Improved Approximations for Guarding 1.5-Dimensional Terrains

We present a 4-approximation algorithm for the problem of placing a fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the currently best approximation factor of 5. Our method is based on rounding the linear programming relaxation of the corresponding covering problem. Besides the simplicity of the analysis, which mainly relies on decomposing the constraint matrix of the LP into totally balanced matrices, our algorithm, unlike previous work, generalizes to the weighted and partial versions of the basic problem.Comment: 10 pages, 1 Postscript figure, uses geometry.st

The Complexity of Guarding Terrains

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 any 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. 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. Our proof uses a reduction from PLANAR 3-SAT. We build gadgets capable of 'mirroring' a consistent variable assignment back and forth across a main valley. The structural simplicity of 1.5-dimensional terrains makes it difficult to build general clause gadgets that do not destroy this assignment when they are evaluated. However, we exploit the structure in instances of PLANAR 3-SAT to find very specific operations involving only 'adjacent' variables. For these restricted operations we can construct gadgets that allow a full reduction to work.Comment: 26 pages, 24 figure