Abstract
The Cable-Trench Problem (CTP) is an optimization problem that generalizes both the Single-Destination Shortest Path Problem and the Minimum Spanning Tree Problem. Given an edge weighted graph with a special root vertex and parameters τ, γ ≥ 0, the objective is to find a rooted spanning tree that minimizes the weight of the tree, scaled by τ, plus the sum of the weights over all shortest paths from the root, scaled by γ. While each of the generalized problems are well-known to be polynomial-time solvable, CTP is NP-hard. In this paper, we show that even finding an approximation with factor of 1.000475 is NP-hard, thus ruling out the existence of a polynomial-time approximation scheme, unless P = NP.
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