A static 2-approximation algorithm for vertex connectivity and incremental approximation algorithms for edge and vertex connectivity

This paper presents insertions-only algorithms for maintaining the exact and/or approximate size of the minimum edge cut and the minimum vertex cut of a graph. The algorithms output the approximate or exact size k in time O(1) and a cut of size k in time linear in its size. For the minimum edge cut problem and for any 0 < epsilon less than or equal to 1, the amortized time per insertion is O(1/epsilon(2)) for a (2 + epsilon)-approximation, O((log lambda)((log n)/epsilon)(2)) for a (1 + epsilon)-approximation, and O(lambda log n) for the exact size, where n is the number of nodes in the graph and lambda is the size of the minimum cut. The (2 + epsilon)-approximation algorithm and the exact algorithm are deterministic; the (1 + epsilon)-approximation algorithm is randomized. We also present a static 2-approximation algorithm for the size kappa of the minimum vertex cut in a graph, which takes time O(n(2) min(root n, kappa)). This is a factor of kappa faster than the best kappa n(2))min(root n, kappa)). We give an insertions-only algorithm for maintaining a (2 + epsilon)-approximation of the minimum vertex cut with amortized insertion time O(n/epsilon). (C) 1997 Academic Press.