Erik Krohn

Terrain Guarding is NP-Hard

A set

An Approximation Scheme for Terrain Guarding

G

GUARDING TERRAINS VIA LOCAL SEARCH

of points on a terrain, also known as an

On Metric Clustering to Minimize the Sum of Radii

x

On Clustering to Minimize the Sum of Radii

-monotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in

Improved Approximations for Guarding 1.5-Dimensional Terrains

G

A Characterization of Visibility Graphs for Pseudo-polygons

. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem asks for a minimum guarding set for the given input terrain. Using a reduction from PLANAR 3-SAT we prove that the decision version of this problem is NP-hard. This solves a significant open problem and complements recent positive approximability results for the optimization problem.

The VC-dimension of visibility on the boundary of a simple polygon

We obtain a polynomial time approximation scheme for the terrain guarding problem improving upon several recent constant factor approximations. Our algorithm is a local search algorithm inspired by the recent results of Chan and Har-Peled [2] and Mustafa and Ray [15]. Our key contribution is to show the existence of a planar graph that appropriately relates the local and global optimum.

Approximate Guarding of Monotone and Rectilinear Polygons

We obtain a polynomial time approximation scheme for the terrain guarding problem improving upon several recent constant factor approximations. Our algorithm is a local search algorithm inspired by the recent results of Chan and Har-Peled (SoCG 2009) and Mustafa and Ray (DCG 2010). Our key contribution is to show the existence of a planar graph that appropriately relates the local and global optimum.

On clustering to minimize the sum of radii

Given an n-point metric (P,d) and an integer k> 0, we consider the problem of covering Pby kballs so as to minimize the sum of the radii of the balls. We present a randomized algorithm that runs in nO(logn·logΔ)time and returns with high probability the optimal solution. Here, Δis the ratio between the maximum and minimum interpoint distances in the metric space. We also show that the problem is NP-hard, even in metrics induced by weighted planar graphs and in metrics of constant doubling dimension.