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 the semistability and $\sigma$-semistability, finding the maximizers of King's criterion, and computing the Harder--Narasimhan filtration. We also investigate a class of polyhedral cones defined by the linear system in King's criterion, which we refer to as King cones. For rank-one representations, we demonstrate that these King cones can be encoded by submodular flow polytopes, enabling us to decide the $\sigma$-semistability in strongly polynomial time. Our approach 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。针对箭图表示半稳定性的若干基本计算问题，我们提出了高效算法：包括判定半稳定性与$\sigma$-半稳定性、求解King准则的极大化子、计算Harder--Narasimhan滤过。我们还研究了由King准则中的线性系统定义的一类多面锥，称之为King锥。对于秩为一的表示，我们证明这些King锥可通过次模流多面体进行编码，从而能够在强多项式时间内判定$\sigma$-半稳定性。我们的方法利用了箭图表示中的次模性质，该性质本身可能具有独立的研究价值。