Separation and Approximation of Polyhedral Objects

Given a family of disjoint polygons P1, P2, : : :, Pk in the plane, and an integer parameter m, it is NP-complete to decide if the Pi's can be pairwise separated by a polygonal family with at most m edges, that is, if there exist polygons R1; R2; : : : ; Rk with pairwise-disjoint boundaries such that Pi Ri and PjRij m. In three dimensions, the problem is NP-complete even for two nested convex polyhedra. Many other extensions and generalizations of the polyhedral separation problem, either to families of polyhedra or to higher dimensions, are also intractable. In this paper, we present ecient approximation algorithms for constructing separating families of near-optimal size. Our main results are as follows. In two dimensions, we give an O(n logn) time algorithm for constructing a separating family whose size is withina constant factor of an optimalseparating family; n is the number of edges in the input family of polygons. In three dimensions, we show how to separate a convex pol...

• Type: Conference
• Year: 1992
• Where: SODA
• Authors: Joseph S. B. Mitchell, Subhash Suri
• Keywords: Algorithms, Convex Polyhedra, Convex Polyhedron, Polyhedron, SODA 1992