The Two Edge Connectivity Survivable Network Design Problem in Planar Graphs
Consider the following problem: given a graph with edge costs and a subset Q of vertices, find a minimum cost 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 arising, for example, in telecommunications applications. We study a more general mixed connectivity formulation, also employed in telecommunications optimization. Given a number (or requirement) r(v) {0, 1, 2} for each vertex v in the graph, find a minimum cost subgraph in which there are min {r(u) r(v) edge disjoint u to v paths for every pair u, v of vertices. 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 max SNP hard in general graphs and strongly NP hard in planar graphs. We give the first polynomial time approximation scheme in planar graphs. The running time is O(nlog n) Under the additional restriction that the requirements are only non zero for vertices on the boundary of a single face of a planar graph, we give a polynomial time algorithm to find the optimal solution.
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