The Generalized Dependency Constrained Spanning Tree Problem

We introduce the Generalized Dependency Constrained Spanning Tree Problem (G-DCST), where an edge can be chosen only if the number of edges chosen from its dependency set lies in a certain interval. The dependency relations between the edges of the input graph G are described by the input digraph D, whose vertices are the edges of G. The in-neighbors of a vertex of D form its dependency set. We show that G-DCST unifies and generalizes some known spanning tree problems that apply conflict constraints over edges or lower and upper bounds on vertex degrees. We show that the feasibility version of G-DCST is NP-complete even under strong restrictions on the structures of G and D as well as on the functions that define the minimum or maximum number of dependencies to be satisfied. We also show that this problem keeps an lnn inapproximability threshold under tight assumptions over G and D. On the other hand, we spot a polynomial case via a matroid intersection argument.