The Two-Edge Connectivity Survivable Network Problem in Planar Graphs

Abstract

Abstract. Consider the following problem: given a graph with edgeweights 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) rv ∈ {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{ru, rv} 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.

Linear-time algorithms for max flow and multiple-source shortest paths in unit-weight planar graphs

“…Fast MSSP plays an essential role in a variety of recent fast algorithms for planar and bounded-genus graphs [5,7,11,22,26,10,20,27,29,28]. Other algorithms, including a series of fast approximation schemes, use variants of MSSP [1,2,4,12,14,24,25,34].…”

Polynomial-Time Approximation Schemes for Subset-Connectivity Problems in Bounded-Genus Graphs

“…The Structure Theorem for Steiner Tree is proved in [5], the case of {0, 1, 2}-edge-connectivity Survivable Network is studied in [4], and we show that the theorem holds for Subset Tsp in Section 5. Note that for Subset Tsp, it is possible to obtain a singly exponential algorithm by following the spanner construction of Klein [15] after performing the planarizing step (Lemma 8).…”

“…Our method is based on the framework of Borradaile et al [5] for planar graphs; in fact, we generalize their work in the sense that basically any problem that is shown to admit a PTAS on planar graphs using their framework easily generalizes to bounded-genus graphs using the methods presented in this work. In particular, this gives rise to PTASes in bounded-genus graphs for Subset Tsp (Section 5), {0, 1, 2}-edge-connected Survivable Network [4], and also Steiner Forest [3].…”

“…For Steiner Tree, at most one copy of each edge of S * is introduced to maintain connectivity [5]. In the case of {0, 1, 2}-edge connected Survivable Network, at most two copies of each edge of S * are required [4]. For Subset Tsp, the method was explained in [15].…”

Dealing with Large Hidden Constants: Engineering a Planar Steiner Tree PTAS

“…This mortar graph/brick-decomposition, in a sense, replaces the need for a spanner and is the centerpiece of the improved PTAS presented in [14]. Very recently, it has been shown that it can also be used to approximate other problems; a PTAS for the minimum 2-edge-connectivity survivable network problem was given in [29] and the traveling salesman problem (TSP) is shown to admit this methodology in [30]. The latter work actually generalizes the concept of a mortar graph to graphs of bounded genus and gives an outlook on even further generalizations.…”