Abstract

In this paper, the rst approach for solving the vertex-biconnectivity aug-mentation problem (V2AUG) to optimality is proposed. Given a spanning subgraph of an edge-weighted graph, we search for the cheapest subset of edges to augment it in order to make it vertex-biconnected. The problem is reduced to the augmentation of the corresponding block-cut tree [16] whose connectivity properties are exploited to develop two minimum-cut-based ILP formulations: a directed and an undirected one. In contrast to the recently obtained result for the more general vertex-biconnected Steiner network problem [2], our theoretical comparison shows that orienting the undirected graph does not help in improving the quality of lower bounds. Hence, starting from the undirected cut formulation, we develop a branch-and-cut-and-price algorithm (BCP) which represents the rst exact approach to V2AUG. Our computational experiments show the practical feasi-bility of the BCP: Complete graphs with more than 400 nodes can be solved to provable optimality. Furthermore, the BCP is even faster than the state-of-the-art metaheuristics and approximation algorithms, as far as graphs with up to 200 nodes are considered. For large graphs with more than 2000 vertices, optimality gaps that are strictly below 2 % are reported. 1 The Vertex-Biconnectivity Augmentation Problem In the design of modern telecommunication networks, in particular in backbones, survivability is an important issue. In many telecommunication applications it is not acceptable that the failure of a single service nodea multiplexer, switch, or router, for exampleleads to a disconnection of other nodes. Redundant connections need to be established to provide alternative routes in case of a temporary malfunction of any one node.