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 xed height h such that each triangle on a given 2.5-dimensional triangulated terrain T is completely visible from at least one guard. We prove this problem to be N P-hard, and we show that it cannot be approximated by a polynomial time algorithm within a ratio of (1 ?) 1 35 ln n for any > 0, unless N P T IM E(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.