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International Symposium on Parallel Architectures, Algorithms and Programming

PAAP 2020: Parallel Architectures, Algorithms and Programming pp 81–92Cite as

Nonsubmodular Maximization with Knapsack Constraint via Multilinear Extension

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Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1362))

Abstract

For the problem of maximizing a monotone submodular function subject to knapsack constraint, there is a \((1-1/e-\epsilon )\)-approximation algorithm running a nearly-linear time. In this paper, we consider the case that the objective function is nonsubmodular. We propose an approximation algorithm with approximation ratio

$$ \kappa \left( 1+\frac{\epsilon }{\kappa ^{2}(1-\epsilon )} \right) ^{-1} \left( 1- e^{-\varOmega \left( \epsilon ^{2} / \lambda \right) }\right) \left( 1-e^{-\kappa ^{3}}-O(\epsilon ) \right) , $$

and complexity \(\tilde{O} (\frac{1}{1-\tau } n^2 (\log n)^{\frac{1}{\epsilon }+2}),\) where \(\kappa \) is the continuous submodularity ratio, \(\tau \) is the curvature and \(\lambda \) is the largest weight. The technology of our algorithm is using continuous greedy to get a fractional solution and then rounding it with the contention resolution scheme.

Supported by National Natural Science Foundation of China (No. 12001335) and Shandong Provincial Natural Science Foundation, China (Nos. ZR2020MA029, ZR2019PA004).

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References

  1. Badanidiyuru, A., Vondr\(\rm {\acute{a}}\)k, J.: Fast algorithms for maximizing submodular functions. In: Proceedings of SODA, pp. 1497–1514 (2013). https://doi.org/10.1137/1.9781611973402.110

  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). https://doi.org/10.5555/3305381.3305433

  3. Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a submodular set function subject to a matroid constraint (extended abstract). In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 182–196. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-72792-7_15

    Chapter  Google Scholar 

  4. Chekuri, C., Vondr\(\rm {\acute{a}}\)k, J., Zenkluse, R.: Submodular function maximization via the multilinear relaxation and contention resolution schemes. SIAM J. Comput. 43(6), 1831–1879 (2014)

    Google Scholar 

  5. Ene, A., Nguy\(\rm {\tilde{\hat{e}}}\)n, H.L.: A nearly-linear time algorithm for submodular maximization with a knapsack constraint. In: Proceedings of ICALP, pp. 53:1–53:12 (2019). https://doi.org/10.4230/LIPIcs.ICALP.2019.53

  6. Feldman, M.: Maximization problems with submodular objective functions. Ph.D. thesis, Computer Science Department, Technion (2013)

    Google Scholar 

  7. Fortuin, C.M., Kasteleyn, P.W.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22(2), 89–103 (1971)

    Article  MathSciNet  Google Scholar 

  8. Gong, S., Nong, Q., Liu, W., Fang, Q.: Parametric monotone function maximization with matroid constraints. J. Global Optim. 75(3), 833–849 (2019)

    Article  MathSciNet  Google Scholar 

  9. Kleywegt, A.J., Papastavrou, J.D.: The dynamic and stochastic knapsack problem. Oper. Res. 46(1), 17–35 (1998)

    Article  MathSciNet  Google Scholar 

  10. Mansini, R., Speranza, M.G.: A multidimensional knapsack model for asset-backed securitization. J. Oper. Res. Soc. 53(8), 822–832 (2002)

    Article  Google Scholar 

  11. Shamir, A.: A polynomial time algorithm for breaking the basic Merkle-Hellman cryptosystem. In: Proceedings of SFCS, pp. 145–152 (1982). https://doi.org/10.1109/SFCS.1982.5

  12. 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 

  13. Sviridenko, M., Vondr\(\rm {\acute{a}}\)k, J., Ward, J.: Optimal approximation for submodular and supermodular optimization with bounded curvature. Math. Oper. Res. 42(4), 1197–1218 (2017)

    Google Scholar 

  14. Yoshida, Y.: Maximizing a monotone submodular function with a bounded curvature under a knapsack constraint. SIAM Discrete Math. 33(3), 1452–1471 (2019)

    Article  MathSciNet  Google Scholar 

  15. Zhang, Z., Liu, B., Wang, Y., Xu, D., Zhang, D.: Greedy algorithm for maximization of non-submodular functions subject to knapsack constraint. In: Du, D.-Z., Duan, Z., Tian, C. (eds.) COCOON 2019. LNCS, vol. 11653, pp. 651–662. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26176-4_54

    Chapter  Google Scholar 

  16. Zhang, Z., Liu, B., Wang, Y., Xu, D., Zhang, D.: Greedy algorithm for maximization of non-submodular functions subject to knapsack constraint. In: Du, D.-Z., Duan, Z., Tian, C. (eds.) COCOON 2019. LNCS, vol. 11653, pp. 651–662. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26176-4_54

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, People’s Republic of China

    Jiachen Ju, Min Li, Jianxin Liu, Qian Liu & Yang Zhou

Authors
  1. Jiachen Ju
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  2. Min Li
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  3. Jianxin Liu
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  4. Qian Liu
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  5. Yang Zhou
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Corresponding author

Correspondence to Qian Liu .

Editor information

Editors and Affiliations

  1. Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China

    Li Ning

  2. Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China

    Vincent Chau

  3. The University of Hong Kong, Hong Kong, China

    Francis Lau

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Ju, J., Li, M., Liu, J., Liu, Q., Zhou, Y. (2021). Nonsubmodular Maximization with Knapsack Constraint via Multilinear Extension. In: Ning, L., Chau, V., Lau, F. (eds) Parallel Architectures, Algorithms and Programming. PAAP 2020. Communications in Computer and Information Science, vol 1362. Springer, Singapore. https://doi.org/10.1007/978-981-16-0010-4_8

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  • DOI: https://doi.org/10.1007/978-981-16-0010-4_8

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-16-0009-8

  • Online ISBN: 978-981-16-0010-4

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