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Approximating Node-Connectivity Augmentation Problems

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Abstract

We consider the (undirected) Node Connectivity Augmentation (NCA) problem: given a graph J=(V,E J ) and connectivity requirements \(\{r(u,v): u,v \in V\}\), find a minimum size set I of new edges (any edge is allowed) such that the graph JI contains r(u,v) internally-disjoint uv-paths, for all u,vV. In Rooted NCA there is sV such that r(u,v)>0 implies u=s or v=s. For large values of k=max  u,vV r(u,v), NCA is at least as hard to approximate as Label-Cover and thus it is unlikely to admit an approximation ratio polylogarithmic in k. Rooted NCA is at least as hard to approximate as Hitting-Set. The previously best approximation ratios for the problem were O(kln n) for NCA and O(ln n) for Rooted NCA. In this paper we give an approximation algorithm with ratios O(kln 2 k) for NCA and O(ln 2 k) for Rooted NCA. This is the first approximation algorithm with ratio independent of n, and thus is a constant for any fixed k. Our algorithm is based on the following new structural result which is of independent interest. If \(\mathcal{D}\) is a set of node pairs in a graph J, then the maximum degree in the hypergraph formed by the inclusion minimal tight sets separating at least one pair in \(\mathcal{D}\) is O( 2), where is the maximum connectivity in J of a pair in \(\mathcal{D}\).

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Correspondence to Zeev Nutov.

Additional information

A preliminary version of this paper is [29].

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Nutov, Z. Approximating Node-Connectivity Augmentation Problems. Algorithmica 63, 398–410 (2012). https://doi.org/10.1007/s00453-011-9538-x

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Keywords

  • Node-connectivity
  • Augmentation problems
  • Approximation algorithms