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Mathematical Programming

, Volume 172, Issue 1–2, pp 505–537 | Cite as

Robust monotone submodular function maximization

  • James B. Orlin
  • Andreas S. Schulz
  • Rajan Udwani
Full Length Paper Series B

Abstract

We consider a robust formulation, introduced by Krause et al. (J Artif Intell Res 42:427–486, 2011), of the classical cardinality constrained monotone submodular function maximization problem, and give the first constant factor approximation results. The robustness considered is w.r.t. adversarial removal of up to \(\tau \) elements from the chosen set. For the fundamental case of \(\tau =1\), we give a deterministic \((1-1/e)-1/\varTheta (m)\) approximation algorithm, where m is an input parameter and number of queries scale as \(O(n^{m+1})\). In the process, we develop a deterministic \((1-1/e)-1/\varTheta (m)\) approximate greedy algorithm for bi-objective maximization of (two) monotone submodular functions. Generalizing the ideas and using a result from Chekuri et al. (in: FOCS 10, IEEE, pp 575–584, 2010), we show a randomized \((1-1/e)-\epsilon \) approximation for constant \(\tau \) and \(\epsilon \le \frac{1}{\tilde{\varOmega }(\tau )}\), making \(O(n^{1/\epsilon ^3})\) queries. Further, for \(\tau \ll \sqrt{k}\), we give a fast and practical 0.387 algorithm. Finally, we also give a black box result result for the much more general setting of robust maximization subject to an Independence System.

Mathematics Subject Classification

90 

Notes

Acknowledgements

This work was partially supported by ONR Grant N00014-17-1-2194. The authors would like to thank all the anonymous reviewers for their useful suggestions and comments on all the versions of the paper so far. In addition, RU would also like to thank Jan Vondrák for a useful discussion and pointing out a relevant result in [10].

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.M.I.T.CambridgeUSA

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