Geometric Multicut: Shortest Fences for Separating Groups of Objects in the Plane

Abstract

AbstractWe study the following separation problem: Given a collection of pairwise disjoint coloured objects in the plane with k different colours, compute a shortest “fence” F, i.e., a union of curves of minimum total length, that separates every pair of objects of different colours. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem as geometrick-cut, as it is a geometric analog to the well-studied multicut problem on graphs. We first give an $$O(n^4\log ^3\!n)$$ O ( n 4 log 3 n ) -time algorithm that computes an optimal fence for the case where the input consists of polygons of two colours with n corners in total. We then show that the problem is NP-hard for the case of three colours. Finally, we give a randomised $$4/3\cdot 1.2965$$ 4 / 3 · 1.2965 -approximation algorithm for polygons and any number of colours.