Steiner type problems for digraphs that are locally semicomplete or extended semicomplete

We consider the following three problems: (P1) Let D be a strong digraph and let X be a non‐empty subset of its vertices. Find a strong subdigraph D ′ of D which contains all vertices of X and has as few arcs as possible. This problem is also known under the name the directed Steiner problem. (P2) Let D be a strong digraph with real‐valued costs on the vertices. Find a strong subdigraph of D of minimum cost. (P3) Let D be a strong digraph with real‐valued costs on the vertices. Find a cycle of minimum cost in D . All three problems are NP‐hard for general digraphs. The second problem generalizes the problem of finding a smallest strong subdigraph (measured in the number of vertices) which covers a given non‐empty set X of vertices. We describe polynomial algorithms for the problems (P1) and (P2) in the case when D is either locally semicomplete or extended semicomplete and for (P3) in the case of locally semicomplete digraphs. A polynomial algorithm for (P3) in the case of extended semicomplete digraphs was given by Bang‐Jensen et al., 2002. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 193–207, 2003