# Geometric Multicut

## TL;DR

This paper introduces the Geometric Multicut problem, providing an optimal algorithm for two colors, proving NP-hardness for three colors, and offering an approximation algorithm for general cases.

## Contribution

It presents the first polynomial-time algorithm for two-color cases, establishes NP-hardness for three colors, and develops a new approximation algorithm for the general problem.

## Key findings

- Optimal fence algorithm for two colors with $O(n^4\log^3 n)$ complexity
- NP-hardness proof for three or more colors
- A $(2-4/3k)$-approximation algorithm for general cases

## Abstract

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.