A near optimal algorithm for vertex connectivity augmentation

Abstract

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 ≤ 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 Jordán [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 me-thod (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.