On feedback vertex set in reducible flow hypergraphs

A directed hypergraph H = (V, A) is a finite set of vertices V and a set of hyper-arcs A, where each hyper-arc is an ordered pair of nonempty subsets of vertices. A flow hypergraph H = (V, A, s) is a triple, such that (V, A) is a directed hypergraph, s is an element of V is a distinguished vertex such that s reaches every vertex of V. Reducible flow hypergraphs are a generalization of Hecht and Ullman's reducible flowgraphs. The FEEDBACK VERTEX SET (FVS) decision problem has a directed hypergraph H and an integer k >= 0 as input and the question is whether there is V' subset of V vertical bar V'vertical bar <= k, such that H \ V' is an acyclic directed hypergraph. It is known that FVS is polynomial time solvable for reducible flowgraphs. In this article we prove that FVS is NP-complete for reducible flow hypergraphs showing a reduction from 3-SATISFIABILITY PROBLEM WITH AT MOST 3 OCCURRENCES PER VARIABLE (3SAT(3)(-)). We exhibit a polynomial-time Delta-approximation for FVS in reducible flow hypergraphs, where Delta is the maximum number of hyper-arcs adjacent to a vertex of H. (C) 2021 The Authors. Published by Elsevier B.V.