Maximal acyclic subgraphs and closest stable matrices

TL;DR

This paper introduces a matrix-based approach to approximate the Maximal Acyclic Subgraph problem by finding the closest nilpotent matrix, demonstrating fast convergence and high approximation rates through numerical experiments.

Contribution

It develops a novel matrix approach to MAS, providing an efficient algorithm and extending it to variants with vertex-specific constraints and applications.

Findings

Algorithm converges quickly with approximation rates over 0.6

Method solves a weakened MAS version exactly

Extensions handle vertex-specific and untouchable edges

Abstract

We develop a matrix approach to the Maximal Acyclic Subgraph (MAS) problem by reducing it to finding the closest nilpotent matrix to the matrix of the graph. Using recent results on the closest Schur stable systems and on minimising the spectral radius over special sets of non-negative matrices we obtain an algorithm for finding an approximate solution of MAS. Numerical results for graphs from 50 to 1500 vertices demonstrate its fast convergence and give the rate of approximation in most cases larger than 0.6. The same method gives the precise solution for the following weakened version of MAS: and the minimal $r$ such that the graph can be made acyclic by cutting at most $r$ incoming edges from each vertex. Several modifications, when each vertex is assigned with its own maximal number $r_i$ of cut edges, when some of edges are "untouchable", are also considered. Some applications are discussed.