  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 $G_{\mathrm{el}}(e)$, $G_{\mathrm{si}}(e)$ of all elementary
cycles or simple cycles running through some arc $e \in E$, 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 $m \leq |E|-|V|
+1$, $n \leq (\Delta(G)-1)|V|-|E| +1$, 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.
