Existence and hardness of conveyor belts

• Main Authors: Demaine, Erik D (Author), Demaine, Martin L (Author)
• Other Authors: Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor), Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory (Contributor)
• Format: Article
• Language: English
• Published: The Electronic Journal of Combinatorics, 2020-12-11T13:59:31Z.
• Subjects: Article
• Online Access: Get fulltext

 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: 1. 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. 2. It is NP-complete to determine whether disks of arbitrary radii have a conveyor belt, and it remains NP-complete when we constrain the belt to touch disks exactly once. 3. 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.