We study the maximum coverage problem with group budget constraints (MCG). The input consists of a ground set X, a collection ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} of subsets of X each of which is associated with a combinatorial structure such that for every set Sj∈ψ\documentclass[12pt]{minimal}
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\begin{document}$$S_j\in \psi $$\end{document}, a cost c(Sj)\documentclass[12pt]{minimal}
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\begin{document}$$c(S_j)$$\end{document} can be calculated based on the combinatorial structure associated with Sj\documentclass[12pt]{minimal}
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\begin{document}$$S_j$$\end{document}, a partition G1,G2,…,Gl\documentclass[12pt]{minimal}
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\begin{document}$$G_1,G_2,\ldots ,G_l$$\end{document} of ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}, and budgets B1,B2,…,Bl\documentclass[12pt]{minimal}
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\begin{document}$$B_1,B_2,\ldots ,B_l$$\end{document}, and B. A solution to the problem consists of a subset H of ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} such that ∑Sj∈Hc(Sj)≤B\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{S_j\in H} c(S_j) \le B$$\end{document} and for each i∈1,2,…,l\documentclass[12pt]{minimal}
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\begin{document}$$i \in {1,2,\ldots ,l}$$\end{document}, ∑Sj∈H∩Gic(Sj)≤Bi\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{S_j \in H\cap G_i}c(S_j)\le B_i$$\end{document}. The objective is to maximize |⋃Sj∈HSj|\documentclass[12pt]{minimal}
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\begin{document}$$|\bigcup _{S_j\in H}S_j|$$\end{document}. In our work we use a new and improved analysis of the greedy algorithm to prove that it is a (α3+2α)\documentclass[12pt]{minimal}
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\begin{document}$$(\frac{\alpha }{3+2\alpha })$$\end{document}-approximation algorithm, where α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is the approximation ratio of a given oracle which takes as an input a subset Xnew⊆X\documentclass[12pt]{minimal}
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\begin{document}$$X^{new}\subseteq X$$\end{document} and a group Gi\documentclass[12pt]{minimal}
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\begin{document}$$G_i$$\end{document} and returns a set Sj∈Gi\documentclass[12pt]{minimal}
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\begin{document}$$S_j\in G_i$$\end{document} which approximates the optimal solution for maxD∈Gi|D∩Xnew|c(D)\documentclass[12pt]{minimal}
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\begin{document}$$\max _{D\in G_i}\frac{|D\cap X^{new}|}{c(D)}$$\end{document}. 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.
