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 < 1, the amortized time per insertion is O(1= ) for a (2 + )-approximation, O((log )((log n)= )) for a (1+ )-approximation, and O( log n) for the exact size, where n is the number of nodes in the graph and is the size of the minimum cut. The (2 + )-approximation algorithm and the exact algorithm are deterministic, the (1 + )-approximation algorithm is randomized. We also present a static 2-approximation algorithm for the size of the minimum vertex cut in a graph, which takes time O(n min( p n; )). This is a factor of faster than the best algorithm for computing the exact size, which takes time O(( n+ n)min( p n; )). We give an insertionsonly algorithm for maintaining a (2 + )-approximation of the minimum vertex cut with amortized insertion time O(n= ).