Minimum Link Fencing

TL;DR

This paper investigates the minimum link fencing problem, aiming to efficiently compute minimal fences that separate colored polygons, with complexity results and an efficient algorithm for specific geometric configurations.

Contribution

It introduces the minimum link fencing problem, proves NP-hardness in general, and provides an XP-time algorithm for cases with limited fence complexity, along with an efficient solution for certain geometric conditions.

Findings

BMLF is NP-hard in general.

XP-time algorithm exists when each fence contains at most two polygons.

An $O(n ext{ log } n)$ algorithm is provided for specific geometric cases.

Abstract

We study a variant of the geometric multicut problem, where we are given a set $\mathcal{P}$ of colored and pairwise interior-disjoint polygons in the plane. The objective is to compute a set of simple closed polygon boundaries (fences) that separate the polygons in such a way that any two polygons that are enclosed by the same fence have the same color, and the total number of links of all fences is minimized. We call this the minimum link fencing (MLF) problem and consider the natural case of bounded minimum link fencing (BMLF), where $\mathcal{P}$ contains a polygon $Q$ that is unbounded in all directions and can be seen as an outer polygon. We show that BMLF is NP-hard in general and that it is XP-time solvable when each fence contains at most two polygons and the number of segments per fence is the parameter. Finally, we present an $O(n \log n)$-time algorithm for the case that the convex hull of $\mathcal{P} \setminus \{Q\}$ does not intersect $Q$.