Fast fencing

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

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review


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 polynomial in both n and k. For the variant with unit cost per curve, or unit disks, we present a 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.
Original languageEnglish
Title of host publicationProceedings of 50th Annual ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Publication date2018
ISBN (Print)9781450355599
Publication statusPublished - 2018
Event50th Annual ACM SIGACT Symposium on Theory of Computing - Los Angeles, United States
Duration: 25 Jun 201829 Jun 2018


Conference50th Annual ACM SIGACT Symposium on Theory of Computing
CountryUnited States
CityLos Angeles
SeriesProceedings of the Annual Acm Symposium on Theory of Computing


  • Geometric clustering
  • Minimum perimeter sum


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