Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth

MohammadHossein Bateni; MohammadTaghi Hajiaghayi; Dániel Marx

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arXiv · 0911.5143

Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth

MohammadHossein Bateni·MohammadTaghi Hajiaghayi·Dániel Marx

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Abstract

We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus. As a first step, we show how to build a Steiner forest spanner for such graphs. The crux of the process is a clustering procedure called prize-collecting clustering that breaks down the input instance into separate subinstances which are easier to handle; moreover, the terminals in different subinstances are far from each other. Each subinstance has a relatively inexpensive Steiner tree connecting all its terminals, and the subinstances can be solved (almost) separately. Another building block is a PTAS for Steiner forest on graphs of bounded treewidth. Surprisingly, Steiner forest is NP-hard even on graphs of treewidth 3. Therefore, our PTAS for bounded treewidth graph needs a nontrivial combination of approximation arguments and dynamic programming on the tree decomposition. We further show that Steiner forest can be solved in polynomial time for series-parallel graphs (graphs of treewidth at most two) by a novel combination of dynamic programming and minimum cut computations, completing our thorough complexity study of Steiner forest in the range of bounded treewidth graphs, planar graphs, and bounded genus graphs.

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MohammadHossein Bateni, MohammadTaghi Hajiaghayi, Dániel Marx. 2009-11-26. Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth. https://arxiv.org/abs/0911.5143

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