Complexity of Partitioning Hypergraphs

For a given π=(π_0, π_1,..., π_k) ∈{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 ⊆ V of size k, X ∈ E if π_|X∩ V_1|=1 and X ∉ E if π_|X∩ V_1|=0. We prove that this problem is either polynomial-time solvable or NP-complete depending on π when k=3 or 4. We also extend this result into k-uniform hypergraphs for k ≥ 5.