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We abstract and study reachability preservers, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph G=(V,E) and a set of demand pairs P⊆VxV, a reachability preserver is a sparse subgraph H that preserves reachability between all demand pairs. Our first contribution is a series of extremal bounds on the size of reachability preservers. Our main result states that, for an n-node graph and demand pairs of the form P⊆SxV for a small node subset S, there is always a reachability preserver on O(n+√n/P//S/) edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which O(n) size reachability preservers are generally possible, in a large range of parameters. The second contribution of this paper is a new connection between extremal graph sparsification results and classical Steiner Network Design problems. Surprisingly, prior to this work, the osmosis of techniques between these two fields had been superficial. This allows us to improve the state of the art approximation algorithms for the most basic Steiner-type problem in directed graphs from the O(n0.6+ε) of Chlamatac, et al. [Approximating spanners and directed steiner forest: Upper and lower bounds, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 534-443] (n4/7+ε). [ABSTRACT FROM AUTHOR]

Given a vertex-weighted directed graph G = (V,E) and a set T = { t1, t2, . . ., tk} 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 ti → tj 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 nO(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 2O(k)·nO(√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)·no(√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 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 Network Problems, 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 gridlike 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 2O(k)·nO(√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 nO(k) algorithm of Feldman and Ruhl: assuming ETH, SCSS in general graphs does not have an f(k)·no(k/ log k) algorithm for any computable function f. Feldman and Ruhl generalized their nO(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 si there is a path to the corresponding terminal ti. We show that, assuming ETH, there is no f(k)·no(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. [ABSTRACT FROM AUTHOR]