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. This is based on the notion of frontier antichains , generalizing the standard notion of right-most maximum antichain .
After revision:  Second, we obtain an O(poly(k)(2} ^k|E| + 4^k|V|) ) time algorithm, based on a new notion of frontier antichains , generalizing the standard notion of right-most maximum antichain .