$O(\log^2{k}/\log\log{k})$-Approximation Algorithm for Directed Steiner Tree: A Tight Quasi-Polynomial Time Algorithm

Abstract

In the directed Steiner tree (DST) problem, we are given an $n$-vertex directed edge-weighted graph, a root $r$, and a collection of $k$ terminal nodes. Our goal is to find a minimum-cost subgraph that contains a directed path from $r$ to every terminal. We present an $O(\log^2 k/\log\log{k})$-approximation algorithm for DST that runs in quasi-polynomial time, i.e., in time $n^{{poly}\log (k)}$. By assuming the projection game conjecture and ${NP}\not\subseteq{\bigcap}_{0<\epsilon<1}{ZPTIME}(2^{n^\epsilon})$ and adjusting the parameters in the hardness result of [Halperin and Krauthgamer, Polylogarithmic inapproximability, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 585--594], we show the matching lower bound of $\Omega(\log^2{k}/\log\log{k})$ for the class of quasi-polynomial time algorithms, meaning that our approximation ratio is asymptotically the best possible. Our algorithm is proceeded by reducing DST to an intermediate problem, namely, the group Steiner tree on trees with dependency constraint problem, which we approximate using the framework developed by [Rothvoß, Directed Steiner Tree and the Lasserre Hierarchy, preprint, arxiv:1111.5473, 2011] and [Friggstad et al., Linear programming hierarchies suffice for directed Steiner tree, in Proceedings of the 17th Annual Conference on Integer Programming and Combinatorial Optimization, 2014, pp. 285--296].