Before revision:  As such, it is enough to sweep the graph once from left to right, paying only f(k) along the way , which is a new surprising insight into the classical MPC problem .
After revision:  As such, it is enough to sweep the graph once from left to right, paying only f(k) along the way , which is a new surprising insight into this classical problem. Our algorithm also allows deciding whether the DAG has width at most w in time O(f(\min(w,k))(|V|+|E|)) .