In the Steiner Tree problem, a set of terminal vertices in an edge-weighted graph needs to be connected in the cheapest possible way. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on the one hand, Steiner Tree is known to be APX-hard, and on the other hand, it is W[2]-hard if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this, we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter.

In the Directed Steiner Network problem we are given an arc-weighted digraph G, a set of terminals T, and an (unweighted) directed request graph R with V(R)=T. Our task is to output a subgraph H of G of the minimum cost such that there is a directed path from s to t in H for all arcs st of R. It is known that the problem can be solved in time |V(G)|^O(|A(R)|) [Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time |V(G)|^o(|A(R)|) even if G is planar, unless Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, the reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time |V(G)|^o(|T|), unless ETH fails. Therefore, there is a significant gap in the complexity with respect to |T| in the exponent. We show that Directed Steiner Network is solvable in time f(|T|) · |V(G)| O(c_g · |T|), where c_g is a constant depending solely on the genus g of G and f is a computable function.