Biconnectivity Approximations and Graph Carvings

A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2-connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified)? Unfortunately, the problem is known to be NP -hard. We consider the problem of finding an approximation to the smallest 2-connected subgraph, by an efficient algorithm. For 2-edge connectivity our algorithm guarantees a solution that is no more than 3/2 times the optimal. For 2-vertex connectivity our algorithm guarantees a solution that is no more than 5/3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP -hard as well. We also consider the case where the graph has edge weights. We show that an approximation factor of 2 is possible in polynomial time for finding a k -edge connected spanning subgraph. This improves an approximation factor of 3 for k =2 due to [FJ81], and extends it for any k (with an increased running time though).

Authors

• Samir Khuller
• Uzi Vishkin

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