NC Approximation Algorithms for 2-Connectivity Augmentation in a Graph

Abst rac t . Given an undirected graph G = (V, E0) with IVI-n, and a feasible set E of m weighted edges on V, the optimal 2-edge (2-vertex) connectivity augmentation problem is to find a subset S* __ E such that G(V, E0 U S*) is 2-edge (2-vertex) connected and the weighted sum of edges in S* is minimized. We devise NC approximation algorithms for the optimal 2-edge connectivity and the optimal 2-vertex connectivity augmentation problems by delivering solutions within (1 + In no)(1 + e) times optimum and within (1 + In nb)(1 + e)log nb times optimum when G is connected, respectively, where nc is the number of 2-edge connected components of G, nb is the number of biconnected components of G, and e is a constant with 0 < e < 1. Consequently, we find an approximation solution for the problem of the minimum 2-edge (biconnected) spanning subgraph on a weighted 2-edge connected (biconnected) graph in the same time and processor bounds.