The Hausdorff Core Problem on Simple Polygons

We present a study of the Hausdorff Core problem on simple polygons. A polygon Q is a k-bounded Hausdorff Core of a polygon P if P contains Q, Q is convex, and the Hausdorff distance between P and Q is at most k. A Hausdorff Core of P is a k-bounded Hausdorff Core of P with the minimum possible value of k, which we denote kmin. Given any k and any ε > 0, we describe an algorithm for computing a k′-bounded Hausdorff Core (if one exists) in O(n3 + n2ε−4(log n+ ε−2)) time, where k′ < k + drad · ε and drad is the radius of the smallest disc that encloses P and whose center is in P . We use this solution to provide an approximation algorithm for the optimization Hausdorff Core problem which results in a solution of size kmin + drad · ε in O(log(ε−1)(n3 + n2ε−4(log n+ ε−2))) time. Finally, we describe an approximation scheme for the k-bounded Hausdorff Core problem which, given a polygon P , a distance k, and any ε > 0, answers true if there is a ((1 + ε)k)bounded Hausdorff Core and false if there is no k-bounded Hausdorff Core. The running time of the approximation scheme is in O(n3 + n2ε−4(log n+ ε−2)).