Complexity of Partitioning 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 $k \geq 5$.