Before revision:  Second, we obtain an O(poly(k)(2}  ^k|E| + 4^k|V|) ) -time algorithmfor the dual problem of computing the width of the DAG.
After revision:  Second, we obtain an O(poly(k)(2} + k2 ^k|E| + 4^k|V|) ) -time algorithmfor the dual problem of computing the width of the DAG.