Abstract
The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform a given multi digraph G = (V, E) into an acyclic graph by deleting as few arcs (vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one of the classic NP-complete problems. An important contribution of this paper is that the subgraphs Gel(e), Gsi(e) of all elementary cycles or simple cycles running through some arc e ∈ E, can be computed in mathcal {O}big (|E|^{2}big ) and mathcal {O}(|E|^{4}), respectively. We use this fact and introduce the notion of the essential minor and isolated cycles, which yield a priori problem size reductions and in the special case of so called resolvable graphs an exact solution in mathcal {O}(|V||E|^{3}). We show that weighted versions of the FASP and FVSP possess a Bellman decomposition, which yields exact solutions using a dynamic programming technique in times mathcal {O}big (2^{m}|E|^{4}log (|V|)big ) and mathcal {O}big (2^{n}Delta (G)^{4}|V|^{4}log (|E|)big ), where m ≤|E|−|V | + 1, n ≤ (Δ(G) − 1)|V |−|E| + 1, respectively. The parameters m, n can be computed in mathcal {O}(|E|^{3}), mathcal {O}(Delta (G)^{3}|V|^{3}), respectively and denote the maximal dimension of the cycle space of all appearing meta graphs, decoding the intersection behavior of the cycles. Consequently, m, n equal zero if all meta graphs are trees. Moreover, we deliver several heuristics and discuss how to control their variation from the optimum. Summarizing, the presented results allow us to suggest a strategy for an implementation of a fast and accurate FASP/FVSP-SOLVER.
Highlights
The feedback arc set problem, shortened Feedback Arc Set Problem (FASP), is to delete as less as possible arcs of a graph such that the resulting subgraph is acyclic, i.e., it contains no directed cycle
The problem is known as maximum acyclic subgraph problem or linear ordering problem
The problem of deleting a smallest subset of vertices to result in an acyclic subgraph is known as feedback vertex set problem (FVSP)
Summary
The feedback arc set problem, shortened FASP, is to delete as less as possible arcs of a graph such that the resulting subgraph is acyclic, i.e., it contains no directed cycle. Another equivalent formulation is to find a linear ordering of the vertices of the graph such that the number of back arcs is minimized. The problem is known as maximum acyclic subgraph problem or linear ordering problem For directed graphs this problem is one of the classic NP-complete problems [22]. The problem of deleting a smallest subset of vertices to result in an acyclic subgraph is known as feedback vertex set problem (FVSP). The FASP or FVSP has a multitude of applications, e.g., retiming synchronous circuitry [25], circuit testing [24], computational biology and neuroscience [19], network analysis and operating systems [33]
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