Abstract
In this paper, we consider a streaming model of maximizing monotone lattice submodular function with a cardinality constraint on the integer lattice. As (lattice) submodularity does not imply the diminishing return property on the integer lattice, we introduce the Sieve-Streaming algorithm combining with a modified binary search subroutine to solve the problem. We also show it is with an approximation ratio \(1/2-\epsilon \), a memory complexity \(O( \epsilon ^{-1} k\log k)\), and a query complexity \(O( \epsilon ^{-2}\log ^2 k )\) per element.
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References
Badanidiyuru, A., Mirzasoleiman, B., Karbasi, A., Krause, A.: Streaming submodular maximization: massive data summarization on the fly. In: Proceedings of SIGKDD, pp. 671–680 (2014)
Gottschalk, C., Peis, B.: Submodular function maximization on the bounded integer lattice. In: Proceedings WAOA, pp. 133–144 (2015)
Huang, C., Kakimura, N.: Improved streaming algorithms for maximizing monotone submodular functions under a knapsack constraint. In: Proceedings of WADS, pp. 438–451 (2019)
E. Kazemi, E., Mitrovic, M., Zadimoghaddam, M., Lattanzi, S., Karbasi, A.: Submodular streaming in all its glory: tight approximation, minimum memory and low adaptive complexity. In: Proceedings of ICML, pp. 3311–3320 (2019)
Kuhnle, A., Smith, J.D., Crawford, V.G., Thai, M.T.: Fast maximization of non-submodular, monotonic functions on the integer lattice. In: Proceedings of ICML, pp. 2791–2800 (2018)
Nong, Q., Fang, J., Gong, S., Du, D., Feng, Y., Qu, X.: A \(1/2\)-approximation algorithm for maximizing a non-monotone weak-submodular function on a bounded integer lattice. J. Comb. Optim. 39, 1208–1220 (2020)
Norouzi-Fard, A., Tarnawski, J., Mitrovic, S., Zandieh, A., Mousavifar, A., Svensson, O.: Beyond 1/2-approximation for submodular maximization on massive data streams. In: Proceedings of ICML, pp. 3829–3838 (2018)
Soma, T., Kakimura, N., Inaba, K., Kawarabayashi, K.: Optimal budget allocation: theoretical guarantee and efficient algorithm. In: Proceedings of ICML, pp. 351–359 (2014)
Soma, T., Yoshida, Y.: A generalization of submodular cover via the diminishing return property on the integer lattice. In: Proceedings of NIPS, pp. 847–855 (2015)
Soma, T., Yoshida, Y.: Maximizing monotone submodular functions over the integer lattice. Math. Program. 172, 539–563 (2018)
Wang, Y., Xu, D., Wang, Y., Zhang, D.: Non-submodular maximization on massive data streams. J. Global Optim. 76(4), 729–743 (2019). https://doi.org/10.1007/s10898-019-00840-8
Acknowledgements
The first author is supported by National Natural Science Foundation of China (No. 12001025) and Science and Technology Program of Beijing Education Commission (No. KM201810005006). The second author is supported by Natural Science Foundation of China (No. 61772005) and Natural Science Foundation of Fujian Province (No. 2017J01753). The third author is supported by Beijing Natural Science Foundation Project No. Z200002 and the National Natural Science Foundation of China (No. 11871081). The fourth author is supported by Natural Science Foundation of China (No. 11801310).
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Zhang, Z., Guo, L., Wang, L., Zou, J. (2021). A Streaming Model for Monotone Lattice Submodular Maximization with a Cardinality Constraint. In: Zhang, Y., Xu, Y., Tian, H. (eds) Parallel and Distributed Computing, Applications and Technologies. PDCAT 2020. Lecture Notes in Computer Science(), vol 12606. Springer, Cham. https://doi.org/10.1007/978-3-030-69244-5_32
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DOI: https://doi.org/10.1007/978-3-030-69244-5_32
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