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

Abstract

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 (FOCS '99; SICOMP '06) 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 log 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.The algorithm eventually relies on the excluded grid theorem for planar graphs, but we stress that it is not simply a straightforward application of treewidth-based techniques: we need several layers of abstraction to arrive to a problem formulation where the speedup due to planarity can be exploited. 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 2O(k log k) · nO(√k) algorithm for planar directed graphs can be generalized to graphs excluding a fixed minor.• 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/logk) algorithm for any computable function f.• Feldman and Ruhl generalized their nO(k) algorithm to the more general Directed Steiner Forest (DSF) 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 that, assuming ETH, there is no f(k) · no(k) time algorithm for DSF on acyclic planar graphs.