Placing observers to cover a polyhedral terrain in polynomial time

The Art Gallery Problem is the problem of determining the number of observers necessary to cover an art gallery room such that every point is seen by at least one observer. This problem is well known and has a linear solution for the 2 dimensional case, but little is known in the 3-D case. In this paper we present a polynomial time solution for the 3-D version of the Art Gallery problem. Because the problem is NP-hard, the solution presented is an approximation, and we present the bounds to our solution. Our solution uses techniques from computational geometry, graph coloring and set coverage. A complexity analysis is presented for each step and an analysis of the overall quality of the solution is given.