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 \(T \subseteq E\) 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. This is a natural generalization of the maximum coverage problem. Our interest in this problem stems from its application to bid optimization in sponsored search auctions. It is easily seen that this problem is at least as hard as budgeted maximum coverage (where the costs are associated with the selected hyperedges instead of the covered vertices). This implies \((1-1/e+\epsilon )\) -inapproximability for any \(\epsilon> 0\) . Furthermore, standard greedy approaches do not yield constant factor approximations for our variant of the problem. In fact, through a reduction from Densest k-Subgraph, it can be established that our problem is inapproximable up to a constant factor, conditional on the exponential time hypothesis. Our main results are as follows: (i.) We obtain a \((1 - 1/\sqrt{e})/2\) -approximation algorithm for graphs. (ii.) 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 extend this result to incidence graphs with a fixed-size feedback hyperedge node set. (iii.) We give a \((1-\varepsilon )/(2d^2)\) -approximation algorithm for all \(\varepsilon> 0\) , where d is the maximum vertex degree.
