Abstract
Constrained submodular maximization (CSM) is widely used in numerous data mining and machine learning applications such as data summarization, network monitoring, exemplar-clustering, and nonparametric learning. The CSM can be described as: Given a ground set, a specified constraint, and a submodular set function defined on the power set of the ground set, the goal is to select a subset that satisfies the constraint such that the function value is maximized. Generally, the CSM is NP-hard, and cardinality constrained submodular maximization is well researched. The greedy algorithm and its variants have good performance guarantees for constrained submodular maximization. When dealing with large input scenario, it is usually formulated as streaming constrained submodular maximization (SCSM), and the classical greedy algorithm is usually inapplicable. The streaming model uses a limited memory to extract a small fraction of items at any given point of time such that the specified constraint is satisfied, and good performance guarantees are also maintained. In this chapter, we list the up-to-date popular algorithms for streaming submodular maximization with cardinality constraint and its variants, and summarize some problems in streaming submodular maximization that are still open.
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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)
Badanidiyuru, A., Mirzasoleiman, B., Karbasi, A., Krause, A.: Streaming submodular maximization: massive data summarization on the fly. In: Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 671–680. ACM, New York (2014)
Bian, A.A., Buhmann, J.M., Krause, A., Tschiatschek, S.: Guarantees for greedy maximization of non-submodular functions with applications. In: Proceedings of the 34th International Conference on Machine Learning, pp. 498–507. ACM, New York (2017)
Chakrabarti, A., Kale, S.: Submodular maximization meets streaming: matchings, matroids, and more. Math. Program. 154(1–2), 225–247 (2015)
Conforti, M., Cornuéjols, G.: Submodular set functions, matroids and the greedy algorithm: tight worst-case bounds and some generalizations of the Rado-Edmonds theorem. Discrete Appl. Math. 7(3), 251–274 (1984)
Das, A., Kempe, D.: Algorithms for subset selection in linear regression. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 45–54. ACM, New York (2008)
Das, A., Kempe, D.: Submodular meets spectral: greedy algorithms for subset selection, sparse approximation and dictionary selection. In: Proceedings of the 28th International Conference on Machine Learning, pp. 1057–1064. ACM, New York (2011)
El-Arini, K., Guestrin, C.: Beyond keyword search: discovering relevant scientific literature. In: Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 439–447. ACM, New York (2011)
Elenberg, E., Dimakis, A.G., Feldman, M., Karbasi, A.: Streaming weak submodularity: interpreting neural networks on the fly. In: Advances in Neural Information Processing Systems, pp. 4044–4054. MIT Press, Cambridge (2017)
Epasto, A., Lattanzi, S., Vassilvitskii, S., Zadimoghaddam, M.: Submodular optimization over sliding windows. In: Proceedings of the 26th International Conference on World Wide Web, pp. 421–430. ACM, New York (2017)
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 25(4), 634–652 (1998)
Feldman, M., Karbasi, A., Kazemi, E.: Do less, get more: streaming submodular maximization with subsampling (2018). Preprint. arXiv:1802.07098
Gomes, R., Krause, A.: Budgeted nonparametric learning from data streams. In: Proceedings of the 27th International Conference on Machine Learning, pp. 391–398. ACM, New York (2010)
Huang, C.C., Kakimura, N., Yoshida, Y.: Streaming algorithms for maximizing monotone submodular functions under a knapsack constraint. In: The 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX and the 21st International Workshop on Randomization and Computation, RANDOM (2017)
Jain, P., Tewari, A., Kar, P.: On iterative hard thresholding methods for high-dimensional m-estimation. In: Advances in Neural Information Processing Systems, pp. 685–693. MIT Press, Cambridge (2014)
Krause, A., Cevher, V.: Submodular dictionary selection for sparse representation. In: Proceedings of the 27th International Conference on Machine Learning, pp. 567–574. ACM, New York (2010)
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. ACM, New York (2011)
Mirzasoleiman, B., Karbasi, A., Krause, A.: Deletion-robust submodular maximization: Data summarization with “the right to be forgotten”. In: Proceedings of the 34th International Conference on Machine Learning, pp. 2449–2458. ACM, New York (2017)
Mitrović, S., Bogunovic, I., Norouzi-Fard, A., Tarnawski, J.: Streaming robust submodular maximization: a partitioned thresholding approach. In: Advances in Neural Information Processing Systems, pp. 4560–4569. MIT Press, Cambridge (2017)
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)
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)
Segui-Gasco, P., Shin, H.S.: Fast non-monotone submodular maximisation subject to a matroid constraint (2017). Preprint. arXiv:1703.06053
Soma, T., Kakimura, N., Inaba, K., Kawarabayashi, K.: Optimal budget allocation: theoretical guarantee and efficient algorithm. In: Proceedings of the 31st International Conference on International Conference on Machine Learning, pp. 351–359. ACM, New York (2014)
Vondrák, J.: Submodularity and curvature: the optimal algorithm (combinatorial optimization and discrete algorithms). RIMS Kokyuroku Bessatsu (2010)
Wang, Y., Li, Y., Tan, K.L.: Efficient streaming algorithms for submodular maximization with multi-knapsack constraints (2017). Preprint. arXiv:1706.04764
Acknowledgements
The first two authors are supported by National Natural Science Foundation of China (Nos. 11531014, 11871081). The third author is supported by the Higher Educational Science and Technology Program of Shandong Province (No. J17KA171). The fourth author is supported by China Postdoctoral Science Foundation funded project (No. 2018M643233) and National Natural Science Foundation of China (Nos.61433012, U1435215).
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Yang, R., Xu, D., Li, M., Xu, Y. (2019). Thresholding Methods for Streaming Submodular Maximization with a Cardinality Constraint and Its Variants. In: Du, DZ., Pardalos, P., Zhang, Z. (eds) Nonlinear Combinatorial Optimization. Springer Optimization and Its Applications, vol 147. Springer, Cham. https://doi.org/10.1007/978-3-030-16194-1_5
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