Approximation algorithms for minimum weight partial connected set cover problem

Abstract

In the Minimum Weight Partial Connected Set Cover problem, we are given a finite ground set $U$, an integer $q\le |U|$, a collection $\mathcal {E}$ of subsets of $U$, and a connected graph $G_{\mathcal {E}}$ on vertex set $\mathcal {E}$, the goal is to find a minimum weight subcollection of $\mathcal {E}$ which covers at least $q$ elements of $U$ and induces a connected subgraph in $G_{\mathcal {E}}$. In this paper, we derive a “partial cover property” for the greedy solution of the Minimum Weight Set Cover problem, based on which we present (a) for the weighted version under the assumption that any pair of sets in $\mathcal {E}$ with nonempty intersection are adjacent in $G_{\mathcal {E}}$ (the Minimum Weight Partial Connected Vertex Cover problem falls into this range), an approximation algorithm with performance ratio $\rho (1+H(\gamma ))+o(1)$, and (b) for the cardinality version under the assumption that any pair of sets in $\mathcal {E}$ with nonempty intersection are at most $d$-hops away from each other (the Minimum Partial Connected $k$-Hop Dominating Set problem falls into this range), an approximation algorithm with performance ratio $2(1+dH(\gamma ))+o(1)$, where $\gamma =\max \{|X|:X\in \mathcal {E}\}$, $H(\cdot )$ is the Harmonic number, and $\rho$ is the performance ratio for the Minimum Quota Node-Weighted Steiner Tree problem.