On the approximability of digraph ordering

Abstract

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 _(u) 0, Max-k-Ordering has an LP integrality gap of 2−ε for nΩ(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. [5]. 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. [16], and Svensson [17].