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
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
u
,
r
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 logn).
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.