The Complexity Of Acyclic Subhypergraph Problems

We investigate the computational complexity of two decision problems concerning the existence of certain acyclic subhypergraphs of a given hypergraph, namely the SPANNING ACYCLIC SUBHYPERGRAPH problem and the MAXIMUM ACYCLIC SUBHYPERGRAPH problem. The former is about the existence of an acyclic subhypergraph such that each vortex of the input hypergraph is contained in at least one hyperedge of the subhypergraph. The latter is about 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, gamma-acyclicity, beta-acyclicity and alpha-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 a:e in P boolean AND RNC and which are NP-complete.