# 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^α), where α≤ 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 α=1- ϵ 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 α=1-1/e-ϵ 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 β≥ 1 to guarantee a 1/2(1-1/e^αβ)-approximation. If we set β=1/α(1/2ϵ), the algorithm achieves an approximation factor of 1/2-ϵ, for any arbitrarily small ϵ>0.

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