# Approximating Subset k-Connectivity Problems

• Zeev Nutov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7164)

## Abstract

A subset T ⊆ V of terminals is k-connected to a root s in a directed/undirected graph J if J has k internally-disjoint vs-paths for every v ∈ T; T is k-connected in J if T is k-connected to every s ∈ T. We consider the Subset k -Connectivity Augmentation problem: given a graph G = (V,E) with edge/node-costs, a node subset T ⊆ V, and a subgraph J = (V,E J ) of G such that T is (k − 1)-connected in J, find a minimum-cost augmenting edge-set F ⊆ E ∖ E J such that T is k-connected in J ∪ F. The problem admits trivial ratio O(|T|2). We consider the case |T| > k and prove that for directed/undirected graphs and edge/node-costs, a ρ-approximation algorithm for Rooted Subset k -Connectivity Augmentation implies the following approximation ratios for Subset k -Connectivity Augmentation:

(i) $$b(\rho+k) + {\left(\frac{|T|}{|T|-k}\right)}^2 O\left(\log \frac{|T|}{|T|-k}\right)$$ and

(ii) $$\rho \cdot O\left(\frac{|T|}{|T|-k} \log k \right)$$,

where b = 1 for undirected graphs and b = 2 for directed graphs. The best known values of ρ on undirected graphs are min {|T|,O(k)} for edge-costs and min {|T|,O(k log|T|)} for node-costs; for directed graphs ρ = |T| for both versions. Our results imply that unless k = |T| − o(|T|), Subset k -Connectivity Augmentation admits the same ratios as the best known ones for the rooted version. This improves the ratios in [19,14].

## Keywords

Directed Graph Undirected Graph Approximation Ratio Survivable Network Connectivity Requirement
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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