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Given an undirected graph G = (V, E0) with |V| = 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  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 + ln nc)(1 + ) times optimum and within (1 + ln nb)(1 + ) 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  is a constant with 0

NC approximation algorithms for 2-connectivity augmentation in a graph