Skip to main content
SpringerLink
Log in

Efficient Algorithms for k-Submodular Function Maximization with p-System and d-Knapsack Constraint

  • Conference paper
  • First Online:
Combinatorial Optimization and Applications (COCOA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14461))

  • 455 Accesses

Abstract

The k-submodular function is a generalization of the submodular function. The k-submodular optimization problems have important applications in influence maximization problems and sensor placement problems with k kinds of sensors. In this paper, we study the problems of maximizing k-submodular functions subject to two kinds of constraints. We set \(\alpha =2\) when f is monotone and \(\alpha =3\) when f is non-monotone. For the p-system constraint, we get a \(\frac{1-\epsilon }{p+\alpha }\)-approximation ratio. For the intersection of p-system and d-knapsack constraints, we get an approximation ratio of \(\frac{1-\epsilon }{p+\alpha +2d}\). And subsequently, we propose an improved algorithm that improves the approximation ratio to \(\frac{1-\epsilon }{p+\alpha +\frac{1+\sqrt{5}}{2}d}\).

This work was supported in part by the National Natural Science Foundation of China (11971447), and the Fundamental Research Funds for the Central Universities (202261097).

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 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
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. Badanidiyuru, A., Mirzasoleiman, B., Karbasi, A., Krause, A.: Streaming submodular maximization: massive data summarization on the fly. In: 20th International Proceedings on KDD, pp. 671–680. ACM, New York, NY, USA (2014)

    Google Scholar 

  2. Badanidiyuru, A., Vondrák, J.: Fast algorithms for maximizing submodular functions. In: 25th International Proceedings on SODA, pp. 1497–1514. SIAM, Portland, Oregon, USA (2014)

    Google Scholar 

  3. Buchbinder, N., Feldman, M., Schwartz, R.: Online submodular maximization with preemption. ACM Trans. Algorithms 15(3), 1–31 (2015)

    Article  MathSciNet  Google Scholar 

  4. Ene, A., Nguyen, H.L.: Streaming algorithm for monotone \(k\)-submodular maximization with cardinality constraints. In: 39th International Proceedings on ICML, pp. 5944–5967. PMLR, Baltimore, Maryland, USA (2022)

    Google Scholar 

  5. 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. Mathematical Programming Studies, vol. 8. Springer, Berlin, Heidelberg (1978). https://doi.org/10.1007/BFb0121195

  6. Gupta, A., Roth, A., Schoenebeck, G., Talwar, K.: Constrained non-monotone submodular maximization: offline and secretary algorithms. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 246–257. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17572-5_20

    Chapter  Google Scholar 

  7. 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 

  8. Iwata, S., ichi Tanigawa, S., Yoshida, Y.: Improved approximation algorithms for \(k\)-submodular function maximization. In: 27th International Proceedings on SODA, pp. 404–413. SIAM, Arlington, VA, USA (2015)

    Google Scholar 

  9. Kazemi, E., Mitrovic, M., Zadimoghaddam, M., Lattanzi, S., Karbasi, A.: Submodular streaming in all its glory: Tight approximation, minimum memory and low adaptive complexity. In: 36th International Proceedings on ICML, pp. 3311–3320. PMLR, Long Beach, California (2019)

    Google Scholar 

  10. Kempe, D., Kleinberg, J.M., Tardos, É.: Maximizing the spread of influence through a social network. In: 20th International Proceedings on KDD, pp. 137–146. ACM, Washington, DC, USA (2003)

    Google Scholar 

  11. Krause, A., McMahan, H.B., Guestrin, C., Gupta, A.: Robust submodular observation selection. J. Mach. Learn. Res. 9(12), 2761–2801 (2008)

    Google Scholar 

  12. Li, W., Feldman, M., Kazemi, E., Karbasi, A.: Submodular maximization in clean linear time. In: 36th International Proceedings on NeurIPS, pp. 7887–7897. New Orleans, LA, USA (2022)

    Google Scholar 

  13. Li, Y., Fan, J., Wang, Y., Tan, K.L.: Influence maximization on social graphs: a survey. IEEE Trans. Knowl. Data Eng. 30(10), 1852–1872 (2018)

    Article  Google Scholar 

  14. Mirzasoleiman, B., Badanidiyuru, A., Karbasi, A.: Fast constrained submodular maximization: personalized data summarization. In: 36th International Proceedings on ICML, pp. 1358–1367. JMLR.org, New York City, NY, USA (2016)

