Generalized Budgeted Submodular Set Function Maximization

In this paper we consider a generalization of the well-known budgeted maximum coverage problem. We are given a ground set of elements and a set of bins. The goal is to find a subset of elements along with an associated set of bins, such that the overall cost is at most a given budget, and the profit is maximized. Each bin has its own cost and the cost of each element depends on its associated bin. The profit is measured by a monotone submodular function over the elements. We first present an algorithm that guarantees an approximation factor of 1/2(1-1/e^alpha), where alpha <= 1 is the approximation factor of an algorithm for a sub-problem. We give two polynomial-time algorithms to solve this sub-problem. The first one gives us alpha=1- epsilon if the costs satisfies a specific condition, which is fulfilled in several relevant cases, including the unitary costs case and the problem of maximizing a monotone submodular function under a knapsack constraint. The second one guarantees alpha=1-1/e-epsilon for the general case. The gap between our approximation guarantees and the known inapproximability bounds is 1/2. We extend our algorithm to a bi-criterion approximation algorithm in which we are allowed to spend an extra budget up to a factor beta >= 1 to guarantee a 1/2(1-1/e^(alpha beta))-approximation. If we set beta=1/(alpha)ln (1/(2 epsilon)), the algorithm achieves an approximation factor of 1/2-epsilon, for any arbitrarily small epsilon>0.