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 sizekin timeO(1) and a cut of sizekin time linear in its size. For the minimum edge cut problem and for any 0 < ε ≤ 1, the amortized time per insertion isO(1/ε2) for a (2 + ε)-approximation,O((log λ)((log n)/ε)2) for a (1 + ε)-approximation, andO(λ log n) for the exact size, wherenis the number of nodes in the graph and λ is the size of the minimum cut. The (2 + ε)-approximation algorithm and the exact algorithm are deterministic; the (1 + ε)-approximation algorithm is randomized. We also present a static 2-approximation algorithm for the size κ of the minimum vertex cut in a graph, which takes time. This is a factor of κ faster than the best algorithm for computing the exact size, which takes time. We give an insertions-only algorithm for maintaining a (2 + ε)-approximation of the minimum vertex cut with amortized insertion timeO(n/ε).