Greedy Set-Cover Algorithms

Years and Authors of Summarized Original Work

1974–1979; Chvátal, Johnson, Lovász, Stein

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 ≥ 0 is also specified, and the goal is to find a set-cover C of minimum total weight $\sum \limits _{S\in C}w_{S}$.

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 nonnegative weight w _s , and a nondecreasing submodular function $f : 2^{\mathcal{S}}\rightarrow\mathbb{R}$, the goal is to find a subcollection $C \subseteq \mathcal{S}$such that $f\left (C\right ) = f\left (\mathcal{S}\right )$ minimizing \(\sum...