Abstract
We consider a two-stage submodular maximization under p-matroid (or p-extendible) constraints. In the model, we are given a collection of submodular functions and some p-matroid (or extendible) system constraints for each of these functions, one need to choose a representative set with a cardinality constraint and simultaneously select a series of subsets that are restricted to the representative set for all functions, the aim is to maximize the average of the summarization of these function values. We extend the two-stage submodular maximization under single matroid to handle p-matroid (or p-extendible) constraints, and derive constant approximation ratio algorithms for the two problems, respectively. In the end, we empirically demonstrate the efficiency of our method on some datasets.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Badanidiyuru, A., Mirzasoleiman, B., Karbasi, A., Krause, A.: Streaming submodular maximization: massive data summarization on the fly. In: Proceedings of SIGKDD, pp. 671–680 (2014)
Balkanski, E., Mirzasoleiman, B., Krause, A., Singer, Y.: Learning sparse combinatorial representations via two-stage submodular maximization. In: Proceedings of ICML, pp. 2207–2216 (2016)
Buchbinder, N., Feldman, M., Garg, M.: Deterministic (\(1/2\)+ \(\varepsilon \))-approximation for submodular maximization over a matroid. In: Proceedings of SODA, pp. 241–254 (2019)
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)
Dasgupta, A., Kumar, R., Ravi, S.: Summarization through submodularity and dispersion. In: Proceedings of ACL, pp. 1014–1022 (2013)
Feldman, M., Harshaw, C., Karbasi, A.: Greed is good: near-optimal submodular maximization via greedy optimization. arXiv: 1704.01652 (2017)
Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions-II. In: Balinski, M.L., Hoffman, A.J. (eds.) Polyhedral Combinatorics, pp. 73–87. Springer, Heidelberg (1978). https://doi.org/10.1007/BFb0121195
Gomes, R., Krause, A.: Budgeted nonparametric learning from data streams. In: Proceedings of ICML, pp. 391–398 (2010)
Kempe, D., Kleinberg, J., Tardos, É.: Maximizing the spread of influence through a social network. In: Proceedings of SIGKDD, pp. 137–146 (2003)
Lee, J., Sviridenko, M., Vondrák, J.: Submodular maximization over multiple matroids via generalized exchange properties. Math. Oper. Res. 35(4), 795–806 (2010)
Lin, H., Bilmes, J.: A class of submodular functions for document summarization. In: Proceedings of ACL, pp. 510–520 (2011)
Mestre, J.: Greedy in approximation algorithms. In: Proceedings of ESA, pp. 528–539 (2006)
Mitrovic, S., Bogunovic, I., Norouzi-Fard, A., Tarnawski, J.M., Cevher, V.: Streaming robust submodular maximization: a partitioned thresholding approach. In: Proceedings of NIPS, pp. 4557–4566 (2017)
Mitrovic, M., Kazemi, E., Zadimoghaddam, M., Karbasi, A.: Data summarization at scale: a two-stage submodular approach. arXiv preprint arXiv:1806.02815 (2018)
Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular setfunctions-I. Math. Program. 14(1), 265–294 (1978)
Sarpatwar, K.K., Schieber, B., Shachnai, H.: Constrained submodular maximization via greedy local search. Oper. Res. Lett. 47(1), 1–6 (2019)
Stan, S., Zadimoghaddam, M., Krause, A., Karbasi, A.: Probabilistic submodular maximization in sub-linear time. In: Proceedings of ICML, pp. 3241–3250 (2017)
Vitter, J.S.: Random sampling with a reservoir. ACM Trans. Math. Softw. 11(1), 37–57 (1985)
Wu, W.L., Zhang, Z., Du, D.Z.: Set function optimization. J. Oper. Res. Soc. China (2018). https://doi.org/10.1007/s40305018-0233-3
Acknowledgements
The first and sixth authors are supported by Natural Science Foundation of China (Nos. 11531014, 11871081). The second and fourth authors are supported by Natural Science Foundation (No. 1747818).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Yang, R., Gu, S., Gao, C., Wu, W., Wang, H., Xu, D. (2019). A Two-Stage Constrained Submodular Maximization. In: Du, DZ., Li, L., Sun, X., Zhang, J. (eds) Algorithmic Aspects in Information and Management. AAIM 2019. Lecture Notes in Computer Science(), vol 11640. Springer, Cham. https://doi.org/10.1007/978-3-030-27195-4_30
Download citation
DOI: https://doi.org/10.1007/978-3-030-27195-4_30
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-27194-7
Online ISBN: 978-3-030-27195-4
eBook Packages: Computer ScienceComputer Science (R0)