# Exact Localisations of Feedback Sets

## TL;DR

This paper introduces exact algorithms and structural insights for the feedback arc and vertex set problems, enabling more efficient solutions for transforming graphs into acyclic structures, with practical heuristics for improved performance.

## Contribution

It presents polynomial-time computable subgraph structures, introduces the concept of essential minors and isolated cycles, and develops dynamic programming methods for exact solutions in special cases.

## Key findings

- Polynomial-time computable subgraphs for cycle analysis
- Exact solutions for resolvable graphs in polynomial time
- Dynamic programming algorithms with exponential parameters for weighted problems

## 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 $G_{\mathrm{el}}(e)$, $G_{\mathrm{si}}(e)$ 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.