A Linear-Time Parameterized Algorithm for Computing the Width of a DAG.

The width k of a directed acyclic graph (DAG) \(G = (V, E)\) equals the largest number of pairwise non-reachable vertices. Computing the width dates back to Dilworth’s and Fulkerson’s results in the 1950s, and is doable in quadratic time in the worst case. Since k can be small in practical applications, research has also studied algorithms whose complexity is parameterized on k. Despite these efforts, it is still open whether there exists a linear-time \(O(f(k)(|V| + |E|))\) parameterized algorithm computing the width . We answer this question affirmatively by presenting an \(O(k^24^k|V| + k2^k|E|)\) time algorithm, based on a new notion of frontier antichains. As we process the vertices in a topological order, all frontier antichains can be maintained with the help of several combinatorial properties, paying only f(k) along the way. The fact that the width can be computed by a single f(k)-sweep of the DAG is a new surprising insight into this classical problem. Our algorithm also allows deciding whether the DAG has width at most w in time \(O(f(\min (w,k))(|V|+|E|))\).