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 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 language | English |
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Title of host publication | Proceedings of 50th Annual ACM Symposium on Theory of Computing |
Publisher | Association for Computing Machinery |
Publication date | 2018 |
Pages | 1319-1332 |
ISBN (Print) | 9781450355599 |
DOIs | |
Publication status | Published - 2018 |
Event | 50th Annual ACM SIGACT Symposium on Theory of Computing - Los Angeles, United States Duration: 25 Jun 2018 → 29 Jun 2018 |
Conference
Conference | 50th Annual ACM SIGACT Symposium on Theory of Computing |
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Country/Territory | United States |
City | Los Angeles |
Period | 25/06/2018 → 29/06/2018 |
Keywords
- Geometric clustering
- Minimum perimeter sum