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arXiv · 2003.05699

On Hop-Constrained Steiner Trees in Tree-Like Metrics

Martin Böhm·Ruben Hoeksma·Nicole Megow·Lukas Nölke·Bertrand Simon
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Abstract

We consider the problem of computing a Steiner tree of minimum cost under a hop constraint which 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 non-metric graphs, we rule out an $o(\log n)$-approximation, assuming P$\neq$NP even when relaxing the hop constraint by any additive constant.

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Martin Böhm, Ruben Hoeksma, Nicole Megow, Lukas Nölke, Bertrand Simon. 2020-03-12. On Hop-Constrained Steiner Trees in Tree-Like Metrics. https://doi.org/10.1137/21m1425487

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