Positioning Guards at Fixed Height above a Terrain - an Optimum Inapproximability Result

. We study the problem of minimizing the number of guards positioned at a fixed height h such that each triangle on a given 2.5dimensional triangulated terrain T is completely visible from at least one guard. We prove this problem to be NP-hard, and we show that it cannot be approximated by a polynomial time algorithm within a ratio of (1 \Gamma ffl) 1 35 ln n for any ffl ? 0, unless NP ` TIME(n O(log log n) ), where n is the number of triangles in the terrain. Since there exists an approximation algorithm that achieves an approximation ratio of ln n+1, our result is close to the optimum hardness result achievable for this problem. 1 Introduction and Problem Definition We study the problem of positioning a minimum number of guards at a fixed height above a terrain. The terrain is given as a finite set of points in the plane, together with a triangulation (of its convex hull), and a height value is associated with each point (a triangulated irregular network (TIN), see e...

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum