Abstract
Submodular function has the property of diminishing marginal gain, and thus it has a wide range of applications in combinatorial optimization and in emerging disciplines such as machine learning and artificial intelligence. For any set S, most of previous works usually do not consider how to compute f(S) , but assume that there exists an oracle that will output f(S) directly. In reality, however, the process of computing the exact f is often inevitably inaccurate or costly. At this point, we adopt the easily available noise version F of f. In this paper, we investigate the problems of maximizing a non-negative monotone normalized submodular function minus a non-negative modular function under the \(\varepsilon \)-multiplicative noise in three situations, i.e., the cardinality constraint, the matroid constraint and the online unconstraint. For the above problems, we design three deterministic bicriteria approximation algorithms using greedy and threshold ideas and furthermore obtain good approximation guarantees.
Similar content being viewed by others
Data availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
References
Bateni MH, Hajiaghayi M, Zadimoghaddam M (2013) Submodular secretary problem and extensions. ACM Trans Algorithms 9(4):1–23
Brualdi RA (1969) Comments on bases in dependence structures. Bull Aust Math Soc 1(2):161–167
Chambers CP, Echenique F (2016) Revealed preference theory. Cambridge University Press, England
Chen Y, Hassani SH, Karbasi A (2015) Sequential information maximization: when is greedy near-optimal? In: Proceedings of The 28th Conference on Learning Theory, pp. 338-363. Paris, France
Das A, Kempe D (2018) Approximate submodularity and its applications: subset selection, sparse approximation and dictionary selection. J Mach Learn Res 19(1):74–107
Du DL, Li Y, Xiu NH (2014) Simultaneous approximation of multi-criteria submodular function maximization. J Oper Res Soc China 2(3):271–290
Edmonds J (2003) Submodular functions, matroids, and certain polyhedra. In: Jünger M, Reinelt G, Rinaldi G (eds) Combinatorial optimization ? Eureka, You Shrink!. Lecture notes in computer science, vol. 2570, pp. 11–26. Springer, Berlin, Heidelberg
Feige U (1998) A threshold of ln n for approximating set cover. J ACM 45(4):634–652
Feldman V (2009) On the power of membership queries in agnostic learning. J Mach Learn Res 10:163–182
Gölz P, Procaccia AD (2019) Migration as submodular optimization. In: Proceedings of the 33rd AAAI conference on artificial intelligence, pp. 549-556. Honolulu, Hawaii, USA
Grötschel M, Lovász L, Schrijver A (2012) Geometric algorithms and combinatorial optimization. Springer-Verlag, Berlin, Germany
Harshaw C, Feldman M, Ward J (2019) Submodular maximization beyond non-negativity: guarantees, fast algorithms, and applications. In: Proceedings of the 36th international conference on machine learning, pp. 2634–2643. Long Beach, California, USA
Horel T, Singer Y (2016) Maximization of approximately submodular functions. In: Proceedings of the 30th international conference on neural information processing systems, pp. 3045-3053. Barcelona, Spain
Iwata S, Fleischer L, Fujishige S (2001) A combinatorial strongly polynomial algorithm for minimizing submodular functions. J ACM 48(4):761–777
Iwata S, Nagano K (2009) Submodular function minimization under covering constraints. In: Proceedings of the 50th annual IEEE symposium on foundations of computer science, pp. 671–680. Atlanta, Georgia, USA
Koufogiannakis C, Young NE (2013) Greedy \(\Delta \)-approximation algorithm for covering with arbitrary constraints and submodular cost. Algorithmica 66(1):113–152
Krause A, Leskovec J, Isovitsch S (2006) Optimizing sensor placements in water distribution systems using submodular function maximization. In: Proceedings of the 8th international conference on water distribution systems analysis symposium. pp. 1–17. Cincinnati, Ohio
Liu MY, Tuzel O, Ramalingam S (2013) Entropy-rate clustering: cluster analysis via maximizing a submodular function subject to a matroid constraint. IEEE Trans Pattern Anal Mach Intell 36(1):99–112
Nikolakaki SM, Ene A, Terzi E (2021) An efficient framework for balancing submodularity and cost. In: Proceedings of the 27th ACM SIGKDD conference on knowledge discovery and data mining, pp. 1256–1266. Virtual Event/Singapore
Papadimitriou CH, Yannakakis M (1991) Optimization, approximation, and complexity classes. J Comput Syst Sci 43(3):425–440
Qian C (2021) Multiobjective evolutionary algorithms are still good: maximizing monotone approximately submodular minus modular functions. Evol Comput 29(4):463–490
Singla A, Tschiatschek S, Krause A (2016) Noisy submodular maximization via adaptive sampling with applications to crowdsourced image collection summarization. In: Proceedings of the 30th AAAI conference on artificial intelligence. pp. 12-17. Phoenix, Arizona, USA
Sviridenko M, Vondrak J, Ward J (2017) Optimal approximation for submodular and supermodular optimization with bounded curvature. Math Oper Res 42(4):1197–1218
Svitkina Z, Fleischer L (2011) Submodular approximation: sampling-based algorithms and lower bounds. SIAM J Comput 40(6):1715–1737
Wang Y, Xu Y, Yang X (2021) On maximizing the difference between an approximately submodular function and a linear function subject to a matroid constraint. In: Proceedings of the 15th international conference on combinatorial optimization and applications, pp. 75–85. Tianjin, China
Xiao D, Guo L, Liao K (2021) Streaming submodular maximization under differential privacy noise. In: Proceedings of the 15th international conference on combinatorial optimization and applications. pp. 431–444. Tianjin, China
Yang R, Xu D, Cheng Y (2019) Streaming submodular maximization under noises. In: Proceedings of the 39th international conference on distributed computing systems, pp. 348–357
Zhang H, Zhang H, Kuhnle A (2016) Profit maximization for multiple products in online social networks. In: Proceedings of the 35th annual IEEE international conference on computer communications, pp. 1–9
Funding
The Funding was provided by National Natural Science Foundation of China (grant no: 11971447, 11871442) and Fundamental Research Funds for the Central Universities.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported in part by the National Natural Science Foundation of China (11971447, 11871442), and the Fundamental Research Funds for the Central Universities.
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
Gong, S., Liu, B., Geng, M. et al. Algorithms for maximizing monotone submodular function minus modular function under noise. J Comb Optim 45, 96 (2023). https://doi.org/10.1007/s10878-023-01026-5
Accepted:
Published:
DOI: https://doi.org/10.1007/s10878-023-01026-5