The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems - Feldmann, Andreas Emil; Marx, Daniel (arXiv.org, 2022)
Given a directed graph \(G\) and a list \((s_1,t_1),…,(s_d,t_d)\) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of \(G\) that contains a directed \(s_i→ t_i\) path for every \(1\le i \le k\). The special case Directed Steiner Tree (when we ask for paths from a root \(r\) to terminals \(t_1,…,t_d\)) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every \(t_i\) to every other \(t_j\)) is known to be W[1]-hard. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if \(\mathcal{H}\) is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list \((s_1,t_1),…,(s_d,t_d)\) of requests form a directed graph that is a member of \(\mathcal{H}\). Our main result is a complete characterization of the classes \(\mathcal{H}\) resulting in fixed-parameter tractable special cases: we show that if every pattern in \(\mathcal{H}\) has the combinatorial property of being "transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges," then the problem is FPT, and it is W[1]-hard for every recursively enumerable \(\mathcal{H}\) not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], \(q\)-Root Steiner Tree is FPT for constant \(q\) [Suchý, WG 2016], Strongly Connected Steiner Subgraph is W[1]-hard [Guo et al., SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before.