Approximating Subset k-Connectivity Problems

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].