On the Approximability of Digraph Ordering [article]

Sreyash Kenkre, Vinayaka Pandit, Manish Purohit, Rishi Saket
2015 arXiv   pre-print
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
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/ 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. (Information Processing Letters, Vol. 115(2), Pages 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 both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any k∈ [n].
arXiv:1507.00662v1 fatcat:izonkxu2x5dijicqixmku4g7mi