We consider the following weakened version of the Maximal Acyclic Subgraph (MAS) problem: given a directed graph, find minimal $r$ such that one can make the graph acyclic by cutting at most $r$ incoming edges from each vertex. We present an efficient algorithm of solution for this problem and for some of its modifications: when each vertex is assigned with its own maximal number $r_i$ of cut edges, when some of edges are "untouchable", etc. The same algorithm can be modified for approximate solution of the classical MAS problem. Numerical results are provided for random graphs with the number of vertices from 50 to 1500. They show the rate of approximation for the MAS problem around 0.6. The main idea is based on finding the closest non-negative Schur stable matrix. In particular, recent methods of minimizing spectral radius over special families of matrices are put to good use in those problems.

中文翻译：

---

最大无环子图和最近稳定矩阵

我们考虑以下最大无环子图 (MAS) 问题的弱化版本：给定一个有向图，找到最小的 $r$，这样可以通过从每个顶点切割最多 $r$ 的传入边来使图无环。我们为这个问题及其一些修改提出了一个有效的解决方案算法：当每个顶点分配有自己的最大切割边数 $r_i$ 时，当一些边是“不可触及的”等时，相同的算法可以是为经典 MAS 问题的近似解而修改。提供了顶点数从 50 到 1500 的随机图的数值结果。它们显示 MAS 问题的近似率约为 0.6。主要思想是基于找到最接近的非负 Schur 稳定矩阵。特别是，