An Approximation Algorithm for the Connected Maximum Coverage Problem in Directed Graphs

Abstract

In the Directed rooted Connected Budgeted maximum Coverage problem (\DRCC), we are given a collection of subsets $\mathcal{S}$, defined over a ground set $X$, and a directed graph $G=(V,E)$, where each node is associated with a set of $\mathcal{S}$. Each set in $\mathcal{S}$ has a different cost and each element of $X$ gives a different prize. The goal is to find a subcollection $\mathcal{S}'\subseteq \mathcal{S}$ such that $\mathcal{S}'$ induces an out-tree rooted at a given node, the total cost of the sets in $\mathcal{S}'$ does not exceed a budget $B$, and the total prize of the elements covered by $\mathcal{S}'$ is maximized. In this paper, we provide an algorithm for \DRCC that guarantees an approximation ratio of $O\left(\frac{\sqrt{|V|}\log^2|X|}{\epsilon^2}\right)$, with a budget violation of a factor $1+\epsilon$, where $\epsilon\in (0,1]$. Our algorithm also implies an improved approximation factor for the budgeted node-weighted Steiner problem in directed graphs, a particular case of \DRCC where the prize function is additive, for which we improve from $O\left(\frac{1}{\epsilon^2}|V|^{2/3}\log|V|\right)$ to $O\left(\frac{1}{\epsilon^2}|V|^{1/2}\log^2|V|\right)$.