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 $$\psi $$ź of subsets of X each of which is associated with a combinatorial structure such that for every set $$S_j\in \psi $$Sjźź, a cost $$c(S_j)$$c(Sj) can be calculated based on the combinatorial structure associated with $$S_j$$Sj, a partition $$G_1,G_2,\ldots ,G_l$$G1,G2,ź,Gl of $$\psi $$ź, and budgets $$B_1,B_2,\ldots ,B_l$$B1,B2,ź,Bl, and B. A solution to the problem consists of a subset H of $$\psi $$ź such that $$\sum _{S_j\in H} c(S_j) \le B$$źSjźHc(Sj)≤B and for each $$i \in {1,2,\ldots ,l}$$iź1,2,ź,l, $$\sum _{S_j \in H\cap G_i}c(S_j)\le B_i$$źSjźHźGic(Sj)≤Bi. The objective is to maximize $$|\bigcup _{S_j\in H}S_j|$$|źSjźHSj|. In our work we use a new and improved analysis of the greedy algorithm to prove that it is a $$(\frac{\alpha }{3+2\alpha })$$(ź3+2ź)-approximation algorithm, where $$\alpha $$ź is the approximation ratio of a given oracle which takes as an input a subset $$X^{new}\subseteq X$$Xnew⊆X and a group $$G_i$$Gi and returns a set $$S_j\in G_i$$SjźGi which approximates the optimal solution for $$\max _{D\in G_i}\frac{|D\cap X^{new}|}{c(D)}$$maxDźGi|DźXnew|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.