Multigraph augmentation under biconnectivity and general edge-connectivity requirements

Given an undirected multigraph G = (V;E) and a requirement function r : 0 V 2 1 ! Z + (where 0 V 2 1 is the set of all pairs of vertices and Z + is the set of nonnegative integers), we consider the problem of augmenting G by the smallest number of new edges so that the local edge-connectivity and vertex-connectivity b e t ween every pair x; y 2 V become at least r (x; y) and two, respectively. In this paper, we show that the problem can be solved in O(n 3 (m + n) log (n 2 =(m + n))) time, where n and m are the numbers of vertices and pairs of adjacent vertices in G, respectively. This time complexity can beimproved to O((nm + n 2 log n) log n), in the case of the uniform requirement r (x; y) = ` for all x; y 2 V. Furthermore, for the general r , we show that the augmentation problem that preserves the simplicity of the resulting graph can be solved in polynomial time for any xed`3 = maxfr (x; y) j x;y 2 V g.