Abstract
This paper studies the problem of maximizing a non-negative monotone k-submodular function. A k-submodular function is a generalization of a submodular function, where the input consists of k disjoint subsets, instead of a single subset. For the problem under a knapsack constraint, we consider the algorithm that returns the better solution between the single element of highest value and the result of the fully greedy algorithm, to which we refer as Greedy+Singleton, and prove an approximation ratio \(\frac{1}{4}(1-\frac{1}{e})\approx 0.158\). Though this ratio is strictly smaller than the best known factor for this problem, Greedy+Singleton is simple, fast, and of special interests. Our experiments demonstrates that the algorithm performs well in terms of the solution quality.
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References
Cohen, R., Katzir, L.: The generalized maximum coverage problem. Inf. Process. Lett. 108(1), 15–22 (2008)
Ene, A., Nguyen, H.L.: A nearly-linear time algorithm for submodular maximization with a knapsack constraint. In: Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP) (2019)
Feldman, M., Nutov, Z., Shoham, E.: Practical budgeted submodular maximization. arXiv preprint arXiv:2007.04937 (2020)
Gridchyn, I., Kolmogorov, V.: Potts model, parametric maxflow and \(k\)-submodular functions. In: Proceedings of the IEEE International Conference on Computer Vision (ICCV), pp. 2320–2327 (2013)
Hirai, H., Iwamasa, Y.: On \(k\)-submodular relaxation. SIAM J. Discret. Math. 30(3), 1726–1736 (2016)
Huang, C.C., Kakimura, N.: Improved streaming algorithms for maximizing monotone submodular functions under a knapsack constraint. Algorithmica 83(3), 879–902 (2021)
Huang, C.C., Kakimura, N., Yoshida, Y.: Streaming algorithms for maximizing monotone submodular functions under a Knapsack constraint. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM) (2017)
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.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 (2016)
Khuller, S., Moss, A., Naor, J.S.: The budgeted maximum coverage problem. Inf. Process. Lett. 70(1), 39–45 (1999)
Kulik, A., Schwartz, R., Shachnai, H.: A refined analysis of submodular greedy. Oper. Res. Lett. 49(4), 507–514 (2021)
Lin, H., Bilmes, J.: Multi-document summarization via budgeted maximization of submodular functions. In: Human Language Technologies: The 2010 Annual Conference of the North American Chapter of the Association for Computational Linguistics, pp. 912–920 (2010)
Nguyen, L., Thai, M.T.: Streaming \(k\)-submodular maximization under noise subject to size constraint. In: Proceedings of the 37th International Conference on Machine Learning (ICML), pp. 7338–7347. PMLR (2020)
Ohsaka, N., Yoshida, Y.: Monotone \(k\)-submodular function maximization with size constraints. In: Proceedings of the 28th International Conference on Neural Information Processing Systems (NeurIPS), vol. 1, pp. 694–702 (2015)
Oshima, H.: Improved randomized algorithm for \(k\)-submodular function maximization. SIAM J. Discret. Math. 35(1), 1–22 (2021)
Sakaue, S.: On maximizing a monotone \(k\)-submodular function subject to a matroid constraint. Discret. Optim. 23, 105–113 (2017)
Soma, T.: No-regret algorithms for online \( k \)-submodular maximization. In: Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), pp. 1205–1214. PMLR (2019)
Sviridenko, M.: A note on maximizing a submodular set function subject to a Knapsack constraint. Oper. Res. Lett. 32(1), 41–43 (2004)
Tang, J., Tang, X., Lim, A., Han, K., Li, C., Yuan, J.: Revisiting modified greedy algorithm for monotone submodular maximization with a Knapsack constraint. Proc. ACM Measure. Anal. Comput. Syst. 5(1), 1–22 (2021)
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 preprint arXiv:2107.07103 (2021)
Ward, J., Živnỳ, S.: Maximizing \(k\)-submodular functions and beyond. ACM Trans. Algorithms 12(4), 1–26 (2016)
Yaroslavtsev, G., Zhou, S., Avdiukhin, D.: “Bring your own greedy”+ max: near-optimal 1/2-approximations for submodular knapsack. In: International Conference on Artificial Intelligence and Statistics, pp. 3263–3274. PMLR (2020)
Acknowledgements
This work is partially supported by Artificial Intelligence and Data Science Research Hub, BNU-HKBU United International College (UIC), No. 2020KSYS007, and by a grant from UIC (No. UICR0400025-21). Zhongzheng Tang is supported by National Natural Science Foundation of China under Grant No. 12101069 and Innovation Foundation of BUPT for Youth (No. 500422309). Chenhao Wang is supported by a grant from UIC (No. UICR0700036-22).
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Chen, J., Tang, Z., Wang, C. (2022). Monotone k-Submodular Knapsack Maximization: An Analysis of the Greedy+Singleton Algorithm. In: Ni, Q., Wu, W. (eds) Algorithmic Aspects in Information and Management. AAIM 2022. Lecture Notes in Computer Science, vol 13513. Springer, Cham. https://doi.org/10.1007/978-3-031-16081-3_13
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