Skip to main content
SpringerLink
Log in

Maximizing monotone submodular functions over the integer lattice

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice. Suppose that a non-negative monotone submodular function \(f:\mathbb {Z}_+^n \rightarrow \mathbb {R}_+\) is given via an evaluation oracle. Assume further that f satisfies the diminishing return property, which is not an immediate consequence of submodularity when the domain is the integer lattice. Given this, we design polynomial-time \((1-1/e-\epsilon )\)-approximation algorithms for a cardinality constraint, a polymatroid constraint, and a knapsack constraint. For a cardinality constraint, we also provide a \((1-1/e-\epsilon )\)-approximation algorithm with slightly worse time complexity that does not rely on the diminishing return property.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Note that f is DR-submodular if and only if it is lattice submodular and satisfies the coordinate-wise concave condition: \(f(\varvec{x}+\varvec{\chi }_e) - f(\varvec{x}) \ge f(\varvec{x}+2\varvec{\chi }_e) - f(\varvec{x}+\varvec{\chi }_e)\) for any \(\varvec{x}\) and \(e \in E\) (see [27, Lemma 2.3]).

  2. If \(n < 3\), by a similar argument, one can show that there exists \(\varvec{x}_0\) in the output of \(\textsf {PartialEnumeration}\) that attains \((1-\epsilon )\)-approximation.

References

  1. Alon, N., Gamzu, I., Tennenholtz, M.: Optimizing budget allocation among channels and influencers. In: Proceedings of the 21st International Conference on World Wide Web (WWW), pp. 381–388 (2012)

  2. Badanidiyuru, A., Vondrák, J.: Fast algorithms for maximizing submodular functions. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1497–1514 (2013)

    Google Scholar 

  3. Buchbinder, N., Feldman, M.: Deterministic algorithms for submodular maximization problems. In: Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 392–403 (2015)

  4. Buchbinder, N., Feldman, M., Naor, J., Schwartz, R.: A tight linear time \((1/2)\)-approximation for unconstrained submodular maximization. In: Proceedings of the IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 649–658 (2012)

  5. Buchbinder, N., Feldman, M., Naor, J.S., Schwartz, R.: Submodular maximization with cardinality constraints. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1433–1452 (2014)

    Google Scholar 

  6. Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput. 40, 1740–1766 (2011)

    Article  MathSciNet  Google Scholar 

  7. Chekuri, C., Vondrák, J., Zenklusen, R.: Dependent randomized rounding via exchange properties of combinatorial structures. In: Proceedings of IEEE 51st Annual Symposium on Foundations of Computer Science, pp. 575–584 (2010)

  8. Cunningham, W.H.: Testing membership in matroid polyhedra. J. Comb. Theory Ser. B 188, 161–188 (1984)

    Article  MathSciNet  Google Scholar 

  9. Demaine, E.D., Hajiaghayi, M., Mahini, H., Malec, D.L., Raghavan, S., Sawant, A., Zadimoghadam, M.: How to influence people with partial incentives. In: Proceedings of the 23rd International World Wide Web Conference, pp. 937–948 (2014)

  10. Feige, U.: A threshold of \(\ln n\) for approximating set cover. J. ACM 45, 634–652 (1998)

    Article  MathSciNet  Google Scholar 

  11. Feige, U., Mirrokni, V.S., Vondrák, J.: Maximizing non-monotone submodular functions. SIAM J. Comput. 40(4), 1133–1153 (2011)

    Article  MathSciNet  Google Scholar 

  12. Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Elsevier, Amsterdam (2005)

    MATH  Google Scholar 

  13. Gottschalk, C., Peis, B.: Submodular function maximization on the bounded integer lattice. In: Approximation and Online Algorithms: 13th International Workshop, WAOA 2015, Patras, Greece, 17–18 Sept 2015. Revised Selected Papers, pp. 133–144 (2015)

    Chapter  Google Scholar 

  14. Iwata, S.: Submodular function minimization. Math. Program. 112(1), 45–64 (2007)

    Article  MathSciNet  Google Scholar 

  15. Iwata, S., Tanigawa, S.I., 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. Society for Industrial and Applied Mathematics, Philadelphia, PA (2016)

  16. Kapralov, M., Post, I., Vondrák, J.: Online submodular welfare maximization: Greedy is optimal. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1216–1225 (2012)

    Chapter  Google Scholar 

  17. Kempe, D., Kleinberg, J., Tardos, E.: Maximizing the spread of influence through a social network. In: Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146 (2003)

  18. Krause, A., Golovin, D.: Submodular function maximization. In: Bordeaux, L., Hamadi, Y., Kohli, P. (eds.) Tractability: Practical Approaches to Hard Problems, pp. 71–104. Cambridge University Press, Cambridge (2014)

    Google Scholar 

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

  20. Lin, H., Bilmes, J.: A class of submodular functions for document summarization. In: Proceedings of the 12th Annual Conference of the North American Chapter of the Association for Computational Linguistics, pp. 510–520 (2011)

  21. Murota, K.: Discrete Convex Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2003)

    Book  Google Scholar 

  22. Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions—II. Math. Program. Stud. 8, 73–87 (1978)

    Article  MathSciNet  Google Scholar 

  23. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1996)

    Google Scholar 

  24. Shioura, A.: On the pipage rounding algorithm for submodular function maximization—a view from discrete convex analysis. Discrete Math. Algorithms Appl. 1(1), 1–23 (2009)

    Article  MathSciNet  Google Scholar 

  25. Singh, A., Guillory, A., Bilmes, J.: On bisubmodular maximization. In: Proceedings of the 15th International Conference on Artificial Intelligence and Statistics, pp. 1055–1063 (2012)

  26. Soma, T., Kakimura, N., Inaba, K., Kawarabayashi, K.: Optimal budget allocation: theoretical guarantee and efficient algorithm. In: Proceedings of the 31st International Conference on Machine Learning (ICML) (2014)

  27. Soma, T., Yoshida, Y.: A generalization of submodular cover via the diminishing return property on the integer lattice. In: Advances in Neural Information Processing Systems (NIPS) (2015)

  28. Soma, T., Yoshida, Y.: Maximizing monotone submodular functions over the integer lattice. In: the Proceedings of the 18th Conference on Integer Programming and Combinatorial Optimization (IPCO), pp. 325–336 (2016)

    MATH  Google Scholar 

  29. 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  Google Scholar 

  30. Ward, J., Živný, S.: Maximizing bisubmodular and \(k\)-submodular functions. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1468–1481 (2014)

Download references

Author information

Authors and Affiliations

  1. Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan

    Tasuku Soma

  2. National Institute of Informatics, Tokyo, Japan

    Yuichi Yoshida

Authors
  1. Tasuku Soma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuichi Yoshida
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tasuku Soma.

Additional information

Tasuku Soma is supported by JSPS Grant-in-Aid for JSPS Fellows. Yuichi Yoshida is supported by JST ERATO Grant Number JPMJER1305 and JSPS KAKENHI Grant Number JP17H04676.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Soma, T., Yoshida, Y. Maximizing monotone submodular functions over the integer lattice. Math. Program. 172, 539–563 (2018). https://doi.org/10.1007/s10107-018-1324-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-018-1324-y

Keywords

Mathematics Subject Classification

Search

Navigation