Surviving in Directed Graphs: A Polylogarithmic Approximation for Two-Connected Directed Steiner Tree.

In this paper, we study a survivable network design problem on directed graphs, 2-Connected Directed Steiner Tree (2-DST): given an $n$-vertex weighted directed graph, a root $r$, and a set of $h$ terminals $S$, find a min-cost subgraph $H$ that has two edge/vertex disjoint paths from $r$ to any $tin S$. 2-DST is a natural generalization of the classical Directed Steiner Tree problem (DST), where we have an additional requirement that the network must tolerate one failure. No non-trivial approximation is known for 2-DST. This was left as an open problem by Feldman et al., [SODAu002709; JCSS] and has then been studied by Cheriyan et al. [SODAu002712; TALG] and Laekhanukit SODAu002714]. However, no positive result was known except for the special case of a $D$-shallow instance [Laekhanukit, ICALPu002716]. present an $O(D^3log Dcdot h^{2/D}cdot log n)$ approximation algorithm for 2-DST that runs in time $O(n^{O(D)})$, for any $Din[log_2h]$. This implies a polynomial-time $O(h^epsilon log n)$ approximation for any constant $epsilonu003e0$, and a poly-logarithmic approximation running in quasi-polynomial time. We remark that this is essentially the best-known even for the classical DST, and the latter problem is $O(log^{2-epsilon}n)$-hard to approximate [Halperin and Krauthgamer, STOCu002703]. As a by product, we obtain an algorithm with the same approximation guarantee for the $2$-Connected Directed Steiner Subgraph problem, where the goal is to find a min-cost subgraph such that every pair of terminals are $2$-edge/vertex connected.