Maximum coverage problem with group budget constraints

We study the maximum coverage problem with group budget constraints (MCG). The input consists of a ground set X, a collection ψ of subsets of X each of which is associated with a combinatorial structure such that for every set S_j∈ ψ, a cost c(S_j) can be calculated based on the combinatorial structure associated with S_j, a partition G₁, G₂, … , G_l of ψ, and budgets B₁, B₂, … , B_l, and B. A solution to the problem consists of a subset H of ψ such that ∑Sj∈Hc(Sj)≤B and for each i∈ 1 , 2 , … , l, ∑Sj∈H∩Gic(Sj)≤Bi. The objective is to maximize |⋃Sj∈HSj|. In our work we use a new and improved analysis of the greedy algorithm to prove that it is a (α3+2α)-approximation algorithm, where α is the approximation ratio of a given oracle which takes as an input a subset X^{n e w}⊆ X and a group G_i and returns a set S_j∈ G_i which approximates the optimal solution for maxD∈Gi|D∩Xnew|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.