On point covers of c-oriented polygons

Let S be any family of n c-oriented polygons of the two-dimensional Euclidean plane E, i.e., bounded intersection of halfplanes whose normal directions of edges belong to a .xed collection of c distinct directions. Let (S) denote the packing number of S, that is the maximum number of pairwise disjoint objects of S. Let (S) be the transversal number of S, that is the minimum number of points required so that each object contains at least one of those points. We prove that (S)6G(2; c) (S) logc−1 2 ( (S)+1), where G(2; c) is the Gallai number of pairwise intersecting c-oriented polygons. Our bound collapses to (S) = O(G(2; c) (S)) if objects are more or less of the same size. We describe a t(n; c)+O(nc log (S))-time algorithm with linear storage that computes such a 0-transversal, where t(n; c) is the time required to pierce pairwise intersecting c-oriented polygons. We provide linear-time algorithms t(n; c) = 7(nc) for -fat c-oriented polytopes, translates or homothets of E proving that G(2; c) = O( ), G(2; c)6d and G(2; c)6(3d) respectively. c © 2001 Elsevier Science B.V. All rights reserved.