Fast Fencing
Mikkel Abrahamsen, Anna Adamaszek, Karl Bringmann, Vincent, Cohen-Addad, Mehran Mehr, Eva Rotenberg, Alan Roytman, Mikkel Thorup

TL;DR
This paper introduces efficient algorithms for minimal fencing problems, showing that certain variants are solvable in polynomial time, contrary to previous conjectures of NP-hardness.
Contribution
The paper presents polynomial-time algorithms for fencing problems with constraints on the number of curves, refuting earlier NP-hardness conjectures.
Findings
Polynomial-time algorithm for at most k curves variant.
Near-linear time algorithm for unit disk enclosure.
Refutation of NP-hardness conjecture for the k curves problem.
Abstract
We consider very natural "fence enclosure" problems studied by Capoyleas, Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a set of points in the plane, we aim at finding a set of closed curves such that (1) each point is enclosed by a curve and (2) the total length of the curves is minimized. We consider two main variants. In the first variant, we pay a unit cost per curve in addition to the total length of the curves. An equivalent formulation of this version is that we have to enclose unit disks, paying only the total length of the enclosing curves. In the other variant, we are allowed to use at most closed curves and pay no cost per curve. For the variant with at most closed curves, we present an algorithm that is polynomial in both and . For the variant with unit cost per curve, or unit disks, we present a near-linear time…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
