The Complexity of Finding Arborescences in Hypergraphs

Basic definitions. For X a finite set of vertices, a hypergraph H on X is a family of subsets (called edges) of X. In a directed hypergraph H, every set h E H contains a distinguished element called the head of h (a set h may appear more than once in H but with different heads). A hypergraph H’ is a subhypergraph of H iff H’ c H. A directed hypergraph H is an arborescence iff (i) H is empty, or if (ii> there exists an edge h, E H such that no other h f h, in H contains the head of h ,, and such that H {h,} is also an arborescence. A cycle in a directed hypergraph is a sequence h,, . . . , h,_, of edges such that hi contains the head of h, + , (indices are taken modulo k). Note that in case a directed hypergraph is an arborescence, it cannot contain any cycle. For the case that all edges in the hypergraph are of cardinality two (i.e. the hypergraph is a