On the Minimum Local-Vertex-Connectivity Augmentation in Graphs

Given a graph G and target values r(u, v) prescribed for each pair of vertices u and v, we consider the problem of augmenting G by a smallest set F of new edges such that the resulting graph G + F has at least r(u,v) internally disjoint paths between each pair of vertices u and v. We show that the problem is NP-hard even if G is (k-1)-vertex-connected and r(u, v) ? {0, k}, u, v ? V holds for a constant k ? 2. We then give a linear time algorithm which delivers a 3/2 -approximation solution to the problem with a connected graph G and r(u, v) ? {0, 2}, u, v ? V.