Optimum inapproximability results for finding minimum hidden guard sets in polygons and terrains

Abstract

We study the problem Minimum Hidden Guard Set, which consists of positioning a minimum number of guards in a given polygon or terrain such that no two guards see each other and such that every point in the polygon or on the terrain is visible from at least one guard. By constructing a gap-preserving reduction from Maximum 5-Ocurrence- 3-Satisfiability, we show that this problem cannot be approximated by a polynomial-time algorithm with an approximation ratio of n1−ɛ for any ɛ > 0, unless NP = P, where n is the number of polygon or terrain vertices. The result even holds for input polygons without holes. This separates the problem from other visibility problems such as guarding and hiding, where strong inapproximability results only hold for polygons with holes. Furthermore, we show that an approximation algorithm achieves a matching approximation ratio of n.