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 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 n O(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.

Exact and heuristic solutions for the prize‐collecting geometric enclosure problem

“…Later, Abrahamsen et al. (2018) considered the PCGEP variants with multiple fences introduced by Arkin et al. (1993), where all points must be enclosed and the total perimeter of the polygons must be minimized.…”

Geometric Multicut: Shortest Fences for Separating Groups of Objects in the Plane

“…Related Work Despite the fact that the problem is natural and fundamental, there is little previous work. The problem of enclosing a set of objects by a shortest system of fences has recently been considered with a single set B 1 by Abrahamsen et al [1].…”