Algorithms for connected set cover problem and fault-tolerant connected set cover problem

Given a set V of elements, S a family of subsets of V, and G a connected graph on vertex set S,a connected set cover (CSC) is a subfamily R of S such that every element in V is covered by at least one set of R, and the subgraph G[R] of G induced by R is connected. If furthermore G[R] is k-connected and every element in V is covered by at least m sets in R, then R is a (k,m)-CSC. In this paper, we present two approximation algorithms for the minimum CSC problem, and one approximation algorithm for the minimum (2,m)-CSC problem. Performance ratios are analyzed. These are the first approximation algorithms for CSC problems in general graphs with guaranteed performance ratios.