# Complexity of Partitioning Hypergraphs

## TL;DR

This paper investigates the computational complexity of partitioning hypergraphs based on a pattern , establishing conditions under which the problem is either polynomial-time solvable or NP-complete for various uniformities.

## Contribution

It characterizes the complexity of hypergraph partitioning problems for different uniformities, extending known results to higher dimensions.

## Key findings

- Complexity depends on the pattern  and the uniformity k.
- The problem is either polynomial-time solvable or NP-complete for k=3,4.
- Results extend to k  for hypergraphs.

## Abstract

For a given $\pi=(\pi_0, \pi_1,..., \pi_k) \in \{0, 1, *\}^{k+1}$, we want to determine whether an input $k$-uniform hypergraph $G=(V, E)$ has a partition $(V_1, V_2)$ of the vertex set so that for all $X \subseteq V$ of size $k$, $X \in E$ if $\pi_{|X\cap V_1|}=1$ and $X \notin E$ if $\pi_{|X\cap V_1|}=0$. We prove that this problem is either polynomial-time solvable or NP-complete depending on $\pi$ when $k=3$ or $4$. We also extend this result into $k$-uniform hypergraphs for $k \geq 5$.