A New Approach for Approximating Directed Rooted Networks

We consider the k-outconnected directed Steiner tree problem (k-DST). Given a directed edge-weighted graph $G=(V,E,w)$, where V=\∪ S ∪ T, and an integer k, the goal is to find a minimum cost subgraph of G in which there are k edge-disjoint $rt$-paths for every terminal t∈ T. The problem is know to be NP-hard. Furthermore, the question on whether a polynomial time, subpolynomial approximation algorithm exists for k-DST was answered negatively by Grandoni et al. (2018), by proving an approximation hardness of Ω (|T|/log |T|) under NP≠ ZPP. Inspired by modern day applications, we focus on developing efficient algorithms for k-DST in graphs where terminals have out-degree $0$, and furthermore constitute the vast majority in the graph. We provide the first approximation algorithm for k-DST on such graphs, in which the approximation ratio depends (primarily) on the size of S. We present a randomized algorithm that finds a solution of weight at most O(k|S|log |T|) times the optimal weight, and with high probability runs in polynomial time.