Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

Fast Fencing

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

Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition

We provide efficient constant factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves partitioning the plane minus the input points into subsets with exactly three boundary components, with approximately minimum total length. In the Euclidean case, these curves are squares; in the hyperbolic case, they combine our Euclidean square pants decomposition with our tree clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now Lemma 5.2, as the previous proof was erroneou