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 $σ$-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 $σ$-semistability in strongly polynomial time. Our approach employs submodularity in quiver representations, which may be of independent interest.