On the Approximability of DAG Edge Deletion

The DAG Edge Deletion problem of k, or DED(k), is to delete the minimum weight set of edges from a directed acyclic graph such that the remaining graph has no path of length k. In 2012, Svensson showed it was hard to approximate the vertex deletion version with a ratio better than k assuming the Unique Games Conjecture is true [14], and in 2015 Kenkre et al. introduced the DED(k) problem and used this result to show that DED(k) is UGC-hard to approximate better than bk2 c for k ≥ 4 [6]. They motivate it as the minimization version of the problem Max-kOrdering on DAGs, which is to assign a labeling ` : V → [k] that maximizes the number of edges (i, j) with `(i) < `(j). This in turn has applications to scheduling jobs with soft precedence constraints. However, the best known approximation algorithm for DED(k), given by Kenkre et al. [6], has a ratio of k. In this work we tighten this gap by giving a ( 2 3k+O(1))-approximation for DED(k), as well as a 1.375-approximation for DED(2), a 2-approximation for DED(3), and a 3 4k-approximation for small k. We then address a combinatorial problem of finding a class of DAGs for which the Max-k-Ordering is as small as possible. We show that a DAG given by Alon et al. [1] that was shown to have a maximum directed cut of size at most ( 14 + o(1))|E| in fact has a Max-k-Ordering of at most (k−1 2k + o(1))|E|. From this result follows a k+1 2 integrality gap for Kenkre et al.’s LP, nearly matching the UGC-hardness factor. We state finding either a k+1 2 rounding for this LP or an improved integrality gap as an open problem.