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Maximizing DR-submodular+supermodular functions on the integer lattice subject to a cardinality constraint

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Abstract

Arising from practical problems such as in sensor placement and influence maximization in social network, submodular and non-submodular maximization on the integer lattice has attracted much attention recently. In this work, we consider the problem of maximizing the sum of a monotone non-negative diminishing return submodular (DR-submodular) function and a supermodular function on the integer lattice subject to a cardinality constraint. By exploiting the special combinatorial structures in the problem, we introduce a decreasing threshold greedy algorithm with a binary search as its subroutine to solve the problem. To avoid introducing the diminishing return ratio and submodularity ratio of the objective function, we generalize the total curvatures of submodular functions and supermodular functions to the integer lattice version. We show that our algorithm has a constant approximation ratio parameterized by the new introduced total curvatures on integer lattice with a polynomial query complexity.

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References

  1. Bai, W., Bilmes, J.: Greed is still good: Maximizing monotone submodular+supermodular (BP) functions. In: Proceedings of ICML, pp. 304–313 (2018)

  2. Bian, A. A., Buhmann, J. M., Krause, A., Tschiatschek, S.: Guarantees for greedy maximization of non-submodular functions with applications. In: Proceedings of ICML, pp. 498–507 (2017)

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

    Article  MathSciNet  Google Scholar 

  4. Conforti, M., Cornuéjols, G.: Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem. Discrete Appl. Math. 7, 251–274 (1984)

    Article  MathSciNet  Google Scholar 

  5. Das, A., Kempe, D.: Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection. In: Proceedings of ICML, 1057–1064 (2011)

  6. Ene, A., Nguyen, H. L.: A reduction for optimizing lattice submodular functions with diminishing returns. ArXiv:1606.08362v2 (2018)

  7. Goldengorin, B.: Maximization of submodular functions: Theory and enumeration algorithms. Eur. J. Oper. Res. 198(1), 102–112 (2009)

    Article  MathSciNet  Google Scholar 

  8. Kempe, D., Kleinberg, J., Tardos, E.: Maximizing the spread of influence through a social network. In: Proceedings of SIGKDD, pp. 137–146 (2003)

  9. Krause, A., Singh, A., Guestrin, C.: Nearoptimal sensor placements in gaussian processes: Theory, efficient algorithms and empirical studies. J. Mach. Learn. Res. 9, 235–284 (2008)

    MATH  Google Scholar 

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

  11. Nemhauser, G., Wolsey, L.: Best algorithms for approximating the maximum of a submodular set function. Math. Oper. Res. 3, 177–188 (1978)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Qian, C., Zhang, Y., Tang, K., Yao, X.: On multiset selection with size constraints. In: Proceedings of AAAI, 1395–1402 (2018)

  14. Soma, T., Kakimura, N., Inaba, K., Kawarabayashi K.: Optimal budget allocation: Theoretical guarantee and efficient algorithm. In: Proceedings of ICML, 351–359 (2014)

  15. Soma, T., Yoshida, Y.: A generalization of submodular cover via the diminishing return property on the integer lattice. In: Proceedings of NIPS, 847–855 (2015)

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

    Article  MathSciNet  Google Scholar 

  17. Sviridenko, M., Vondrák, J., Ward, J.: Optimal approximation for submodular and supermodular optimization with bounded curvature. Math. Oper. Res. 42, 1197–1218 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Vondrák, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: Proceedings of STOC, 67–74 (2008)

  20. Yoshida, Y.: Maximizing a monotone submodular function with a bounded curvature under a knapsack constraint. SIAM J. Discrete Math. 33, 1452–1471 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author is supported by National Natural Science Foundation of China (Nos. 12001025 and 11871081) and Science and Technology Program of Beijing Education Commission (No. KM201810005006). The second author is partially supported by the Natural Sciences and Engineering Research Council of Canada grant (No. 283106), and National Natural Science Foundation of China (Nos. 11771386 and 11728104). The third author is supported by National Natural Science Foundation of China (No. 11801251). The fourth author is supported by National Natural Science Foundation of China (No. 11971349).

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Authors and Affiliations

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

    Zhenning Zhang

  2. Faculty of Business Administration, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

    Donglei Du

  3. School of Mathematics and Statistics Science, Ludong University, 186 Hongqi Middle Road, Zhifu District, Yantai, Shandong, 264025, People’s Republic of China

    Yanjun Jiang

  4. College of Science, Tianjin University of Technology, NO. 391, Binshui West Street, Xiqing District, Tianjin, People’s Republic of China

    Chenchen Wu

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  1. Zhenning Zhang
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  2. Donglei Du
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  3. Yanjun Jiang
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  4. Chenchen Wu
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Correspondence to Yanjun Jiang.

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Zhang, Z., Du, D., Jiang, Y. et al. Maximizing DR-submodular+supermodular functions on the integer lattice subject to a cardinality constraint. J Glob Optim 80, 595–616 (2021). https://doi.org/10.1007/s10898-021-01014-1

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  • DOI: https://doi.org/10.1007/s10898-021-01014-1

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