Minimum Perimeter-Sum Partitions in the Plane

Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P-1 and P-2 such that the sum of the perimeters of ch(P-1) and CH(P-2) is minimized, where CH(P-i) denotes the convex hull of P-i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n(2)) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(n log(2) n) time and a (1 + epsilon)-approximation algorithm running in O(n + 1/epsilon(2) . log(2)(1/epsilon)) time.