On Hop-Constrained Steiner Trees in Tree-Like Metrics

We consider the problem of computing a Steiner tree of minimum cost under a hop constraint that requires the depth of the tree to be at most k. Our main result is an exact algorithm for metrics induced by graphs with bounded treewidth that runs in time n^O(k). For the special case of a path, we give a simple algorithm that solves the problem in polynomial time, even if k is part of the input. The main result can be used to obtain, in quasi-polynomial time, a near-optimal solution that violates the k-hop constraint by at most one hop for more general metrics induced by graphs of bounded highway dimension and bounded doubling dimension. For nonmetric graphs, we rule out an o(log n)-approximation, assuming P ≠ NP even when relaxing the hop constraint by any additive constant.