Complexity of the Steiner Network Problem with Respect to the Number of Terminals
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In the Directed Steiner Network problem we are given an arc-weighted digraph G, a set of terminals T ⊆ V(G), and an (unweighted) directed request graph R with V(R)=T. Our task is to output a subgraph G' ⊆ G of the minimum cost such that there is a directed path from s to t in G' for all st ∈ A(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, as this 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, 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(R)· |V(G)|^O(c_g · |T|), where c_g is a constant depending solely on the genus of G and f is a computable function. We complement this result by showing that there is no f(R)· |V(G)|^o(|T|^2/ |T|) algorithm for any function f for the problem on general graphs, unless ETH fails.
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