A finite dominating set of cardinality O(k) and a witness set of cardinality O(n) for 1.5D terrain guarding problem

Abstract

1.5 dimensional (1.5D) terrain is characterized by a piecewise linear curve. Locating minimum number of guards on the terrain (T) to cover/guard the whole terrain is known as 1.5D terrain guarding problem. Approximation algorithms and a polynomial-time approximation scheme have been presented for the problem. The problem has been shown to be NP-Hard. In the problem, the set of possible guard locations and the set of points to be guarded are uncountable. To solve the problem to optimality, a finite dominating set (FDS) of size $\hbox {O}(n^{2})$ and a witness set of size $\hbox {O}(n^{3})$ have been presented, where n is the number of vertices on T. We show that there exists an even smaller FDS of cardinality $\hbox {O}(k)$ and a witness set of cardinality O(n), where k is the number of convex points. Convex points are vertices with the additional property that between any two convex points the piecewise linear curve representing the terrain is convex. Since it is always true that $k \le n$ for $n \ge$ 2 and since it is possible to construct terrains such that $n=2^{k}$, the existence of an FDS with cardinality O(k) and a witness set of cardinality of $\hbox {O}(n)$ leads to the reduction of decision variables and constraints respectively in the zero-one integer programming formulation of the problem.