A simple approximation algorithm for minimum weight partial connected set cover

Abstract

This paper studies the minimum weight partial connected set cover problem (PCSC). Given an element set U, a collection ${\mathcal {S}}$ of subsets of U, a weight function c :  ${\mathcal {S}} \rightarrow {\mathbb {Q}}^{+}$, a connected graph $G_{{\mathcal {S}}}$ on vertex set ${\mathcal {S}}$, and a positive integer $k\le |U|$, the goal is to find a minimum weight subcollection ${\mathcal {S}}' \subseteq {\mathcal {S}}$ such that the subgraph of G induced by ${\mathcal {S}}'$ is connected, $|\bigcup _{S\in {\mathcal {S}}^{\prime }}S| \ge k$, and the weight of ${\mathcal {S}}'$ is minimum. If the graph $G_{{\mathcal {S}}}$ has the property that any two sets with a common element has hop distance at most r in $G_{{\mathcal {S}}}$, then the problem is called r-hop PCSC. We presented an $O(\ln (m+n))$-approximation algorithm for the minimum weight 1-hop PCSC problem and an $O(\ln (m+n))$-approximation algorithm for the minimum cardinality r-hop PCSC problem. Our performance ratio improves previous results and our method is much simpler than previous methods.