# Parameterized Approximation Algorithms for Directed Steiner Network Problems

(See paper for full abstract) Given an edge-weighted directed graph G=(V,E) and a set of k terminal pairs {(s_i, t_i)}_i=1^k, the objective of the Directed Steiner Network (DSN) problem is to compute the cheapest subgraph N of G such that there is an s_i→ t_i path in N for each i∈ [k]. This problem is notoriously hard: there is no polytime O(2^^1-εn)-approximation [Dodis & Khanna, STOC '99], and it is W[1]-hard [Guo et al., SIDMA '11] for the well-studied parameter k. To circumvent these hardness results we consider parameterized approximations. Our first result is that under Gap-ETH there is no k^o(1)-approximation for DSN in f(k)· n^O(1) time, for any function f. Can we obtain faster algorithms or better approximation ratios for other special cases of DSN when parameterizing by k? A special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, where the goal is to pairwise connect all k given terminals. We show that under Gap-ETH there is no (2-ε)-approximation in f(k)· n^O(1) time, for any function f. This implies a tight parameterized approximation factor. Next we consider bidirected input graphs, i.e., for every edge uv the reverse edge vu exists and has the same weight. We call the respective problems bi-DSN and bi-SCSS and show that bi-SCSS is FPT, while bi-DSN is W[1]-hard. The former is the first special case where SCSS remains NP-hard but turns out to be FPT. A standard way to obtain special cases is to restrict the input graphs. We generalize this concept: for a class K of graphs, we define the bi-DSN_K problem as asking to find an optimum solution N such that N∈K. We give a parameterized approximation scheme for minor-closed K, and prove that no efficient parameterized approximation scheme exists.

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