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.
| Originalsprog | Engelsk |
|---|---|
| Titel | STOC 2018 - Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing |
| Forlag | Association for Computing Machinery |
| Publikationsdato | 2018 |
| Sider | 564-573 |
| ISBN (Elektronisk) | 9781450355599 |
| DOI | |
| Status | Udgivet - 2018 |
| Begivenhed | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 - Los Angeles, USA Varighed: 25 jun. 2018 → 29 jun. 2018 |
Konference
| Konference | 50th Annual ACM Symposium on Theory of Computing, STOC 2018 |
|---|---|
| Land/Område | USA |
| By | Los Angeles |
| Periode | 25/06/2018 → 29/06/2018 |
| Sponsor | ACM Special Interest Group on Algorithms and Computation Theory (SIGACT) |