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Many practical problems emphasize the importance of not only knowing whether an element is selected but also deciding to what extent it is selected, which imposes a challenge on submodule optimization. In this study, we consider the monotone, nondecreasing, and non-submodular maximization on the integer lattice with a cardinality constraint. We first design a two-pass streaming algorithm by refining the estimation interval of the optimal value. For each element, the algorithm not only decides whether to save the element but also gives the number of reservations. Then, we introduce the binary search as a subroutine to reduce the time complexity. Next, we obtain a one-pass streaming algorithm by dynamically updating the estimation interval of optimal value. Finally, we improve the memory complexity of this algorithm.


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Streaming Algorithms for Non-Submodular Maximization on the Integer Lattice

Show Author's information Jingjing Tan1Yue Sun2Yicheng Xu3Juan Zou4( )
School of Mathematics and Information Science, Weifang University, Weifang 261061, China
Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China
Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Abstract

Many practical problems emphasize the importance of not only knowing whether an element is selected but also deciding to what extent it is selected, which imposes a challenge on submodule optimization. In this study, we consider the monotone, nondecreasing, and non-submodular maximization on the integer lattice with a cardinality constraint. We first design a two-pass streaming algorithm by refining the estimation interval of the optimal value. For each element, the algorithm not only decides whether to save the element but also gives the number of reservations. Then, we introduce the binary search as a subroutine to reduce the time complexity. Next, we obtain a one-pass streaming algorithm by dynamically updating the estimation interval of optimal value. Finally, we improve the memory complexity of this algorithm.

Keywords: integer lattice, non-submodular, streaming algorithm, cardinality constraint

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Received: 04 April 2022
Revised: 18 June 2022
Accepted: 08 August 2022
Published: 19 May 2023
Issue date: October 2023

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© The author(s) 2023.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11871081), the Natural Science Foundation of Shandong Province (No. ZR2022MA034), the Guangxi Key Laboratory of Cryptography and Information Security (No. GCIS202116), the Fundamental Research Project of Shenzhen City (No. JCYJ20210324102012033), the National Natural Science Foundation of China (No. 11901558), and the National Natural Science Foundation of China (No. 11801310).

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