O(log² k/ log log k)-approximation algorithm for directed steiner tree: A tight quasi-polynomial-time algorithm

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(log² 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 ⊈ ^\₀<ϵ <₁ZPTIME(2^nϵ ), and adjusting the parameters in the hardness result of Halperin and Krauthgamer [STOC’03], we show the matching lower bound of Ω(log² k/ log log k) for the class of quasi-polynomial-time algorithms, meaning that our approximation ratio is asymptotically the best possible. This is the first improvement on the DST problem since the classical quasi-polynomial-time O(log³ k) approximation algorithm by Charikar et al. [SODA’98 & J. Algorithms’99]. (The paper erroneously claims an O(log² k) approximation due to a mistake in prior work.) Our approach is based on two main ingredients. First, we derive an approximation preserving reduction to the Group Steiner Tree on Trees with Dependency Constraint (GSTTD) problem. Compared to the classic Group Steiner Tree on Trees problem, in GSTTD we are additionally given some dependency constraints among the nodes in the output tree that must be satisfied. The GSTTD instance has quasi-polynomial size and logarithmic height. We remark that, in contrast, Zelikovsky’s heigh-reduction theorem [Algorithmica’97] used in all prior work on DST achieves a reduction to a tree instance of the related Group Steiner Tree (GST) problem of similar height, however losing a logarithmic factor in the approximation ratio. Our second ingredient is an LP-rounding algorithm to approximately solve GSTTD instances, which is inspired by the framework developed by [Rothvoß, Preprint’11; Friggstad et al., IPCO’14]. We consider a Sherali-Adams lifting of a proper LP relaxation of GSTTD. Our rounding algorithm proceeds level by level from the root to the leaves, rounding and conditioning each time on a proper subset of label variables. The limited height of the tree and small number of labels on root-to-leaf paths guarantee that a small enough (namely, polylogarithmic) number of Sherali-Adams lifting levels is sufficient to condition up to the leaves. We believe that our basic strategy of combining label-based reductions with a round-and-condition type of LP-rounding over hierarchies might find applications to other related problems.