  Let $M$ be a representation of an acyclic quiver $Q$ over an infinite field
$k$. We establish a deterministic algorithm for computing the Harder-Narasimhan
filtration of $M$. The algorithm is polynomial in the dimensions of $M$, the
weights that induce the Harder-Narasimhan filtration of $M$, and the number of
paths in $Q$. As a direct application, we also show that when $k$ is
algebraically closed and when $M$ is unstable, the same algorithm produces
Kempf's maximally destabilizing one parameter subgroups for $M$.
