Tight bounds for planar strongly connected Steiner subgraph with fixed number of terminals (and extensions)

Given a vertex-weighted directed graph G=(V,E) and a set T={t₁ ,t₂ ,…t_k } of k terminals, the objective of the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem is to find a vertex set H⊆V of minimum weight such that G[H] contains a t_i →t_j path for each i≠j . The problem is NP-hard, but Feldman and Ruhl [SIAM J. Comput., 36 (2006), pp. 543--561] gave a novel n^O(k) algorithm for the SCSS problem, where n is the number of vertices in the graph and k is the number of terminals. We explore how much easier the problem becomes on planar directed graphs. Our main algorithmic result is a 2^O(k)⋅n^O(√k)   algorithm for planar SCSS, which is an improvement of a factor of O(√k)  in the exponent over the algorithm of Feldman and Ruhl. Our main hardness result is a matching lower bound for our algorithm: we show that planar SCSS does not have an f(k)⋅n^o(√k) algorithm for any computable function f , unless the exponential time hypothesis (ETH) fails. To obtain our algorithm, we first show combinatorially that there is a minimal solution whose treewidth is of O(√k) , and then use the dynamic-programming based algorithm for finding bounded-treewidth solutions due to Feldmann and Marx [The Complexity Landscape of Fixed-Parameter Directed Steiner NetworkProblems, preprint, https://arxiv.org/abs/1707.06808]. To obtain the lower bound matching the algorithm, we need a delicate construction of gadgets arranged in a grid-like fashion to tightly control the number of terminals in the created instance. The following additional results put our upper and lower bounds in context: our  2^O(k)⋅n^O(√k) algorithm for planar directed graphs can be generalized to graphs excluding a fixed minor. Additionally, we can obtain this running time for the problem of finding an optimal planar solution even if the input graph is not planar. In general graphs, we cannot hope for such a dramatic improvement over the n^O(k) algorithm of Feldman and Ruhl: assuming ETH, SCSS in general graphs does not have an f(k)⋅n^{o(k/ log k)} algorithm for any computable function f. Feldman and Ruhl generalized their n^O(k) algorithm to the more general Directed Steiner Network (DSN) problem; here the task is to find a subgraph of minimum weight such that for every source s_i there is a path to the corresponding terminal t_i. We show that, assuming ETH, there is no f(k)⋅n^o(k) time algorithm for DSN on acyclic planar graphs. All our lower bounds hold for the integer weighted edge version, while the algorithm works for the more general unweighted vertex version.