Abstract
In the problem of maximizing non-monotone k-submodular function f under individual size constraints, the goal is to maximize the value of k disjoint subsets with size upper bounds \(B_1,B_2,\ldots ,B_k\), respectively. This problem generalized both submodular maximization and k-submodular maximization problem with total size constraint. In this paper, we propose two results about this kind of problem. One is a \(\frac{1}{B_m+4}\)-approximation algorithm, where \(B_m=\max \{B_1,B_2,\ldots ,B_k\}\). The other is a bi-criteria algorithm with approximation ratio \(\frac{1}{4}\), where each subset is allowed to exceed the size constraint by up to \(B_m\), and in the worst case, only one subset will exceed \(B_m\).
Supported by Natural Science Foundation of Shandong Province of China (Nos. ZR2020MA029, ZR2021MA100) and National Science Foundation of China (No. 12001335).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Ene, A., Nguyen, H.: Streaming algorithm for monotone \(k\)-submodular maximization with cardinality constraints. In: Proceedings of ICML, pp. 5944–5967 (2022)
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
Iwata, S., Tanigawa, S., Yoshida, Y.: Improved approximation algorithms for \(k\)-submodular function maximization. In: Proceedings of SODA, pp. 404–413 (2016)
Krause, A., Singh, A., Guestrin, C.: Near-optimal sensor placements in Gaussian processes: theory, efficient algorithms and empirical studies. J. Mach. Learn. Res. 9(8), 235–284 (2008)
Nguyen, L., Thai, M.: Streaming \(k\)-submodular maximization under noise subject to size constraint. In: Proceedings of ICML, pp. 7338–7347 (2020)
Ohsaka, N., Yoshida, Y.: Monotone \(k\)-submodular function maximization with size constraints. In: Proceedings of NeurIPS, pp. 694–702 (2015)
Oshima, H.: Improved randomized algorithm for \(k\)-submodular function maximization. SIAM J. Discrete Math. 35(1), 1–22 (2021)
Pham, C.V., Vu, Q.C., Ha, D.K.T., Nguyen, T.T.: Streaming algorithms for budgeted k-submodular maximization problem. In: Mohaisen, D., Jin, R. (eds.) CSoNet 2021. LNCS, vol. 13116, pp. 27–38. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-91434-9_3
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). https://doi.org/10.1007/s10878-022-00858-x
Qian, C., Shi, J., Tang, K., Zhou, Z.: Constrained monotone \(k\)-submodular function maximization using multiobjective evolutionary algorithms with theoretical guarantee. IEEE Trans. Evol. Comput. 22, 595–608 (2018)
Rafiey, A., Yoshida, Y.: Fast and private submodular and \(k\)-submodular functions maximization with matroid constraints. In: Proceeding of ICML, pp. 7887–7897 (2020)
Sakaue, S.: On maximizing a monotone \(k\)-submodular function subject to a matroid constraint. Discrete Optim. 23, 105–113 (2017)
Shi, G., Gu, S., Wu, W.: \(k\)-submodular maximization with two kinds of constraints. Discrete Math. Algorithms Appl. 13(4), 2150036 (2021)
Sun, Y., Liu, Y., Li, M.: Maximization of \(k\)-submodular function with a matroid constraint. In: Du, D.Z., Du, D., Wu, C., Xu, D. (eds.) TAMC 2022. LNCS, vol. 13571, pp. 1–10. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-20350-3_1
Tang, Z., Wang, C., Chan, H.: Monotone \(k\)-submodular secretary problems: cardinality and knapsack constraints. Theor. Comput. Sci. 921, 86–99 (2022)
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)
Wang, B., Zhou, H.: Multilinear extension of \(k\)-submodular functions. arXiv:2107.07103 (2021)
Ward, J., Živný, S.: Maximizing \(k\)-submodular functions and beyond. ACM Trans. Algorithms 12(4), 1–26 (2016). Article 47
Zheng, L., Chan, H., Loukides, G., Li, M.: Maximizing approximately \(k\)-submodular functions. In: Proceeding of SDM, pp. 414–422 (2021)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Xiao, H., Liu, Q., Zhou, Y., Li, M. (2023). Non-monotone k-Submodular Function Maximization with Individual Size Constraints. In: Dinh, T.N., Li, M. (eds) Computational Data and Social Networks . CSoNet 2022. Lecture Notes in Computer Science, vol 13831. Springer, Cham. https://doi.org/10.1007/978-3-031-26303-3_24
Download citation
DOI: https://doi.org/10.1007/978-3-031-26303-3_24
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-26302-6
Online ISBN: 978-3-031-26303-3
eBook Packages: Computer ScienceComputer Science (R0)