Abstract
In the Directed Steiner Network problem, the input is a directed graph G, a
subset T of k vertices of G called the terminals, and a demand graph D on T.
The task is to find a subgraph H of G with the minimum number of edges such
that for every edge (s,t) in D, the solution H contains a directed s to t path.
In this paper we investigate how the complexity of the problem depends on the
demand pattern when G is planar. Formally, if \mathcal{D} is a class of
directed graphs closed under identification of vertices, then the
\mathcal{D}-Steiner Network (\mathcal{D}-SN) problem is the special case where
the demand graph D is restricted to be from \mathcal{D}. For general graphs,
Feldmann and Marx [ICALP 2016] characterized those families of demand graphs
where the problem is fixed-parameter tractable (FPT) parameterized by the
number k of terminals. They showed that if \mathcal{D} is a superset of one of
the five hard families, then \mathcal{D}-SN is W[1]-hard parameterized by k,
otherwise it can be solved in time f(k)n^{O(1)}.
For planar graphs an interesting question is whether the W[1]-hard cases can
be solved by subexponential parameterized algorithms. Chitnis et al. [SICOMP
2020] showed that, assuming the ETH, there is no f(k)n^{o(k)} time algorithm
for the general \mathcal{D}-SN problem on planar graphs, but the special case
called Strongly Connected Steiner Subgraph can be solved in time f(k)
n^{O(\sqrt{k})} on planar graphs. We present a far-reaching generalization and
unification of these two results: we give a complete characterization of the
behavior of every $\mathcal{D}$-SN problem on planar graphs. We show that
assuming ETH, either the problem is (1) solvable in time 2^{O(k)}n^{O(1)}, and
not in time 2^{o(k)}n^{O(1)}, or (2) solvable in time f(k)n^{O(\sqrt{k})}, but
not in time f(k)n^{o(\sqrt{k})}, or (3) solvable in time f(k)n^{O(k)}, but not
in time f(k)n^{o({k})}.
