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Conference Papers Year : 2018

Fast fencing

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 S of n 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 n unit disks, paying only the total length of the enclosing curves. In the other variant, we are allowed to use at most k closed curves and pay no cost per curve. For the variant with at most k closed curves,we present an algorithm that is polynomialin bothn andk. For the variant with unit cost per curve, or unit disks, we presenta near-linear time algorithm. Capoyleas, Rote, and Woeginger solved the problem with at most k curves in nO(k) time. Arkin, Khuller, and Mitchell used this to solve the unit cost per curve version in exponential time. At the time, they conjectured that the problem with k curves is NP-hard for general k. Our polynomial time algorithm refutes this unless P equals NP.

Dates and versions

hal-02169561 , version 1 (01-07-2019)

Identifiers

Cite

Mikkel Abrahamsen, Anna Adamaszek, Karl Bringmann, Vincent Cohen-Addad, Mehran Mehr, et al.. Fast fencing. The 50th Annual ACM SIGACT Symposium on Theory of Computing, Jun 2018, Los Angeles, United States. pp.564-573, ⟨10.1145/3188745.3188878⟩. ⟨hal-02169561⟩
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