Maximum coverage problem with group budget constraints

Abstract

We study the maximum coverage problem with group budget constraints (MCG). The input consists of a ground set X, a collection $\psi$ of subsets of X each of which is associated with a combinatorial structure such that for every set $S_j\in \psi$, a cost $c(S_j)$ can be calculated based on the combinatorial structure associated with $S_j$, a partition $G_1,G_2,\ldots ,G_l$ of $\psi$, and budgets $B_1,B_2,\ldots ,B_l$, and B. A solution to the problem consists of a subset H of $\psi$ such that $\sum _{S_j\in H} c(S_j) \le B$ and for each $i \in {1,2,\ldots ,l}$, $\sum _{S_j \in H\cap G_i}c(S_j)\le B_i$. The objective is to maximize $|\bigcup _{S_j\in H}S_j|$. In our work we use a new and improved analysis of the greedy algorithm to prove that it is a $(\frac{\alpha }{3+2\alpha })$-approximation algorithm, where $\alpha$ is the approximation ratio of a given oracle which takes as an input a subset $X^{new}\subseteq X$ and a group $G_i$ and returns a set $S_j\in G_i$ which approximates the optimal solution for $\max _{D\in G_i}\frac{|D\cap X^{new}|}{c(D)}$. This analysis that is shown here to be tight for the greedy algorithm, improves by a factor larger than 2 the analysis of the best known approximation algorithm for MCG.