On the complexity of the Cable-Trench Problem - Benedito, Marcelo P.L.; Pedrosa, Lehilton L.C.; Rosado, Hugo K.K. (Discrete Applied Mathematics, 2023)
The Cable-Trench Problem (CTP) is a common generalization of the Single-Source Shortest Paths Problem (SSSP) and the Minimum Spanning Tree Problem (MST): given an edge-weighted graph with a special root vertex and parameters τ,γ≥0, the goal is to find a spanning tree that minimizes the total edge costs plus the total cost of the paths from each vertex to the root, scaled by τ and γ, respectively. While it is well known that both SSSP and MST can be solved in polynomial time, CTP is NP-hard. We show that computing an approximate solution with factor less than 1.000475 is NP-hard, thus ruling out a polynomial-time approximation scheme, unless P=NP. We also consider the more general Steiner Cable-Trench Problem (SCTP), for which only a given subset of terminal vertices must be spanned by a solution. The tree might include non-terminal vertices, known as Steiner vertices, although only paths from terminals to the root are considered in the total cost. For this problem, we present a (2.88+ϵ)-approximation based on a counting argument, for any ϵ>0; also, we give a simple parameterized algorithm with the number of terminals as parameter.