Skip to main content
SpringerLink
Log in

An Improved Analysis of the Greedy+Singleton Algorithm for k-Submodular Knapsack Maximization

  • Conference paper
  • First Online:
Frontiers of Algorithmics (IJTCS-FAW 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13933))

Included in the following conference series:

Abstract

We focus on maximizing a non-negative k-submodular function under a knapsack constraint. As a generalization of submodular functions, a k-submodular function considers k distinct, non-overlapping subsets instead of a single subset as input. We explore the algorithm of Greedy+Singleton, which returns the better one between the best singleton solution and the fully greedy solution. When the function is monotone, we prove that Greedy+Singleton achieves an approximation ratio of \(\frac{1}{4}(1-\frac{1}{e^2})\approx 0.216\), improving the previous analysis of 0.158 in the literature. Further, we provide the first analysis of Greedy+Singleton for non-monotone functions, and prove an approximation ratio of \(\frac{1}{6}(1-\frac{1}{e^3})\approx 0.158\).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 74.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Chen, J., Tang, Z., Wang, C.: Monotone \(k\)-submodular knapsack maximization: an analysis of the Greedy+Singleton algorithm. In: Ni, Q., Wu, W. (eds.) AAIM 2022. LNCS, vol. 13513, pp. 144–155. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-16081-3_13

    Chapter  Google Scholar 

  2. Ene, A., Nguyen, H.: Streaming algorithm for monotone k-submodular maximization with cardinality constraints. In: Proceedings of the 39th International Conference on Machine Learning (ICML), pp. 5944–5967. PMLR (2022)

    Google Scholar 

  3. Feldman, M., Nutov, Z., Shoham, E.: Practical budgeted submodular maximization. Algorithmica 1–40 (2022)

    Google Scholar 

  4. Gridchyn, I., Kolmogorov, V.: Potts model, parametric maxflow and \(k\)-submodular functions. In: Proceedings of the IEEE International Conference on Computer Vision (ICCV), pp. 2320–2327 (2013)

    Google Scholar 

  5. Hirai, H., Iwamasa, Y.: On \(k\)-submodular relaxation. SIAM J. Discret. Math. 30(3), 1726–1736 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Huber, A., Kolmogorov, V.: Towards minimizing k-submodular functions. In: Mahjoub, A.R., Markakis, V., Milis, I., Paschos, V.T. (eds.) ISCO 2012. LNCS, vol. 7422, pp. 451–462. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32147-4_40

    Chapter  Google Scholar 

  7. Iwata, S., Tanigawa, S., Yoshida, Y.: Improved approximation algorithms for \(k\)-submodular function maximization. In: Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 404–413 (2016)

    Google Scholar 

  8. Khuller, S., Moss, A., Naor, J.S.: The budgeted maximum coverage problem. Inf. Process. Lett. 70(1), 39–45 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kudla, J., Živnỳ, S.: Sparsification of monotone \( k \)-submodular functions of low curvature. arXiv preprint arXiv:2302.03143 (2023)

  10. Kulik, A., Schwartz, R., Shachnai, H.: A refined analysis of submodular greedy. Oper. Res. Lett. 49(4), 507–514 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lin, H., Bilmes, J.: Multi-document summarization via budgeted maximization of submodular functions. In: Human Language Technologies: The 2010 Annual Conference of the North American Chapter of the Association for Computational Linguistics, pp. 912–920 (2010)

    Google Scholar 

  12. Nguyen, L., Thai, M.T.: Streaming \(k\)-submodular maximization under noise subject to size constraint. In: Proceedings of the 37th International Conference on Machine Learning (ICML), pp. 7338–7347. PMLR (2020)

    Google Scholar 

  13. Ohsaka, N., Yoshida, Y.: Monotone \(k\)-submodular function maximization with size constraints. In: Proceedings of the 28th International Conference on Neural Information Processing Systems (NeurIPS), vol. 1, pp. 694–702 (2015)

