Consider the following problem: given a graph with edge-weights and a subset Q of vertices, find a minimum-weight subgraph in which there are two edge-disjoint paths connecting every pair of vertices in Q. The problem is a failure-resilient analog of the Steiner tree problem, and arises in telecommunications applications. A more general formulation, also employed in telecommunications optimization, assigns a number (or requirement) r{sub}v ∈{0, 1, 2} to each vertex v in the graph; for each pair u, v of vertices, the solution network is required to contain min{r{sub}u, r{sub}v} edge-disjoint u-to-v paths. We address the problem in planar graphs, considering a popular relaxation in which the solution is allowed to use multiple copies of the input-graph edges (paying separately for each copy). The problem is SNP-hard in general graphs and NP-hard in planar graphs. We give the first polynomial-time approximation scheme in planar graphs. The running time is O(n log n). Under the additional restriction that the requirements are in {0, 2} for vertices on the boundary of a single face of a planar graph, we give a linear-time algorithm to find the optimal solution.
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