An approximation algorithm for maximum weight budgeted connected set cover

Abstract

This paper studies approximation algorithm for the maximum weight budgeted connected set cover (MWBCSC) problem. Given an element set $X$, a collection of sets ${\mathcal {S}}\subseteq 2^X$, a weight function $w$ on $X$, a cost function $c$ on ${\mathcal {S}}$, a connected graph $G_{\mathcal {S}}$ (called communication graph) on vertex set ${\mathcal {S}}$, and a budget $L$, the MWBCSC problem is to select a subcollection ${\mathcal {S'}}\subseteq {\mathcal {S}}$ such that the cost $c({\mathcal {S'}})=\sum _{S\in {\mathcal {S'}}}c(S)\le L$, the subgraph of $G_{\mathcal {S}}$ induced by ${\mathcal {S'}}$ is connected, and the total weight of elements covered by ${\mathcal {S'}}$ (that is $\sum _{x\in \bigcup _{S\in {\mathcal {S'}}}S}w(x)$) is maximized. We present a polynomial time algorithm for this problem with a natural communication graph that has performance ratio $O((\delta +1)\log n)$, where $\delta$ is the maximum degree of graph $G_{\mathcal {S}}$ and $n$ is the number of sets in ${\mathcal {S}}$. In particular, if every set has cost at most $L/2$, the performance ratio can be improved to $O(\log n)$.