We study the semistability of quiver representations from an algorithmic perspective. We present efficient algorithms for several fundamental computational problems on the semistability of quiver representations: deciding semistability and $\sigma$-semistability, finding maximizers of King's criterion, and finding the Harder-Narasimhan filtration. We also investigate a class of polyhedral cones defined by the linear system in King's criterion, which we call King cones. We demonstrate that the King cones for rank-one representations can be encoded by submodular flow polytopes, allowing us to decide the $\sigma$-semistability of rank-one representations in strongly polynomial time. Our argument employs submodularity in quiver representations, which may be of independent interest.

翻译：本文从算法角度研究箭图表示的半稳定性。我们针对箭图表示半稳定性的若干基本计算问题提出了高效算法：判定半稳定性与$\sigma$-半稳定性、寻找King准则的极大化子、以及计算Harder-Narasimhan滤过。同时，我们研究了一类由King准则中的线性系统定义的多面体锥，称之为King锥。我们证明了秩1表示的King锥可通过次模流多胞形进行编码，从而能够在强多项式时间内判定秩1表示的$\sigma$-半稳定性。我们的证明过程运用了箭图表示中的次模性质，该性质可能具有独立的研究价值。