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 $\ell : V \rightarrow [k]$ maximizing the number of forward edges, i.e. edges (u, v) such that $\ell (u) < \ell (v)$. 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 $k=\{2,\ldots , n\}$, improving on the known $\left. 2k\big /(k-1)\right.$-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any $k=\{2,\ldots , n\}$ and constant $\varepsilon > 0$, Max-$k$-Ordering has an LP integrality gap of $2 - \varepsilon$ for $n^{\varOmega \left( \left. 1\big /\log \log k\right. \right) }$ 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 $S_v \subseteq \mathbb {Z}^+$. We prove an LP rounding based $\left. 4\sqrt{2}\big /\left( \sqrt{2}+1\right) \right. \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ 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 $k\in [n]$. 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).