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Algorithms for Cardinality-Constrained Monotone DR-Submodular Maximization with Low Adaptivity and Query Complexity

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Abstract

Submodular maximization is a NP-hard combinatorial optimization problem regularly used in machine learning and data mining with large-scale data sets. To quantify the running time of approximation algorithms, the query complexity and adaptive complexity are two important measures, where the adaptivity of an algorithm is the number of sequential rounds it makes when each round can execute polynomially many function evaluations in parallel. These two concepts reasonably quantify the efficiency and practicability of parallel computation. In this paper, we consider the problem of maximizing a nonnegative monotone DR-submodular function over a bounded integer lattice with a cardinality constraint in a value oracle model. Prior to our work, Soma and Yoshida (Math Program 172:539–563, 2018) have studied this problem and present an approximation algorithm with almost optimal approximate ratio and the same adaptivity and query complexity. We develop two novel algorithms, called low query algorithm and low adaptivity algorithm, which have approximation ratios equal to Soma and Yoshida (2018), but reach a new record of query complexity and adaptivity. As for methods, compared with the threshold greedy in Soma and Yoshida (2018), the low query algorithm reduces the number of possible thresholds and improves both the adaptivity and query complexity. The low adaptivity algorithm further integrates a vector sequencing technique to accelerate adaptive complexity in an exponential manner while only making logarithmic sacrifices on oracle queries.

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Acknowledgements

This research was supported in part by the National Natural Science Foundation of China under Grant Nos. 12171444 and 11871442 and was also supported in part by the Natural Science Foundation of Shandong Province under Grant No. ZR2019MA052 and the Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, China

    Suning Gong

  2. School of Mathematical Science, Ocean University of China, Qingdao, 266100, China

    Qingqin Nong & Jiazhu Fang

  3. Department of Computer Science, University of Texas, Dallas, 75083, USA

    Ding-Zhu Du

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  1. Suning Gong
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  2. Qingqin Nong
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Correspondence to Qingqin Nong.

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Communicated by Antoine Deza.

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Gong, S., Nong, Q., Fang, J. et al. Algorithms for Cardinality-Constrained Monotone DR-Submodular Maximization with Low Adaptivity and Query Complexity. J Optim Theory Appl 200, 194–214 (2024). https://doi.org/10.1007/s10957-023-02353-7

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