  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.
