Robust monotone submodular function maximization
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
90Notes
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].
References
- 1.Badanidiyuru, A., Vondrák, J.: Fast algorithms for maximizing submodular functions. In SODA ’14, pages 1497–1514. SIAM, (2014)Google Scholar
- 2.Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust optimization. Princeton University Press, Princeton (2009)CrossRefGoogle Scholar
- 3.Bertsimas, D., Brown, D., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)MathSciNetCrossRefGoogle Scholar
- 4.Bertsimas, D., Sim, M.: Robust discrete optimization and network flows. Math. Program. 98(1–3), 49–71 (2003)MathSciNetCrossRefGoogle Scholar
- 5.Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)MathSciNetCrossRefGoogle Scholar
- 6.Bogunovic, I., Mitrovic, S., Scarlett, J., Cevher, V.: Robust submodular maximization: a non-uniform partitioning approach. In: ICML (2017)Google Scholar
- 7.Buchbinder, N., Feldman, M.: Deterministic algorithms for submodular maximization problems. CoRR. arXiv:1508.02157 (2015)
- 8.Buchbinder, N., Feldman, M., Naor, J.S., Schwartz, R.: A tight linear time (1/2)-approximation for unconstrained submodular maximization. In: FOCS ’12, pp. 649–658 (2012)Google Scholar
- 9.Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput. 40(6), 1740–1766 (2011)MathSciNetCrossRefGoogle Scholar
- 10.Chekuri, C., Vondrák, J., Zenklusen, R.: Dependent randomized rounding via exchange properties of combinatorial structures. In: FOCS 10, pp. 575–584. IEEE (2010)Google Scholar
- 11.Dobzinski, S., Vondrák, J.: From query complexity to computational complexity. In: STOC ’12, pp. 1107–1116. ACM (2012)Google Scholar
- 12.Feige, U.: A threshold of ln n for approximating set cover. J. ACM (JACM) 45(4), 634–652 (1998)CrossRefGoogle Scholar
- 13.Feige, U., Mirrokni, V.S., Vondrak, J.: Maximizing non-monotone submodular functions. SIAM J. Comput. 40(4), 1133–1153 (2011)MathSciNetCrossRefGoogle Scholar
- 14.Feldman, M., Naor, J.S., Schwartz, R.: A unified continuous greedy algorithm for submodular maximization. In: FOCS ’11, pp. 570–579. IEEE (2011)Google Scholar
- 15.Feldman, M., Naor, J.S., Schwartz, R.: Nonmonotone submodular maximization via a structural continuous greedy algorithm. In: Automata, Languages and Programming, pp. 342–353. Springer (2011)Google Scholar
- 16.Gharan, S.O., Vondrák, J.: Submodular maximization by simulated annealing. In: SODA ’11, pp. 1098–1116. SIAM (2011)Google Scholar
- 17.Globerson, A., Roweis, S.: Nightmare at test time: robust learning by feature deletion. In: Proceedings of the 23rd International Conference on Machine learning, pp. 353–360. ACM (2006)Google Scholar
- 18.Golovin, D., Krause, A.: Adaptive submodularity: Theory and applications in active learning and stochastic optimization. J. Artif. Intell. Res. 42, 427–486 (2011)MathSciNetzbMATHGoogle Scholar
- 19.Guestrin, C., Krause, A., Singh, A.P.: Near-optimal sensor placements in gaussian processes. In: Proceedings of the 22nd International Conference on Machine learning, pp. 265–272. ACM (2005)Google Scholar
- 20.Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM (JACM) 48(4), 761–777 (2001)MathSciNetCrossRefGoogle Scholar
- 21.Krause, A., Guestrin, C., Gupta, A., Kleinberg, J.: Near-optimal sensor placements: maximizing information while minimizing communication cost. In: Proceedings of the 5th International Conference On Information processing in Sensor Networks, pp. 2–10. ACM (2006)Google Scholar
- 22.Krause, A., McMahan, H.B., Guestrin, C., Gupta, A.: Robust submodular observation selection. J. Mach. Learn. Res. 9, 2761–2801 (2008)zbMATHGoogle Scholar
- 23.Leskovec, J., Krause, A., Guestrin, C., Faloutsos, C., VanBriesen, J., Glance, N.: Cost-effective outbreak detection in networks. In: Proceedings of the 13th ACM SIGKDD International Conference on Knowledge discovery and Data Mining, pp. 420–429. ACM (2007)Google Scholar
- 24.Liu, Y., Wei, K., Kirchhoff, K., Song, Y., Bilmes, J.: Submodular feature selection for high-dimensional acoustic score spaces. In: 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 7184–7188. IEEE (2013)Google Scholar
- 25.Nemhauser, G.L., Wolsey, L.A.: Best algorithms for approximating the maximum of a submodular set function. Math. Oper. Res. 3(3), 177–188 (1978)MathSciNetCrossRefGoogle Scholar
- 26.Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions–i. Math. Program. 14(1), 265–294 (1978)MathSciNetCrossRefGoogle Scholar
- 27.Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory, Ser. B 80(2), 346–355 (2000)MathSciNetCrossRefGoogle Scholar
- 28.Sviridenko, M.: A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett. 32(1), 41–43 (2004)MathSciNetCrossRefGoogle Scholar
- 29.Thoma, M., Cheng, H., Gretton, A., Han, J., Kriegel, H.P., Smola, A.J., Song, L., Philip, S.Y., Yan, X., Borgwardt, K.M.: Near-optimal supervised feature selection among frequent subgraphs. In: SDM, pp. 1076–1087. SIAM (2009)Google Scholar
- 30.Vondrák, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: STOC ’08, pp. 67–74. ACM (2008)Google Scholar
- 31.Vondrák, J.: Symmetry and approximability of submodular maximization problems. SIAM J. Comput. 42(1), 265–304 (2013)MathSciNetCrossRefGoogle Scholar
- 32.Vondrák, J., Chekuri, C., Zenklusen, R.: Submodular function maximization via the multilinear relaxation and contention resolution schemes. In: STOC ’11, pp. 783–792. ACM (2011)Google Scholar