Subexponential Parameterized Directed Steiner Network Problems on Planar Graphs: a Complete Classification

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})}.