On the complexity of two acyclic subhypergraphs problems

We investigate the computational complexity of two decision problems on hypergraphs, namely the Spanning Acyclic Subhypergraph problem and the Maximal Acyclic Subhypergraph problem. The former is the existence of an acyclic subhypergraph such that each vertex of the input hypergraph is contained in at least one hyperedge of the subhypergraph. The latter is the existence of an acyclic subhypergraph with k hyperedges where k is part of the input. For each of these problems, we consider different notions of acyclicity of hypergraphs: Berge-acyclicity, γ-acyclicity, β-acyclicity and α-acyclicity. We are also concerned with the size of the hyperedges of the input hypergraph. Depending on these two parameters (notion of acyclicity and size of the hyperedges), we try to determine which instances of the two problems are in RP and which are NP-complete.