Fast Approximation Algorithms for Art Gallery Problems in Simple Polygons

We present approximation algorithms with O(n) processing time for the minimum vertex and edge guard problems in simple polygons. It is improved from previous O(n) time algorithms of Ghosh. For simple polygon, there are O(n) visibility regions, thus any approximation algorithm for the set covering problem with approximation ratio of log(n) can be used for the approximation of n vertex and edge guard problems with O(n) visibility sequence. We prove that the visibility of all points in simple polygons is guaranteed by covering O(n) sinks from vertices and edges : It comes to O(n) time bound.