Connected Set-Cover and Group Steiner Tree

Years and Authors of Summarized Original Work

• 2012; Zhang, Wu, Lee, Du

• 2012; Elbassion, Jelic, Matijevic

• 2013; Wu, Du, Wu, Li, Lv, Lee

Problem Definition

Given a collection $\mathcal{C}$ of subsets of a finite set X, find a minimum subcollection $\mathcal{C}^{{\prime}}$ of $\mathcal{C}$ such that every element of X appears in some subset in $\mathcal{C}^{{\prime}}$. This problem is called the minimum set-cover problem. Every feasible solution, i.e., a subcollection $\mathcal{C}^{{\prime}}$ satisfying the required condition, is called a set-cover. The minimum set-cover problem is NP-hard, and the complexity of approximation for it is well solved. It is well known that (1) the minimum set-cover problem has a polynomial-time $(1 +\ln n)$-approximation where n =  | X | [2, 7, 8], and moreover (2) if the minimum set-cover problem has a polynomial-time $(\rho \ln n)$-approximation for any 0 < ρ < 1, then $NP \subseteq DTIME(n^{O(\log \log n)})$ [4].

The minimum connected...