Abstract
We introduce the Scenario Submodular Cover problem. In this problem, the goal is to produce a cover with minimum expected cost, with respect to an empirical joint probability distribution, given as input by a weighted sample of realizations. The problem is a counterpart to the Stochastic Submodular Cover problem studied by Golovin and Krause [6], which assumes independent variables. We give two approximation algorithms for Scenario Submodular Cover. Assuming an integer-valued utility function and integer weights, the first achieves an approximation factor of \(O(\log Qm)\), where m is the sample size and Q is the goal utility. The second, simpler algorithm achieves an approximation factor of \(O(\log QW)\), where W is the sum of the weights. We achieve our bounds by building on previous related work (in [4, 6, 15]) and by exploiting a technique we call the Scenario-OR modification. We apply these algorithms to a new problem, Scenario Boolean Function Evaluation. Our results have applciations to other problems involving distributions that are explicitly specified by their support.
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Notes
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In the Operations Research literature, Stochastic Function Evaluation is often called Sequential Testing or Sequential Diagnosis.
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Acknowledgements
The work in this paper was supported by NSF Grant 1217968. L. Hellerstein thanks Andreas Krause for useful discussions at ETH, and for directing our attention to the bound of Streeter and Golovin for min-sum submodular cover. We thank an anonymous referee for suggesting the Kosaraju trick.
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Grammel, N., Hellerstein, L., Kletenik, D., Lin, P. (2017). Scenario Submodular Cover. In: Jansen, K., Mastrolilli, M. (eds) Approximation and Online Algorithms. WAOA 2016. Lecture Notes in Computer Science(), vol 10138. Springer, Cham. https://doi.org/10.1007/978-3-319-51741-4_10
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