Greedy Set-Cover Algorithms

Keywords and Synonyms

Dominating set; Greedy algorithm; Hitting set; Set cover ; Minimizing a linear function subject to a submodular constraint          

Problem Definition

Given a collection ${ \mathcal{S} }$ of sets over a universe U, a set cover ${ C\subseteq\mathcal{S} }$ is a subcollection of the sets whose union is U. The set-cover problem is, given ${ \mathcal{S} }$, to find a minimum-cardinality set cover. In the weighted set-cover problem, for each set ${ s\in\mathcal{S} }$ a weight ${ w_s \ge 0 }$ is also specified, and the goal is to find a set cover C of minimum total weight ${ \sum_{s\in C} w_s }$.

Weighted set cover is a special case of minimizing a linear function subject to a submodular constraint, defined as follows. Given a collection ${ \mathcal{S} }$ of objects, for each object s a non-negative weight w _s , and a non-decreasing submodular function ${ f:2^\mathcal{S}\rightarrow\mathbb{R} }$, the goal is to find a subcollection \( {...