# The Landscape of Minimum Label Cut (Hedge Connectivity) Problem

## TL;DR

This paper investigates the Minimum Label Cut problem, clarifies its definitions, corrects errors in prior research, and explores relationships and approximations between different versions of the problem, with implications for network design and image segmentation.

## Contribution

It provides a unified framework, corrects previous inaccuracies, and offers new insights and results on the complexity and approximation of the problem.

## Key findings

- Established the relationship between overlapping and non-overlapping label versions.
- Analyzed hardness and approximation bounds for weighted and unweighted cases.
- Corrected errors and unified definitions in existing literature.

## Abstract

Minimum Label Cut (or Hedge Connectivity) problem is defined as follows: given an undirected graph $G=(V, E)$ with $n$ vertices and $m$ edges, in which, each edge is labeled (with one or multiple labels) from a label set $L=\{\ell_1,\ell_2, ..., \ell_{|L|}\}$, the edges may be weighted with weight set $W =\{w_1, w_2, ..., w_m\}$, the label cut problem(hedge connectivity) problem asks for the minimum number of edge sets(each edge set (or hedge) is the edges with the same label) whose removal disconnects the source-sink pair of vertices or the whole graph with minimum total weights(minimum cardinality for unweighted version). This problem is more general than edge connectivity and hypergraph edge connectivity problem and has a lot of applications in MPLS, IP networks, synchronous optical networks, image segmentation, and other areas. However, due to limited communications between different communities, this problem was studied in different names, with some important existing literature citations missing, or sometimes the results are misleading with some errors. In this paper, we make a further investigation of this problem, give uniform definitions, fix existing errors, provide new insights and show some new results. Specifically, we show the relationship between non-overlapping version(each edge only has one label) and overlapping version(each edge has multiple labels), by fixing the error in the existing literature; hardness and approximation performance between weighted version and unweighted version and some useful properties for further research.