A near optimal algorithm for vertex connectivity augmentation

Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G + F is k-vertex-connected. Polynomial algorithms for this problem are known only for k less than or equal to 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general. For arbitrary k, a previous result of Jordan [14] gives a polynomial algorithm which adds an augmenting set F of size at most k - 3 more than the optimum, provided G is (k - 1)-vertex-connected. In this paper we develop a polynomial algorithm which makes an l-connected graph G k-vertex-connected by adding an augmenting set of size at most ((k - l)(k - 1) + 4)/2 more than (a new lower bound for) the optimum. This extends the main results of [14,15]. We partly follow and generalize the approach of [14] and we adapt the splitting off method (which worked well on edge-connectivity augmentation problems) to vertex-connectivity. A key point in our proofs, which may also find applications elsewhere, is a new tripartite submodular inequality for the sizes of neighbour-sets in a graph.