Practical Budgeted Submodular Maximization
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We consider the Budgeted Submodular Maximization problem, that seeks to maximize an increasing submodular function subject to budget constraints. Extending a result of Khuller, Moss, and Naor for the Budgeted Coverage problem, Sviridenko showed that the greedy algorithm combined with guessing 3 most profitable elements of an optimal solution has approximation ratio α=1-1/e≈ 0.632. We show that just 2 guesses suffice to achieve ratio α, 1 guess suffices to achieve ratio 0.899 α≈ 0.568, while ratio 0.68α≈ 0.43 can be achieved without any guessing. We note that ratio α-ϵ can be achieved using (1/ϵ)^O(1/ϵ^4)nlog n value oracle calls, but this algorithm is impractical already for large values of ϵ. Among practical algorithms, our is the currently best known one.
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