# Optimal segmentation of directed graph and the minimum number of feedback arcs

## TL;DR

This paper introduces a novel approach to the minimum feedback arc set problem in directed graphs by using layered segmentation, mean field theory, belief propagation, and a divide-and-conquer algorithm to improve efficiency and performance.

## Contribution

It presents a new layered segmentation framework combined with statistical physics methods and a divide-and-conquer algorithm for the feedback arc set problem.

## Key findings

- Derived the minimum feedback arc density for random digraphs.
- Developed a highly efficient divide-and-conquer algorithm.
- Validated the approach with strong performance results.

## Abstract

The minimum feedback arc set problem asks to delete a minimum number of arcs (directed edges) from a digraph (directed graph) to make it free of any directed cycles. In this work we approach this fundamental cycle-constrained optimization problem by considering a generalized task of dividing the digraph into D layers of equal size. We solve the D-segmentation problem by the replica-symmetric mean field theory and belief-propagation heuristic algorithms. The minimum feedback arc density of a given random digraph ensemble is then obtained by extrapolating the theoretical results to the limit of large D. A divide-and-conquer algorithm (nested-BPR) is devised to solve the minimum feedback arc set problem with very good performance and high efficiency.