The Two-Edge Connectivity Survivable Network Problem in Planar Graphs

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

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.