Terrain Guarding is NP-Hard

Abstract

A set G of points on a 1.5-dimensional terrain, also known as an x-monotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in G. 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.

Guarding Monotone Art Galleries with Sliding Cameras in Linear Time

“…Indeed, even for x-monotone orthogonal polygons there is only an approximation algorithm for the problem. Recall that the classical art-gallery problem is np-hard on simple orthogonal polygons [11], simple monotone polygons [8] and even on terrains [7].…”

Algorithms for Tower Placement on Terrain

“…If the length of the tower is not long enough it would be interesting to develop an algorithm for covering the whole terrain by placing only a small number of towers. If the length of the tower is zero then the problem of finding the minimum number of tower placement is NP-Hard [KK11].…”

“…In this thesis, we examine algorithmic approaches for the placement of towers in terrain. Some versions of tower placement problems are known to be intractable [KK11] and consequently our motivation is in the development of tower placement algorithms that are efficient and easy to implement. In chapter 2, we present a critical review of groundbreaking algorithms reported in publication avenues.…”

The Complexity of Separating Points in the Plane

“…PLANAR-3-SAT is NP-complete [12], and it remains so when the 3-legged representation is given as part of the input. Several NP-hardness proofs of geometric problems have used PLANAR-3-SAT; see for example [1], [5], [8], [9], and [14].…”