# Approximating unweighted connectivity problems in parallel

## Abstract

Given an integer *k* and a *k*-edge-connected graph *G* = (*V, E*), we wish to find an *E*^{1} ⊆ *E* of minimum size such that the graph (*V, E*^{1}) is *k*-edge-connected. This problem is NP-hard and it is open whether there is an NC approximation algorithm with a constant performance ratio smaller than 2. Previously, the special case where the input integer *k* is fixed to be 2 was considered. This paper considers a much more general case where *k* is polylogarithmic in the size of the input graph, and presents the *first* NC approximation algorithm with a constant performance ratio smaller than 2 for this case. Unlike all the previous approximation algorithms and their analysis for this problem, ours need to deal with multiple edges in the input graph. We also consider the vertexanalogue of this problem in which we require *k*-vertex-connectivity instead of *k*-edge-connectivity. We present the *first* NC approximation algorithm with a constant performance ratio smaller than 2 for the special case where the input integer *k* is fixed to be 3. Previously, only the special case where k is fixed to be 2 was known to have an NC approximation algorithm with a constant performance ratio smaller than 2.

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