On Hop-Constrained Steiner Trees in Tree-Like Metrics

We consider the problem of computing a Steiner tree minimum cost under hop constraint which requires depth 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 special case path, we give simple solves polynomial time, even if $k$ part input. The can used obtain, quasi-polynomial near-optimal solution violates $k$-hop one more general highway dimension and doubling dimension. non-metric graphs, rule out $o(\log n)$-approximation, assuming P$\neq$NP when relaxing any additive constant.