Abstract
Submodular optimization is a classical problem of combinatorial optimization. The objective functions of many combinatorial optimization problems are submodular functions and they also have significant applications in real life. Since some practical problems such as budget allocations that are hard to be modeled over set functions, submodular functions over the integer lattice have been widely and intensively studied subject to various classical constraints for decades. In this paper we study the robustness of maximizing a monotone diminishing return submodular function over the integer lattice under the cardinality constraint. We propose a robustness model over the integer lattice and design algorithms under this specific model by utilizing stochastic strategy combining with binary search approach. The algorithms we designed in centralized settings can achieve a \((1/2-\delta )\)-approximation and maintain robustness against deleting any d elements adversarially. While in streaming settings the algorithms we designed can still achieve the same approximation and be robust against deleting any d elements adversarially as well.
This work was supported in part by the National Natural Science Foundation of China (11971447), and the Fundamental Research Funds for the Central Universities.
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References
Alon, N., Gamzu, I., Tennenholtz, M.: Optimizing budget allocation among channels and influencers. In: Proceedings of the 21st International Conference on World Wide Web, pp. 381–388. ACM, New York (2012)
Cunningham, W.H.: On submodular function minimization. Combinatorica 5(3), 185–192 (1985)
Charikar, M., Khuller, S., Mount, D.M., Narasimhan, G.: Algorithms for facility location problems with outliers. In: Proceedings of ACM-SIAM Symposium on Discrete Algorithms, pp. 642–651. SIAM, Philadelphia (2001)
Feldman, M., Norouzi-Fard, A., Svensson, O., Zenklusen, R.: The one-way communication complexity of submodular maximization with applications to streaming and robustness. ACM 70(4), 1–52 (2023)
Gao, H., Xu, H., Vucetic, S.: Sample efficient decentralized stochastic Frank-Wolfe methods for continuous DR-submodular Maximization. In: Proceedings of International Joint Conferences on Artificial Intelligence, pp. 3501–3507. AAAI Press, Palo Alto (2021)
Gong, S., Nong, Q., Bao, S., Fang, Q., Du, D.Z.: A fast and deterministic algorithm for knapsack-constrained monotone DR-submodular maximization over an integer lattice. J. Global Optim. 85(1), 15–38 (2023)
Ji, S., Xu, D., Li, M., Zhang, D.: Stochastic greedy algorithms for maximizing constrained submodular+ supermodular functions. Concurr. Comput. 35(17), e6575 (2023)
Khuller, S., Moss, A., Naor, J.: The budgeted maximum coverage problem. Inform. Process. Lett. 70(1), 39–45 (1999)
Kazemi, E., Zadimoghaddam, M., Karbasi, A.: Scalable deletion-robust submodular maximization: data summarization with privacy and fairness constraints. In: Proceedings of the 35th International Conference on Machine Learning, pp. 2544–2553. ICML, Stockholm (2018)
Kothawade, S., Beck, N., Killamsetty, K., Iyer, R.: Similar: submodular information measures based active learning in realistic scenarios. NIPS 34, 18685–18697 (2021)
Lin, H., Bilmes, J.: Multi-document summarization via budgeted maximization of submodular functions. In: Proceedings of the 2010 Annual Conference of the North American Chapter of the Association for Computational Linguistics, pp. 912–920. ACL, Stroudsburg (2010)
Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions. Math. Program. 14(1), 265–294 (1978)
Orlin, J.B., Schulz, A.S., Udwani, R.: Robust monotone submodular function maximization. Math. Program. 172(1–2), 505–537 (2018)
Queyranne, M.: Minimizing symmetric submodular functions. Math. Program. 82, 3–12 (1998)
Sviridenko, M.: A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett. 32(1), 41–43 (2004)
Soma, T., Kakimura, N., Inaba, K., Kawarabayashi, K.: Optimal budget allocation: theoretical guarantee and efficient algorithm. In: Proceedings of the International Conference on Machine Learning, pp. 351–359. JMLR, New York (2014)
Soma, T., Yoshida, Y.: Maximizing monotone submodular functions over the integer lattice. Math. Program. 172(1–2), 539–563 (2018)
Yang, R., Xu, D., Guo, L., Zhang, D.: Regularized two-stage submodular maximization under streaming. Sci. China Inf. Sci. 65(4), 140602 (2022)
Zhang, Z., Guo, L., Wang, Y., Zhang, D.: Streaming algorithms for maximizing monotone DR-submodular functions with a cardinality constraint on the integer lattice. Asia. Pac. J. Oper. Res. 38(05), 2140004 (2021)
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Zhou, G., Liu, B., Qiang, Y. (2024). Deletion-Robust Submodular Maximization Under the Cardinality Constraint over the Integer Lattice. In: Hà, M.H., Zhu, X., Thai, M.T. (eds) Computational Data and Social Networks. CSoNet 2023. Lecture Notes in Computer Science, vol 14479. Springer, Singapore. https://doi.org/10.1007/978-981-97-0669-3_22
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DOI: https://doi.org/10.1007/978-981-97-0669-3_22
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