Packing d-dimensional balls into a d+1-dimensional container

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 $d+1$ 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 $d$-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 $n$ such balls, a container of volume $O(n^{\frac{d-1}{d}})$ is always sufficient and sometimes necessary. As a byproduct, this implies that for $d \geq 2$ there is no finite size $(d+1)$-dimensional convex body into which all $d$-dimensional unit-radius balls can be packed simultaneously.