    Google Scholar 

  15. Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions-i. Math. Program. 14(1), 265–294 (1978)

    Article  MathSciNet  Google Scholar 

  16. Nguyen, L.N., Thai, M.T.: Streaming \(k\)-submodular maximization under noise subject to size constraint. In: 37th International Proceedings on ICML, pp. 7338–7347. PMLR, Virtual Event (2020)

    Google Scholar 

  17. Ohsaka, N., Yoshida, Y.: Monotone \(k\)-submodular function maximization with size constraints. In: 28th International Proceedings on NIPS, pp. 7694–702. Montreal, Quebec, Canada (2015)

    Google Scholar 

  18. Orlin, J.B., Schulz, A.S., Udwani, R.: Robust monotone submodular function maximization. Math. Program. 172(1–2), 505–537 (2018)

    Article  MathSciNet  Google Scholar 

  19. Oshima, H.: Derandomization for k-submodular maximization. In: Brankovic, L., Ryan, J., Smyth, W.F. (eds.) IWOCA 2017. LNCS, vol. 10765, pp. 88–99. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78825-8_8

    Chapter  Google Scholar 

  20. Pham, C.V., Vu, Q.C., Ha, D.K.T., 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  Google Scholar 

  21. Qian, C., Shi, J.C., Tang, K., Zhou, Z.H.: Constrained monotone \(k\)-submodular function maximization using multiobjective evolutionary algorithms with theoretical guarantee. IEEE Trans. Evol. Comput. 22(4), 595–608 (2018)

    Article  Google Scholar 

  22. Rafiey, A., Yoshida, Y.: Fast and private submodular and \( k \)-submodular functions maximization with matroid constraints. In: 28th International Proceedings on ICML, pp. 7887–7897. PMLR, Virtual Event (2020)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Singh, A.P., Guillory, A., Bilmes, J.A.: On bisubmodular maximization. In: 15th International Proceedings on AISTATS, pp. 1055–1063. JMLR.org, La Palma, Canary Islands, Spain (2012)

    Google Scholar 

  25. Sun, Y., Liu, Y., Li, M.: Maximization of \(k\)-submodular function with a matroid constraint. In: Du, DZ., Du, D., Wu, C., Xu, D. (eds.) Theory and Applications of Models of Computation. TAMC 2022. LNCS, vol. 13571. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-20350-3_1

  26. 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  Google Scholar 

  27. Wang, B., Zhou, H.: Multilinear extension of \(k\)-submodular functions. ArXiv abs/2107.07103 (2021)

    Google Scholar 

  28. Ward, J., Zivný, S.: Maximizing bisubmodular and \(k\)-submodular functions. In: 16th International Proceedings on SODA, pp. 1468–1481. SIAM, Portland, Oregon, USA (2014)

    Google Scholar 

  29. Ward, J., Zivný, S.: Maximizing \(k\)-submodular functions and beyond. ACM Trans. Algorithms 12(4), 1–26 (2014)

    Article  MathSciNet  Google Scholar 

  30. 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  Google Scholar 

  31. Zhang, Y., Li, M., Yang, D., Xue, G.: A budget feasible mechanism for \(k\)-topic influence maximization in social networks. In: 19th International Proceedings on GLOBECOM, pp. 1–6. IEEE, Waikoloa, HI, USA (2019)

    Google Scholar 

  32. Zheng, L., Chan, H., Loukides, G., Li, M.: Maximizing approximately \(k\)-submodular functions. In: 15th International Proceedings on SIAM, pp. 414–422. SIAM, Virtual Event (2021)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Ocean University of China, Qingdao, 266100, China

    Wenzhe Zhang, Shufang Gong & Bin Liu

Authors
  1. Wenzhe Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shufang Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bin Liu .

Editor information

Editors and Affiliations

  1. University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

  2. Beijing Normal University, Zhuhai, China

    Jianxiong Guo

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Zhang, W., Gong, S., Liu, B. (2024). Efficient Algorithms for k-Submodular Function Maximization with p-System and d-Knapsack Constraint. In: Wu, W., Guo, J. (eds) Combinatorial Optimization and Applications. COCOA 2023. Lecture Notes in Computer Science, vol 14461. Springer, Cham. https://doi.org/10.1007/978-3-031-49611-0_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-49611-0_19

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-49610-3

  • Online ISBN: 978-3-031-49611-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation