Edge-Connectivity Augmentation and Network Matrices

We study the following NP-hard graph augmentation problem: Given a weighted graph G and a connected spanning subgraph H of G, find a minimum weight set of edges of G to be added to H so that H becomes 2-edge-connected. We provide a formulation of the problem as a set covering problem, and we analyze the conditions for which the linear programming relaxation of our formulation always gives an integer solution. This yields instances of the problem that can be solved in polynomial time. As we will show in the paper, these particular instances have not only theoretical but also practical interest, since they model a wide range of survivability problems in communication networks.

[1]  Joseph Naor,et al.  Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality , 1993, Math. Program..

[2]  András Frank.  Augmenting Graphs to Meet Edge-Connectivity Requirements , 1992, SIAM J. Discret. Math..

[3]  Guido Proietti,et al.  Polynomial Time Algorithms for 2-Edge-Connectivity Augmentation Problems , 2003, Algorithmica.

[4]  Harold N. Gabow,et al.  Applications of a poset representation to edge connectivity and graph rigidity , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[5]  Hiroshi Nagamochi,et al.  An approximation for finding a smallest 2-edge-connected subgraph containing a specified spanning tree , 1999, Discret. Appl. Math..

[6]  Akira Nakamura,et al.  Edge-Connectivity Augmentation Problems , 1987, J. Comput. Syst. Sci..

[7]  S. Khuller.  Approximation algorithms for finding highly connected subgraphs , 1996 .

[8]  D. Hochbaum.  Approximating covering and packing problems: set cover, vertex cover, independent set, and related problems , 1996 .

[9]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[10]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[11]  S. Khuller,et al.  Approximation Algorithms for Graph Augmentation , 1992, J. Algorithms.

[12]  Jon Feldman,et al.  A 3/2-Approximation Algorithm for Augmenting the Edge-Connectivity of a Graph from 1 to 2 Using a Subset of a Given Edge Set , 2001, RANDOM-APPROX.

[13]  Joseph JáJá,et al.  Approximation Algorithms for Several Graph Augmentation Problems , 1981, SIAM J. Comput..