Two-Connected Networks with Rings of Bounded Cardinality

We study the problem of designing at minimum cost a two-connected network such that each edge belongs to a cycle using at most K edges. This problem is a particular case of the two-connected networks with bounded meshes problem studied by Fortz, Labbé and Maffioli (Operations Research, vol. 48, no. 6, pp. 866–877, 2000).In this paper, we compute a lower bound on the number of edges in a feasible solution, we show that the problem is strongly NP-complete for any fixed K, and we derive a new class of facet defining inequalities. Numerical results obtained with a branch-and-cut algorithm using these inequalities show their effectiveness for solving the problem.

[1]  Martine Labbé,et al.  Polyhedral results for two-connected networks with bounded rings , 2002, Math. Program..

[2]  William H. Cunningham,et al.  Optimal attack and reinforcement of a network , 1985, JACM.

[3]  Stefan Thienel,et al.  ABACUS - a branch-and-CUt system , 1995 .

[4]  M. Stoer.  Design of Survivable Networks , 1993 .

[5]  T. C. Hu,et al.  Multi-Terminal Network Flows , 1961 .

[6]  Sylvia C. Boyd,et al.  An Integer Polytope Related to the Design of Survivable Communication Networks , 1993, SIAM J. Discret. Math..

[7]  Martin Grötschel,et al.  Computational Results with a Cutting Plane Algorithm for Designing Communication Networks with Low-Connectivity Constraints , 1992, Oper. Res..

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

[9]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[10]  Martin Grötschel,et al.  Integer Polyhedra Arising from Certain Network Design Problems with Connectivity Constraints , 1990, SIAM J. Discret. Math..

[11]  Bernard Fortz,et al.  Design of Survivable Networks with Bounded Rings , 2000, Network Theory and Applications.

[12]  Francisco Barahona,et al.  Separating from the dominant of the spanning tree polytope , 1992, Oper. Res. Lett..

[13]  Francesco Maffioli,et al.  Solving the Two-Connected Network with Bounded Meshes Problem , 2000, Oper. Res..

[14]  C. Monma,et al.  Methods for Designing Communications Networks with Certain Two-Connected Survivability Constraints , 1989, Oper. Res..

[15]  K. Steiglitz,et al.  The Design of Minimum-Cost Survivable Networks , 1969 .

[16]  Bernard Fortz,et al.  A tabu search algorithm for self-healing ring network design , 2003, Eur. J. Oper. Res..