On the Intractability Landscape of Digraph Intersection Representations

We study the classical graph intersection number problem [Erdos et al., CJM1966] for directed acyclic graphs as recently proposed in [Kostochka et al, ISIT2019]. We prove a strong inapproximability result for arbitrary DAGs. We show that the problem is NP-hard when restricted to arborescences, which strongly contrasts with the existence of a trivial linear time solution for the corresponding problem on undirected trees. For the restriction of the problem to the case of arborescences, we complement the hardness result with an asymptotic FPTAS, which significantly improves on a previously known 2-approximation algorithm.