Approximation Algorithms for Maximum Coverage with Group Budget Constraints

Abstract

In this paper, we study the maximum coverage problem with group budget constraints (MCG) that generalizes the maximum coverage problem. Given a ground set U in which $i\in U$ has a non-negative weight $w_{i}$, a positive integer k and a collection of sets $\mathcal{S}$, the maximum coverage problem is to pick k sets of $\mathcal{S}$ to maximize the total weight of their union. In MCG, $\mathcal{S}$ is partitioned into groups $\mathcal{G}_{1},\,\dots ,\,\mathcal{G}_{q}$, and the goal is to pick k sets from $\mathcal{S}$ to maximize the total weight of their union, with at most $n_{l}\in \mathbb {Z}_{0}^{+}$ sets being picked from group $\mathcal{G}_{l}$. For MCG with $n_{l}=1$, $\forall l$, we first present a factor $1-\frac{1}{e}$ approximation algorithm which runs in exponential time. Then we improve the runtime of the algorithm to $O((m+n+q)^{3.5}L+k^{3.5}q^{7}L)$ where $\vert \mathcal{S}\vert =m$, $\vert U\vert =n$, q is the number of groups, and L is the length of the input. The key idea of the improvement is to model selecting groups for MCG as computing a constrained flow in a corresponding auxiliary graph. It is also shown that the algorithm can be extended to solve MCG with general $n_{l}$. Later, based on the main idea of partition we further improve the runtime of the algorithm to $O((m+n+q)^{3.5}L+k\delta ^{10.5}L)$ , while compromise the approximation ratio to $1-e^{\frac{1}{\delta }-1}$, where $\delta \ge 2$ is any fixed integer. Consequently, we can balance approximation ratio and runtime of the algorithm by setting the value of $\delta$. This improves the previous best ratio of 0.5 on MCG due to Chekuri and Kumar [4].