On the Approximability of Digraph Ordering

Given an n -vertex digraph D = (V, A) the Max - k - Ordering problem is to compute a labeling ℓ : V → [k] maximizing the number of forward edges, i.e. edges ( u , v ) such that ℓ (u) < ℓ (v) . For different values of k , this reduces to maximum acyclic subgraph ( k=n ), and Max-DiCut ( k=2 ). This work studies the approximability of Max - k - Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max - k - Ordering for any k={2,… , n} , improving on the known . 2k /(k-1). -approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,… , n} and constant ε > 0 , Max - k - Ordering has an LP integrality gap of 2 - ε for n^( . 1 /loglog k. ) rounds of the Sherali-Adams hierarchy. A further generalization of Max - k - Ordering is the restricted maximum acyclic subgraph problem or RMAS , where each vertex v has a finite set of allowable labels S_v ⊆ℤ^+ . We prove an LP rounding based . 4√(2) /( √(2)+1) . ≈ 2.344 approximation for it, improving on the 2√(2)≈ 2.828 approximation recently given by Grandoni et al. (Inf Process Lett 115(2): 182–185, 2015 ). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED (k) , which requires deleting the minimum number of edges from an n -vertex directed acyclic graph (DAG) to remove all paths of length k . We show that a simple rounding of the LP relaxation as well as a local ratio approach for DED (k) yields k -approximation for any k∈ [n] . A vertex deletion version was studied earlier by Paik et al. (IEEE Trans Comput 43(9): 1091–1096, 1994 ), and Svensson (Proceedings of the APPROX, 2012 ).