O(log⁡2k/log⁡log⁡k)O(\log^2k/\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 arborescence 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. By adjusting the parameters in the hardness result of Halperin and Krauthgamer, we show the matching lower bound of $\Omega(\log^2{k}/\log\log{k})$ for the class of quasi-polynomial-time algorithms. This is the first improvement on the DST problem since the classical quasi-polynomial-time $O(\log^3 k)$ approximation algorithm by Charikar et al. (The paper erroneously claims an $O(\log^2k)$ 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 Label-Consistent Subtree (LCST) problem. The LCST instance has quasi-polynomial size and logarithmic height. We remark that, in contrast, Zelikovsky's heigh-reduction theorem 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 LCST instances, which is inspired by the framework developed by Rothvo{\ss}. We consider a Sherali-Adams lifting of a proper LP relaxation of LCST. 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. A small enough (namely, polylogarithmic) number of Sherali-Adams lifting levels is sufficient to condition up to the leaves

Tree Embeddings for Two-Edge-Connected Network Design

The group Steiner problem is a classical network design problem where we are given a graph and a collection of groups of vertices, and want to build a min-cost subgraph that connects the root vertex to at least one vertex from each group. What if we wanted to build a subgraph that two-edge-connects the root to each group—that is, for every group g ⊆ V, the subgraph should contain two edge-disjoint paths from the root to some vertex in g? What if we wanted the two edge-disjoint paths to end up at distinct vertices in the group, so that the loss of a single member of the group would not destroy connectivity? In this paper, we investigate tree-embedding techniques that can be used to solve these and other 2-edge-connected network design problems. We illustrate the potential of these techniques by giving poly-logarithmic approximation algorithms for two-edge-connected versions of the group Steiner, connected facility location, buy-at-bulk, and the k-MST problems

Tree embeddings for two-edge-connected network design

The group Steiner problem is a classical network design problem where we are given a graph and a collection of groups of vertices, and want to build a min-cost subgraph that connects the root vertex to at least one vertex from each group. What if we wanted to build a subgraph that two-edge-connects the root to each group—that is, for every group g ⊆ V, the subgraph should contain two edge-disjoint paths from the root to some vertex in g? What if we wanted the two edgedisjoint paths to end up at distinct vertices in the group, so that the loss of a single member of the group would not destroy connectivity? In this paper, we investigate tree-embedding techniques that can be used to solve these and other 2-edgeconnected network design problems. We illustrate the potential of these techniques by giving poly-logarithmic approximation algorithms for two-edge-connected versions of the group Steiner, connected facility location, buy-at-bulk, and the k-MST problems.