Packing d-dimensional balls into a d+1-dimensional container
Helmut Alt, Sergio Cabello, Otfried Cheong, Ji-won Park, Nadja Seiferth

TL;DR
This paper presents approximation algorithms for packing d-dimensional balls into minimal-volume (d+1)-dimensional containers, revealing volume bounds and limitations on universal packing bodies.
Contribution
It introduces constant-factor approximation algorithms for packing problems and establishes volume bounds and impossibility results for universal containers.
Findings
Approximation algorithms based on shortest Hamiltonian path reduction.
A volume bound of O(n^{(d-1)/d}) for packing d-balls in (d+1)-dimensional containers.
Existence of no finite universal container for all d-dimensional balls when d ≥ 2.
Abstract
In this article, we consider the problems of finding in dimensions a minimum-volume axis-parallel box, a minimum-volume arbitrarily-oriented box and a minimum-volume convex body into which a given set of -dimensional unit-radius balls can be packed under translations. The computational problem is neither known to be NP-hard nor to be in NP. We give a constant-factor approximation algorithm for each of these containers based on a reduction to finding a shortest Hamiltonian path in a weighted graph, which in turn models the problem of stabbing the centers of the input balls while keeping them disjoint. We also show that for such balls, a container of volume is always sufficient and sometimes necessary. As a byproduct, this implies that for there is no finite size -dimensional convex body into which all -dimensional unit-radius balls…
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Taxonomy
TopicsAdvanced Manufacturing and Logistics Optimization · Optimization and Packing Problems · Modular Robots and Swarm Intelligence
