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.
Abrahamsen et al. (2018) studied this question.