Fast algorithms for k-shredders and k-node connectivity augmentation

Abstract

A k-separator (k-shredder) of a graph is a set of k nodes whose removal results in two or more (three or more) connected components. Let the given (undirected) graph be k-node connected, and let n denote the number of nodes. Solving an open question, we show that the problem of counting the number of k-separators is #P-complete. However, we present an O(k2n2 + k3n1.5)-time (deterministic) algorithm for finding all the k-shredders. This solves an open question: efficiently find a k-separator whose removal maximizes the number of connected components. For k ≥ 4, our running time equals that of the fastest algorithm known for testing k-node connectivity. One application of shredders is in increasing the node connectivity from k to (k + 1) by efficiently adding an (approximately) minimum number of new edges. Jordán [JCT(B) 1995] gave an O(n5)-time augmentation algorithm such that the number of new edges is within an additive term of (k - 2) from a lower bound. We improve the running time to O(min(k, √n)k2n2 + (log n)kn2), while achieving the same performance guarantee. For k ≥ 4, the running time compares favorably with the running time for testing k-node connectivity.