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Title: Existence and Hardness of Conveyor Belts
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: 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. 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. 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.  more » « less
Award ID(s):
2007275 1764012
PAR ID:
10282136
Author(s) / Creator(s):
; ; ; ; ; ; ; ; ;
Date Published:
Journal Name:
The Electronic Journal of Combinatorics
Volume:
27
Issue:
4
ISSN:
1077-8926
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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