Approximating Minimum Cost Steiner Forests

We consider the k-Directed Steiner Forest (k-DSF) problem: given a directed graph G = (V, E) with edge costs, a collection D ⊆ V × V of ordered node pairs, and an integer k ≤ |D|, find a minimum cost subgraph H of G that contains an st-path for (at least) k pairs (s, t) ∈ D. When k = |D|, we get the Directed Steiner Forest (DSF) problem. The best known approximation ratios for these problems are: Õ(k2/3) for k-DSF by Charikar et al. [3], and O(k1/2+ε) for DSF by Chekuri et al. [4]. We improve these approximation ratios as follows. For DSF we give an Õ(n4/5+ε)-approximation scheme using a novel LP-relaxation that seeks to connect pairs with “cheap” paths. This is the first sub-linear (in terms of n = |V |) approximation ratio for the problem; all previous algorithm had ratio Ω(n1+ε), which can be trivially derived from known algorithms for the Directed Steiner Tree problem. For k-DSF we give a simple greedy O(k1/2+ε)-approximation algorithm. This improves the best known ratio Õ(k2/3) by Charikar et al. [3], and (almost) matches in terms of k the best ratio known for the undirected variant [2]. Even when used for the particular case of DSF, our algorithm favorably compares to the one of [4], which repeatedly solves linear programs and uses complex space and time consuming transformations. Our algorithm is much simpler and faster, since it essentially reduces k-DSF to a variant of the Directed Steiner Tree problem. The simplification is due to a new notion of “junction star-tree” – a union of an in-star and an out-branching having the same root, which is of independent interest.