  We study the following separation problem: Given a collection of colored
objects in the plane, compute a shortest "fence" $F$, i.e., a union of curves
of minimum total length, that separates every two objects of different colors.
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 GEOMETRIC $k$-CUT, where $k$ is the number of different colors, as it can be
seen as a geometric analogue to the well-studied multicut problem on graphs. We
first give an $O(n^4\log^3 n)$-time algorithm that computes an optimal fence
for the case where the input consists of polygons of two colors and $n$ corners
in total. We then show that the problem is NP-hard for the case of three
colors. Finally, we give a $(2-4/3k)$-approximation algorithm.
