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

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 graphs needs a nontrivial combination of approximation arguments and dynamic programming on the tree decomposition. We further show that Steiner forest can be solved...

• Type: Conference
• Year: 2010
• Where: STOC
• Authors: MohammadHossein Bateni, MohammadTaghi Hajiaghayi and Dániel Marx
• Keywords: Planar Graphs, Steiner Forest, STOC 2010, Treewidth Graphs