Existence and hardness of conveyor belts

TL;DR

This paper investigates the existence and computational complexity of conveyor belts connecting disjoint disks, providing efficient solutions for special cases, NP-completeness results for general cases, and a method to augment disk sets for guaranteed belts.

Contribution

It proves existence and efficient algorithms for certain disk configurations, establishes NP-completeness for general cases, and introduces a method to augment disks for guaranteed conveyor belts.

Findings

Conveyor belts exist and can be found efficiently for x- and y-monotone disk centers.

Determining conveyor belts for disks with varying radii is NP-complete.

Augmenting disks with guide disks guarantees a conveyor belt touching each disk once.

Abstract

An open problem of Manuel Abellanas asks whether every set of disjoint closed unit disks in the plane can be connected by a conveyor belt, which means a tight simple closed curve that touches the boundary of each disk, possibly multiple times. We prove three main results. First, for unit disks whose centers are both $x$-monotone and $y$-monotone, or whose centers have $x$-coordinates that differ by at least two units, a conveyor belt always exists and can be found efficiently. Second, it is NP-complete to determine whether disks of varying radii have a conveyor belt, and it remains NP-complete when we constrain the belt to touch disks exactly once. Third, any disjoint set of $n$ disks of arbitrary radii can be augmented by $O(n)$ "guide" disks so that the augmented system has a conveyor belt touching each disk exactly once, answering a conjecture of Demaine, Demaine, and Palop.