Minimum Path Cover in Parameterized Linear Time
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A minimum path cover (MPC) of a directed acyclic graph (DAG) G = (V,E) is a minimum-size set of paths that together cover all the vertices of the DAG. Computing an MPC is a basic polynomial problem, dating back to Dilworth's and Fulkerson's results in the 1950s. Since the size k of an MPC (also known as the width) can be small in practical applications, research has also studied algorithms whose running time is parameterized on k. We obtain a new MPC parameterized algorithm for DAGs running in time O(k^2|V| + |E|). Our algorithm is the first solving the problem in parameterized linear time. Additionally, we obtain an edge sparsification algorithm preserving the width of a DAG but reducing |E| to less than 2|V|. This algorithm runs in time O(k^2|V|) and requires an MPC of a DAG as input, thus its total running time is the same as the running time of our MPC algorithm.
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