Tight Localizations of Feedback Sets

The classical NP–hard feedback arc set problem (FASP) and feedback vertex set problem (FVSP) ask for a minimum set of arcs ε ⊆ E or vertices ν ⊆ V whose removal G ∖ ε, G ∖ ν makes a given multi–digraph G =( V , E ) acyclic, respectively. Though both problems are known to be APX–hard, constant ratio approximations or proofs of inapproximability are unknown. We propose a new universal O (| V || E | 4 )–heuristic for the directed FASP. While a ratio of r ≈ 1.3606 is known to be a lower bound for the APX–hardness, at least by empirical validation we achieve an approximation of r ≤ 2. Most of the relevant applications, such as circuit testing , ask for solving the FASP on large sparse graphs, which can be done efficiently within tight error bounds with our approach.