A Static 2-Approximation Algorithm for Vertex Connectivity and Incremental Approximation Algorithms for Edge and Vertex Connectivity

Abstract

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((logλ)((logn)/ε)2) for a (1+ε)-approximation, andO(λlogn) 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 timeO(n2min(n,κ)). This is a factor of κ faster than the best algorithm for computing the exact size, which takes timeO((κ3n+κn2)min(n,κ)). We give an insertions-only algorithm for maintaining a (2+ε)-approximation of the minimum vertex cut with amortized insertion timeO(n/ε).