The two-edge connectivity survivable-network design problem in planar graphs

TL;DR

This paper presents a polynomial-time approximation scheme for the two-edge connectivity survivable-network design problem in planar graphs, addressing a failure-resilient network design challenge with practical telecommunications applications.

Contribution

It introduces the first polynomial-time approximation scheme for this problem in planar graphs and provides an exact polynomial-time algorithm under specific boundary conditions.

Findings

First PTAS with O(n log n) time in planar graphs.

NP-hardness established for general and planar cases.

Exact polynomial-time solution for boundary-restricted instances.

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) in {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.