A PTAS for the continuous 1.5D terrain guarding problem

Abstract

In the continuous 1.5-dimensional terrain guarding problem we are given an x-monotone chain (the terrain T) and ask for the minimum number of point guards (located anywhere on T), such that all points of T are covered by at least one guard. It has been shown that the 1.5-dimensional terrain guarding problem is NP- hard. The currently best known approximation algorithm achieves a factor of 4. For the discrete problem version with a finite set of guard candidates and a finite set of points on the terrain that need to be monitored, a polynomial time approximation scheme (PTAS) has been presented [11]. We show that for the general problem we can construct finite guard and witness sets, G and W, such that there exists an optimal guard cover G∗ ⊆ G that covers T, and when these guards monitor all points in W the entire terrain is guarded. This leads to a PTAS as well as an (exact) IP formulation for the continuous terrain guarding problem.