# 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 π is a class of directed graphs closed under identification of vertices, then the π-Steiner Network (π-SN) problem is the special case where the demand graph D is restricted to be from π. 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 π is a superset of one of the five hard families, then π-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 π-SN problem on planar graphs, but the special case called Strongly Connected Steiner Subgraph can be solved in time f(k) n^O(β(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 π-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(β(k)), but not in time f(k)n^o(β(k)), or (3) solvable in time f(k)n^O(k), but not in time f(k)n^o(k).

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