# Approximating minimum cuts under insertions

## Abstract

This paper presents insertions-only algorithms for maintaining the exact and 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) or *O*(log *n*) and a cut of size *k* in time linear in its size. The amortized time per insertion is *O*(1/*ε*^{2}) for a (2+*ε*)-approximation, *O*((log λ)((log *n)/ε*)^{2}) for a (1+*ε*)-approximation, and *O*(λ log *n*) for the exact size of the minimum edge cut, where *n* is the number of nodes in the graph, λ is the size of the minimum cut and *ε*>0. The (2+*ε*)-approximation algorithm and the exact algorithm are deterministic, the (1+*ε*)-approximation algorithm is randomized. The algorithms are optimal in the sense that the time needed for *m* insertions matches the time needed by the best static algorithm on a *m*-edge graph. We also present a static 2-approximation algorithm for the size *κ* of the minimum vertex cut in a graph, which takes time *O(n*^{2}*min(√n,κ*)). This is a factor of *κ* faster than the best algorithm for computing the exact size, which takes time *O(κ*^{2}*n*^{2}*+κ*^{3}*n*^{1.5}). We give an insertionsonly algorithm for maintaining a (2+*ε*)-approximation of the minimum vertex cut with amortized insertion time *O(n(logκk)/ε)*.

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