Skip to main content

On streaming algorithms for maximizing a supermodular function plus a MDR-submodular function on the integer lattice

Journal of Combinatorial Optimization volume 45, Article number: 55 (2023) Cite this article

Abstract

In this paper, we provide a streaming algorithm for the problem of maximizing the sum of a supermodular function and a nonnegative monotone diminishing return submodular (MDR-submodular) function with a knapsack constraint on the integer lattice. Inspired by the SIEVE-STREAMING algorithm, we present a two-pass streaming algorithm by using the threshold technique. Then, we improve the two-pass streaming algorithm to one-pass and further reduce its space complexity. The proposed algorithms are proved to have polynomial time and space complexity, and a performance guarantee dependent on the curvature of the supermodular function. Finally, we carry out numerical experiments to verify the performance of the algorithm.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2

Data availability

Enquiries about data availability should be directed to the authors.

References

  • Badanidiyuru A, Mirzasoleiman B, Karbasi A, Krause A (2014) Streaming submodular maximization: massive data summarization on the fly. In: Proceedings of KDD, pp 671–680

  • Bai W, Bilmes JA (2018) Greed is still good: maximizing monotone submodular+supermodular functions. In: Proceedings of ICML, pp 1–10

  • Buchbinder N, Feldman M, Schwartz R (2015) Onling submodular maximization with preemption. In: Proceedings of SODA, pp. 1202–1216

  • Chakrabarti A, Kale S (2015) Submodular maximization meets streaming: matchings, matroids, and more. Math Program 154:225–247

    Article  MATH  Google Scholar 

  • Das A, Kempe D (2011) Submodular meets spectral: greedy algorithms for subset selection, sparse approximation and dictionary selection. In: Proceedings of ICML, pp 1057–1064

  • He S, Zhang J, Zhang S (2012) Polymatroid optimization, submodularity, and joint replenishment games. Oper Res 60:128–137

    Article  MATH  Google Scholar 

  • Huang C, Kakimura N (2019) Improved streaming algorithms for maximising monotone submodular functions under a knapsack constraint. In: Proceedings of WADS, pp 438–451

  • Ji S, Xu DC, Li M, Wang YJ, Zhang DM (2020) Stochastic greedy algorithm is still good: maximizing submodular + supermodular functions. In: Optimization of complex systems: theory, models, algorithms and applications, pp 488–497

  • Jiang YJ, Wang YS, Xu DC, Yang RQ, Zhang Y (2020) Streaming algorithm for maximizing a monotone non-submodular function under d-knapsack constraint. Optim Lett 14:1235–1248

    Article  MATH  Google Scholar 

  • Kazemi E, Mitrovic M, Zadimoghaddam M, Lattanzi S, Karbasi A (2019) Submodular streaming in all its glory: tight approximation. Minimum memory and low adaptive complexity, In: Proceedings of ICML, pp 3311–3320

  • Khanna R, Elenberg ER, Dimakis AG, Negahban S, Ghosh J (2017) Scalable greedy feature selection via weak submodularity. In: Proceedings of ICAIS, pp 1560–1568

  • Krause A, Singh A, Guestrin C (2008) Near-optimal sensor placements in Gaussian processes: theory, efficient algorithms and empirical studies. J Mach Learn Res 9:235–284

    MATH  Google Scholar 

  • Kuhnle A, Smith J, Crawford V, Thai M (2018) Fast maximization of non-submodular, monotonic functions on the integer lattice. In: Proceedings of ICML, pp 2791–2800

  • Norouzi-Fard A, Tarnawski J, Mitrovic S, Zandieh A, Mousavifar A, Svensson O (2018) Beyond 1/2-approximation for submodular maximization on massive data streams. In: Proceedings of ICML, pp 3829–3838

  • Sarpatwar KK, Schieber B, Shachnai H (2019) Constrained submodular maximization via greedy local search. Oper Res Lett 47:1–6

    Article  MATH  Google Scholar 

  • Soma T, Yoshida Y (2018) Maximization monotone submodular functions over the integer lattice. Math Program 172:539–563

    Article  MATH  Google Scholar 

  • Sviridenko M, Vondrak J, Ward J (2015) Optimal approximation for submodular and supermodular optimization with bounded curvature. In: Proceedings of SODA, pp 1134–1148

  • Wang YJ, Xu DC, Wang YS, Zhang DM (2020) Non-submodular maximization on massive data streams. J Global Optim 76:729–743

    Article  MATH  Google Scholar 

  • Wolsey L (1982) Maximising real-valued submodular set function: primal and dual heuristics for location problems. Math Oper Res 7:410–425

    Article  MATH  Google Scholar 

  • Yang RQ, Xu DC, Li M, Xu YC (2019a) Thresholding methods for streaming submodular maximization with a cardinality constraint and its variants. Nonlinear Comb Optim 2019:123–140

    MATH  Google Scholar 

  • Yang RQ, Xu DC, Du DL, Xu YC, Yan XH (2019b) Maximization of constrained non-submodular functions. In: Proceedings of COCOON, pp 615–626

  • Yu QL, Xu EL, Cui SG (2016) Streaming algorithms for news and scientific literature recommendation: submodular maximization with a d-knapsack constraint. In: Proceedings of IEEE GCSI, https://arxiv.org/abs/1603.05614

  • Zhang ZZ, Du DL, Jiang YJ, Wu CC (2021) Maximizing DR-submodular+supermodular functions on the integer lattice subject to a cardinality constraint. J Global Optim 80:595–616

    Article  MATH  Google Scholar 

Download references

Funding

The first author is supported by Natural Science Foundation of Shandong Province (Nos. ZR2022MA034, ZR2019MA022), and Doctoral research foundation of Weifang University (No. 2017BS02). The second author is supported by Fundamental Research Project of Shenzhen City (No. JCYJ20210324102012033) and National Natural Science Foundation of China (No. 11901158), and Guangxi Key Laboratory of Cryptography and Information Security (No. GCIS202116). The third author is supported by National Natural Science Foundation of China (No. 11871081). The fourth author is supported by National Natural Science Foundation of China (No. 12131003).

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Weifang University, Weifang, 261061, People’s Republic of China

    Jingjing Tan

  2. Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen, 518055, People’s Republic of China

    Yicheng Xu

  3. Guangxi Key Laboratory of Cryptography and Information Security, Guilin, 541004, People’s Republic of China

    Yicheng Xu

  4. School of Computer Science and Technology, Shandong Jianzhu University, Jinan, 250101, People’s Republic of China

    Dongmei Zhang

  5. Department of Operations Research and Information Engineering, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Xiaoqing Zhang

Authors
  1. Jingjing Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yicheng Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dongmei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiaoqing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yicheng Xu.

Ethics declarations

Conflict of interest

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 10th International Conference on Computational Data and Social Networks (CSoNet2021), 2021, pp. 68-75.

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tan, J., Xu, Y., Zhang, D. et al. On streaming algorithms for maximizing a supermodular function plus a MDR-submodular function on the integer lattice. J Comb Optim 45, 55 (2023). https://doi.org/10.1007/s10878-023-00986-y

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10878-023-00986-y

Keywords

  • DR-submodular function
  • Supermodular function
  • Knapsack constraint
  • Threshold greedy algorithm
  • Integer lattice