A Pseudo - Polynomial Time O ( log 2 n ) - Approximation Algorithm for Art Gallery Problems

Master of Science in Mechanical Engineering and Master of Science in Electrical Engineering and Computer Science In this thesis, we give a pseudo-polynomial time o (log2 n)-approximation algorithm for a variant of the art gallery problem the point-guard problem. The point-guard problem involves finding the minimum number of points and their positions so that guards located at these points cover the interior of the art gallery. Our algorithm is pseudo-polynomial in the sense that it is polynomial in the number of walls of the art gallery but is possibly exponential in the number of bits required to represent the positions of the vertices of the art gallery. Our approach involves reducing the point-guard problem to a new problem of choosing a minimum number of guard-locations from a finite set obtained by a special subdivision procedure. The new problem has the optimal solution at most three times the optimal solution of the point-guard problem. We further reduce the new problem to the set cover problem and obtain an approximate solution to the set cover problem. Thesis Supervisor: Sanjay E. Sarma Title: Associate Professor of Mechanical Engineering Thesis Reader: Erik Demaine Title: Associate Professor of Electrical Engineering and Computer Science