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 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. * For the omitted proofs and further discussion, refer to the full version of the paper [6] .
Bateni et al. REF give approximation schemes for the classical Steiner Forest Problem on bounded treewidth graphs.
Approximation schemes for steiner forest on planar graphs and graphs of bounded treewidth
STOC '10
Mathematics Computer Science