Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

The DIRECTED STEINER NETWORK (DSN) problem takes as input a directed edge-weighted graph G = (V, E) and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N ⊆ G for which there is an s → t path for each (s,t) ∈ D. It is known that this problem is notoriously hard as there is no k^1/4-o(1)-approximation algorithm under Gap-ETH, even when parameterizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k.

For the BI-DSN_PLANAR problem, the aim is to compute a planar optimum solution N ⊆ G in a bidirected graph G, i.e. for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of BI-DSN_PLANAR, unless FPT=W[1]. Additionally we study several generalizations of BI-DSN_PLANAR and obtain upper and lower bounds on obtainable runtimes parameterized by k.

One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-approximation algorithm exists under Gap-ETH. Additionally, we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k.