Effectiveness of Local Search for Art Gallery Problems

Abstract

We study the variant of the art gallery problem where we are given an orthogonal polygon P (possibly with holes) and we want to guard it with the minimum number of sliding cameras. A sliding camera travels back and forth along an orthogonal line segment s in P and a point p in P is said to be visible to the segment s if the perpendicular from p onto s lies in P. Our objective is to compute a set containing the minimum number of sliding cameras (orthogonal segments) such that every point in P is visible to some sliding camera. We study the following two variants of this problem: Minimum Sliding Cameras problem, where the cameras can slide along either horizontal or vertical segments in P, and Minimum Horizontal Sliding Cameras problem, where the cameras are restricted to slide along horizontal segments only. In this work, we design local search PTASes for these two problems improving over the existing constant factor approximation algorithms. We note that in the first problem, the polygons are not allowed to contain holes. In fact, there is a family of polygons with holes for which the performance of our local search algorithm is arbitrarily bad.