A multi-objective perspective on the cable-trench problem

The cable-trench problem is defined as a linear combination of the shortest path and the minimum spanning tree problem. In particular, the goal is to find a spanning tree that simultaneously minimizes its total length and the total path length from a pre-defined root to all other vertices. Both, the minimum spanning tree and the shortest path problem are known to be efficiently solvable. However, a linear combination of these two objectives results in a highly complex problem. In this article, we introduce the bi-objective cable-trench problem which separates the two cost functions. We show that in general, the bi-objective formulation has additional compromise solutions compared to the cable-trench problem in its original formulation. To determine the set of non-dominated points and efficient solutions, we use \(\varepsilon \)-constraint scalarizations in combination with a problem-specific cutting plane. Moreover, we present numerical results on different types of graphs analyzing the impact of density and cost structure on the cardinality of the non-dominated set and the solution time.