
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Taxonomy
TopicsOptimization and Search Problems · Optimization and Packing Problems · Advanced Manufacturing and Logistics Optimization