    Google Scholar 

  14. Oshima, H.: Improved randomized algorithm for \(k\)-submodular function maximization. SIAM J. Discret. Math. 35(1), 1–22 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pham, C.V., Vu, Q.C., Ha, D.K., Nguyen, T.T., Le, N.D.: Maximizing k-submodular functions under budget constraint: applications and streaming algorithms. J. Comb. Optim. 44(1), 723–751 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sakaue, S.: On maximizing a monotone \(k\)-submodular function subject to a matroid constraint. Discret. Optim. 23, 105–113 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Soma, T.: No-regret algorithms for online \( k \)-submodular maximization. In: Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), pp. 1205–1214. PMLR (2019)

    Google Scholar 

  18. Sun, Y., Liu, Y., Li, M.: Maximization of \(k\)-submodular function with a matroid constraint. In: Du, D.Z., Du, D., Wu, C., Xu, D. (eds.) TAMC 2022. LNCS, vol. 13571, pp. 1–10. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-20350-3_1

    Chapter  Google Scholar 

  19. Sviridenko, M.: A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett. 32(1), 41–43 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tang, Z., Wang, C., Chan, H.: Monotone k-submodular secretary problems: cardinality and knapsack constraints. Theor. Comput. Sci. 921, 86–99 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tang, Z., Wang, C., Chan, H.: On maximizing a monotone k-submodular function under a knapsack constraint. Oper. Res. Lett. 50(1), 28–31 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, B., Zhou, H.: Multilinear extension of \( k \)-submodular functions. arXiv preprint arXiv:2107.07103 (2021)

  23. Ward, J., Živnỳ, S.: Maximizing \(k\)-submodular functions and beyond. ACM Trans. Algorithms 12(4), 1–26 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xiao, H., Liu, Q., Zhou, Y., Li, M.: Small notes on \(k\)-submodular maximization with a knapsack constraint. Technical report (2023)

    Google Scholar 

  25. Yu, K., Li, M., Zhou, Y., Liu, Q.: Guarantees for maximization of \(k\)-submodular functions with a knapsack and a matroid constraint. In: Ni, Q., Wu, W. (eds.) AAIM 2022. LNCS, vol. 13513, pp. 156–167. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-16081-3_14

    Chapter  Google Scholar 

  26. Yu, K., Li, M., Zhou, Y., Liu, Q.: On maximizing monotone or non-monotone k-submodular functions with the intersection of knapsack and matroid constraints. J. Comb. Optim. 45(3), 1–21 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yu, Q., Küçükyavuz, S.: An exact cutting plane method for \(k\)-submodular function maximization. Discret. Optim. 42, 100670 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is partially supported by Artificial Intelligence and Data Science Research Hub, BNU-HKBU United International College (UIC), No. 2020KSYS007, and by a grant from UIC (No. UICR0400025-21). Zhongzheng Tang is supported by National Natural Science Foundation of China under Grant No. 12101069. Chenhao Wang is supported by NSFC under Grant No. 12201049, and is also supported by UIC grants of UICR0400014-22, UICR0200008-23 and UICR0700036-22.

Author information

Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Zhongzheng Tang

  2. BNU-HKBU United International College, Zhuhai, 519087, China

    Jingwen Chen & Chenhao Wang

  3. Beijing Normal University, Zhuhai, 519087, China

    Chenhao Wang

Authors
  1. Zhongzheng Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jingwen Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chenhao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chenhao Wang .

Editor information

Editors and Affiliations

  1. City University of Hong Kong, Hong Kong SAR, China

    Minming Li

  2. Chinese Academy of Sciences, Beijing, China

    Xiaoming Sun

  3. University of Macau, Macau, China

    Xiaowei Wu

Rights and permissions

Reprints and permissions

Copyright information

© 2023 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Tang, Z., Chen, J., Wang, C. (2023). An Improved Analysis of the Greedy+Singleton Algorithm for k-Submodular Knapsack Maximization. In: Li, M., Sun, X., Wu, X. (eds) Frontiers of Algorithmics. IJTCS-FAW 2023. Lecture Notes in Computer Science, vol 13933. Springer, Cham. https://doi.org/10.1007/978-3-031-39344-0_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-39344-0_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-39343-3

  • Online ISBN: 978-3-031-39344-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation