On k-Guarding Polygons

We describe a polynomial time O(k log log OPTk(P ))approximation algorithm for the k-guarding problem of finding a minimum number, OPTk(P ), of vertex guards of an n-vertex simple polygon P so that for every point p ∈ P , the number of guards that see p is at least the minimum of k and the number of vertices that see p. Our approach finds O ( k ε log log 1 ε ) size (k, ε)-nets for instances of the k-hitting set problem arising from the k-guarding problem. These nets contain k distinct elements (or the entire set if it has fewer than k elements) from any set that has at least an ε fraction of the total weight of all elements. To find a nearly optimal k-guarding, we slightly modify the technique of Brönnimann and Goodrich [4] so that the weights of all elements remain small, which is necessary for our (k, ε)net finder. Our approach, generalizes, simplifies, and corrects a subtle flaw in the technique introduced by King and Kirkpatrick [11] to find small ε-nets for set systems arising from 1-guarding instances.