Abstract
The concept of submodularity has wide applications in data science, artificial intelligence, and machine learning, providing a boost to the investigation of new ideas, innovative techniques, and creative algorithms to solve different submodular optimization problems arising from a diversity of applications. However pure submodular problems only represent a small portion of the problems we are facing in real life applications. In this paper, we further discuss the two-stage submodular maximization problem under a \(\ell \)-matroid constraint. We design an approximation algorithm with constant approximation ratio with respect to the curvature, which improves the previous bound. In addition, we generalize our algorithm to the two-stage submodular maximization problem under a \(\ell \)-exchange system constraint.
Similar content being viewed by others
Data Availability
Enquiries about data availability should be directed to the authors.
References
Balkanski E, Krause A, Mirzasoleiman B, Singer Y (2016) Learning sparse combinatorial representations via two-stage submodular maximization. In ICML, pp 2207–2216
Conforti M, Cornuejols G (1984) Submodular set functions, matroids and the greedy algorithm: tight worst-case bounds and some generalizations of the Rado-Edmonds theorem. Discr Appl Math 7(3):251–274
Fisher M L, Nemhauser G L, Wolsey L A (1978) An analysis of approximations for maximizing submodular set functions–ii. Polyhedral combinatorics, pp 73–87
Feldman M, Naor J, Schwartz R, Ward J (2011) Improved approximations for k-exchange systems. In: Proceedings of ESA, pp 784–798
Krause A, Guestrin A (2005) Near-optimal nonmyopic value of information in graphical models. In UAI, 5
Lee J, Mirrokni VS, Nagarajan V, Sviridenko M (2010) Maximizing nonmonotone submodular functions under matroid or knapsack constraints. SIAM J Discr Math 23(4):2053–2078
Laitila J, Moilanen A (2017) New performance guarantees for the greedy maximization of submodular set functions. Optim Lett 11:655–665
Mitrovic M, Kazemi E, Zadimoghaddam M, Karbasi A (2018) Data summarization at scale: a two-stage submodular approach. In ICML, pp 3593–3602
Nemhauser GL, Wolsey LA, Fisher ML (1978) An analysis of approximations for maximizing submodular set functions-i. Math Program 14(1):265–294
Schulz AS, Uhan NA (2013) Approximating the least core value and least core of cooperative games with supermodular costs. Discr Optim 10(2):163–180
Stan S, Zadimoghaddam M, Krause A, Karbasi A (2017) Probabilistic submodular maximization in sub-linear time. In ICML, pp 3241–3250
Yang R, Gu S, Gao C, Wu W, Wang H, Xu D (2021) A constrained two-stage submodular maximization. Theor Comput Sci 853:57–64
Zhou M, Chen H, Ren L, Sapiro G, Carin L, Paisley JW (2009) Non-parametric bayesian dictionary learning for sparse image representations. In NIPS, pp 2295–2303
Acknowledgements
The authors would like to thank the referees for giving this paper a careful reading and many valuable comments and useful suggestions.
Funding
The research is supported by NSFC (Nos.12271259, 11971349, 12101314, 12131003), Qinglan Project, Natural Science Foundation of Jiangsu Province (No. BK20200723), and Jiangsu Province Higher Education Foundation (No.20KJB110022).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing Interests
The authors have not disclosed any competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
A preliminary version of this paper appeared in Proceedings of the 15th Annual International Conference on Combinatorial Optimization and Applications (COCOA).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Li, Y., Liu, Z., Xu, C. et al. Two-stage submodular maximization under curvature. J Comb Optim 45, 77 (2023). https://doi.org/10.1007/s10878-023-01001-0
Accepted:
Published:
DOI: https://doi.org/10.1007/s10878-023-01001-0