Approximating Node Connectivity Problems via Set Covers

Abstract

We generalize and unify techniques from several papers to obtain relatively simple and general technique for designing approximation algorithms for finding min-cost k-node connected spanning subgraphs. For the general instance of the problem, the previously best known algorithm has approximation ratio 2k. For k ≤ 5, algorithms with approximation ratio [(k+1)/2] are known. For metric costs Khuller and Raghavachari gave a $\left( {2 + \frac{{2(k - 1)}} {n}} \right)$-approximation algorithm. We obtain the following results.

• (i) An I(k-k ₀)-approximation algorithm for the problem of making a k0-connected graph k-connected by adding a minimum cost edge set, where $I\left( k \right) = 2 + \sum\nolimits_{j = 1}^{\left\lfloor {\tfrac{k} {2}} \right\rfloor - 1} {\tfrac{1} {j}\left\lfloor {\tfrac{k} {{j + 1}}} \right\rfloor }$

• (ii) A $(2 + {\mathbf{ }}\frac{{k - 1}} {n})$-approximation algorithm for metric costs.

• (iv) A [(k + 1)/2]-approximation algorithm fork= 6, 7.

• (v) A fast [(k + 1)/2]-approximation algorithm fork= 4. The multiroot problem generalizes the min-cost k-connected subgraph problem. In the multiroot problem, requirements ku for every node u are given, and the aim is to find a minimum-cost subgraph that contains max{ku, kv} internally disjoint paths between every pair of nodes u, v. For the general instance of the problem, the best known algorithm has approximation ratio 2k, wherek= maxku. For metric costs there is a 3-approximation algorithm. We consider the case of metric costs, and, using our techniques, improve fork= 7 the approximation guarantee from 3 to $2 + \frac{{\left\lfloor {\left( {k - 1} \right)/2} \right\rfloor }} {k} < 2.5$