The ground-set-cost budgeted maximum coverage problem

We study the following natural variant of the budgeted maximum coverage problem: We are given a budget B and a hypergraph G = (V,E), where each vertex has a non-negative cost and a non-negative profit. The goal is to select a set of hyperedges T c E such that the total cost of the vertices covered by T is at most B and the total profit of all covered vertices is maximized. Besides being a natural generalization of the well-studied maximum coverage problem, our motivation for investigating this problem originates from its application in the context of bid optimization in sponsored search auctions, such as Google AdWords. It is easily seen that this problem is strictly harder than budgeted max coverage, which means that the problem is (1 - 1/e)-inapproximable. The difference of our problem to the budgeted maximum coverage problem is that the costs are associated with the covered vertices instead of the selected hyperedges. As it turns out, this difference refutes the applicability of standard greedy approaches which are used to obtain constant factor approximation algorithms for several other variants of the maximum coverage problem. Our main results are as follows: We obtain a (1 - 1/ p e)/2-approximation algorithm for graphs. We derive a fully polynomial-time approximation scheme (FPTAS) if the incidence graph of the hypergraph is a forest (i.e., the hypergraph is Berge-acyclic). We also extend this result to incidence graphs with a fixed-size feedback hyperedge node set. We give a (1 - ϵ)/(2d²)-approximation algorithm for every ϵ > 0, where d is the maximum degree of a vertex in the hypergraph.