Experimental Evaluation of an Exact Algorithm for the Orthogonal Art Gallery Problem

We consider the Orthogonal Art Gallery problem (OAGP) whose goal is to minimize the number of vertex guards required to watch an art gallery whose boundary is an n-vertex orthogonal polygon P. Here, we explore an exact algorithm for OAGP, which we proposed in [1], that iteratively computes optimal solutions to Set Cover problems (SCPs) corresponding to discretizations of P. While it is known [1] that this procedure converges to an exact solution of the original continuous problem, the number of iterations executed is highly dependent on the way we discretize P. Although the best theoretical bound for convergence is θ(n3) iterations, we show that, in practice, it is achieved after only a few of them, even for random polygons of hundreds of vertices. As each iteration involves the solution of an SCP, the strategy for discretizing P is of paramount importance. In this paper, we carry out an extensive empirical investigation with five alternative discretization strategies to implement the algorithm. A broad range of polygon classes is tested. As a result, we are able to significantly improve the performance of the algorithm, while maintaining low execution times, to the point that we achieve a fivefold increase in polygon size, compared to the literature.