### On the Approximability of Digraph Ordering

Sreyash Kenkre, Vinayaka Pandit, Manish Purohit, Rishi Saket
2016 Algorithmica
Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling : V → [k] maximizing the number of forward edges, i.e. edges (u, v) such that (u) < (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 2approximation algorithm
more » ... 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 Ω( 1/log log 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 Sv ⊆ 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. [7]. 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 both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any k ∈ [n]. A vertex deletion version was studied earlier by Paik et al. [17], and Svensson [18].