The Two-Edge Connectivity Survivable-Network Design Problem in Planar Graphs

Abstract

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 ( n log 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.