Placing your Coins on a Shelf

TL;DR

This paper studies the problem of optimally placing disks on a shelf to minimize the span, proving NP-hardness, and providing approximation and exact algorithms for specific cases.

Contribution

It establishes NP-hardness for the disk placement problem and offers a 4/3-approximation algorithm along with an exact solution for cases with limited radius ratios.

Findings

NP-hardness of the placement problem.

A 4/3-approximation algorithm with O(n log n) complexity.

An exact algorithm for cases with radius ratio at most four.

Abstract

We consider the problem of packing a family of disks "on a shelf", that is, such that each disk touches the $x$-axis from above and such that no two disks overlap. We prove that the problem of minimizing the distance between the leftmost point and the rightmost point of any disk is NP-hard. On the positive side, we show how to approximate this problem within a factor of 4/3 in $O(n \log n)$ time, and provide an $O(n \log n)$-time exact algorithm for a special case, in particular when the ratio between the largest and smallest radius is at most four.