Approximation Algorithms for k-Submodular Maximization Subject to a Knapsack Constraint

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  • School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, Shandong, China

Received date: 2023-10-10

  Revised date: 2024-01-03

  Online published: 2026-07-06

Supported by

This research was supported by the Natural Science Foundation of Shandong Province of China (Nos. ZR2020MA029 and ZR2021MA100) and the National Natural Science Foundation of China (No. 12001335).

Abstract

In this paper, we study the problem of maximizing $k$-submodular functions subject to a knapsack constraint. For monotone objective functions, we present a $\frac{1}{2}\left(1-\mathrm{e}^{-2}\right) \approx$ 0.432 greedy approximation algorithm, improving the previous best-known ratio $\frac{1}{2}(1- \left.\mathrm{e}^{-1}\right) \approx 0.316$. We also consider the non-monotone knapsack problem and provide two algorithms. The first is a greedy-type combinatorial algorithm with approximation ratio $\frac{1}{3}\left(1-\mathrm{e}^{-3}\right) \approx 0.317$, while the second is a multilinear-extension-based algorithm with approximation ratio $\frac{1}{3}-\varepsilon$, where $\varepsilon>0$.

Cite this article

Hao Xiao, Qian Liu, Yang Zhou, Min Li . Approximation Algorithms for k-Submodular Maximization Subject to a Knapsack Constraint[J]. Journal of the Operations Research Society of China, 2026 , 14(2) : 397 -412 . DOI: 10.1007/s40305-024-00539-y

References

[1] Ene, A., Nguyen, H.: Streaming algorithm for monotone k-submodular maximization with cardinality constraints. In: Chaudhuri, K., Jegelka, S., Song, L., Szepesvari, C., Niu, G., & Sabato, S. (Eds.), Proceedings of the 39th International Conference on Machine Learning, pp. 5944–5967(2022)
[2] Huber, A., Kolmogorov, V.: Towards minimizing k-submodular functions. In: Mahjoub, A.R., Markakis, V., Milis, I. & Paschos V.T. (Eds.), Combinatorial optimization, Lecture notes in computer science, pp. 451–462. Springer, Heidelberg (2012)
[3] Iwata, S., Tanigawa, S., Yoshida, Y.: Improved approximation algorithms for k-submodular function maximization. In: Kraughgamer, R. (Eds.), Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 404–413. Society for Industrial and Applied Mathematics (2016)
[4] Kulik, A., Shachnai, H., Tamir, T.: Approximations for monotone and non-monotone submodular maximization with knapsack constraints. Math. Op. Res. 38(4), 729–739(2013)
[5] Li, J., Liu, Y.: Approximation algorithms for stochastic combinatorial optimization problems. J. Op. Res. Soc. China 4, 1–47(2016)
[6] Liu, Q., Yu, K., Li, M., Zhou, Y.: k-submodular maximization with a knapsack constraint and p matroid constraints. Tsinghua Sci. Technol. 28(5), 896–905(2023)
[7] Matsuoka,T.,Ohsaka,N.:Maximizationofmonotonek-submodularfunctionswithboundedcurvature and non-k-submodular functions. In: V. N. Balasubramanian, & I. Tsang (Eds.), Proceedings of the 13th Asian Conference on Machine Learning, pp. 1707–1722(2021)
[8] Nguyen, L., Thai, M.: Streaming k-submodular maximization under noise subject to size constraint. In: H. Daumé, & A. Singh (Eds.), Proceedings of the 37th International Conference on Machine Learning, pp. 7338–7347(2020)
[9] Ohsaka, N., Yoshida, Y.: Monotone k-submodular function maximization with size constraints. In: C. Cortes, N. Lawrence, D. Lee and M. Sugiyama, & R. Garnett (Eds.), Proceedings of the 29th Conference on Neural Information Processing Systems, pp. 694–702. Curran Associates, Inc. (2015)
[10] Oshima, H.: Improved randomized algorithm for k-submodular function maximization. SIAM J. Discr. Math. 35(1), 1–22(2021)
[11] Pham, C., Vu, Q., Ha, D., Nguyen, T., Le, N.: Maximizing k-submodular functions under budget constraint: applications and streaming algorithms. J. Comb. Optim. 44, 723–751(2022)
[12] Quoc, H., Tam, N., Nhan, T.: The continuous knapsack problem with capacities. J. Op. Res. Soc. China 9, 713–721(2021)
[13] Rafiey, A., Yoshida, Y.: Fast and private submodular and k-submodular functions maximization with matroid constraints. In: H. Daumé, & A. Singh (Eds), Proceedings of the 37th International Conference on Machine Learning, pp. 7887–7897(2020)
[14] Sakaue, S.: On maximizing a monotone k-submodular function subject to a matroid constraint. Discr. Optim. 23, 105–113(2017)
[15] Shi, G., Gu, S., Wu, W.: k-submodular maximization with two kinds of constraints. Discr. Math. Algorithms Appl. 13(4), 2150036(2021)
[16] Sun, Y., Liu, Y., Li, M.: Maximization of k-submodular function with a matroid constraint. In: Du, D., Du, D., Wu, C.,& Xu, D. (Eds.), Proceedings of the 17th Annual Conference on Theory and Applications of Models of Computation, pp. 1–10. Springer, Cham (2022)
[17] Sviridenko, M.: A note on maximizing a submodular set function subject to a knapsack constraint. Op. Res. Lett. 32(1), 41–43(2004)
[18] Tang, Z., Wang, C., Chan, H.: On maximizing a monotone k-submodular function under a knapsack constraint. Op. Res. Lett. 50(1), 28–31(2022). https://doi.org/10.1016/j.orl.2021.11.010
[19] Tang,Z.,Wang,C.:Improvedanalysisofgreedyalgorithmonk-submodularknapsack.In:Proceedings of the 26th European Conference on Artificial Intelligence, pp.2338–2345(2023)
[20] Wang, B., Zhou, H.: Multilinear extension of k-submodular functions (2021). https://doi.org/10. 48550/arXiv.2107.07103
[21] Ward, J., Živný, S.: Maximizing k-submodular functions and beyond. ACM Trans. Algorithms 47, 1–26(2016)
[22] Wolsey, L.: Maximising real-valued submodular functions: primal and dual heuristics for location problems. Math. Op. Res. 7(3), 410–425(1982)
[23] Xiao, H., Liu, Q., Zhou, Y., Li, M: Non-monotone k-submodular function maximization with individual size constraints. In: T. N. Dinh, & M. Li (Eds.), Proceedings of the 11th International Conference on Computational Data and Social Networks, pp. 268–279. Springer, Cham (2022)
[24] 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, 93(2023)
[25] Zheng, L., Chan, H., Loukides, G., Li, M.: Maximizing approximately k-submodular functions. In: C. Demeniconi, I. Davidson, L. Akoglu, & E. Terzi (Eds.), Proceedings of the 2021 SIAM International Conference on Data Mining, pp.414–422. Society for Industrial and Applied Mathematics (2021)
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