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Two approximation algorithms for maximizing nonnegative weakly monotonic set functions

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Abstract

Many combinatorial optimization problems can be reduced to submodular optimization problems. However, many cases in practical applications do not fully comply with the diminishing returns characteristic. This paper studies the problem of maximizing weakly-monotone non-submodular non-negative set functions without constraints, and extends the normal submodular ratio to weakly-monotone submodular ratio \(\widehat{\gamma }\). In this paper, two algorithms are given for the above problems. The first one is a deterministic greedy approximation algorithm, which realizes the approximation ratio of \(\frac{\widehat{\gamma }}{\widehat{\gamma }+2}\). When \(\widehat{\gamma }=1\), the approximate ratio is 1/3, which matches the ratio of the best deterministic algorithm known for the maximization of submodular function without constraints. The second algorithm is a random greedy algorithm, which improves the approximation ratio to \(\frac{\widehat{\gamma }}{\widehat{\gamma }+1}\). When \(\widehat{\gamma }=1\), the approximation ratio is 1/2, the same as the best algorithm for the maximization of unconstrained submodular set functions.

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Acknowledgements

The first and third authors are supported by National Natural Science Foundation of China (No. 12131003). The second author is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant 06446, and Natural Science Foundation of China (Nos. 11771386, 11728104). The fourth author is supported by National Natural Foundation of China (No. 12101587), China Postdoctoral Science Foundation (No. 2022M720329).

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Authors and Affiliations

  1. Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Min Cui, Dachuan Xu & Ruiqi Yang

  2. Faculty of Management, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

    Donglei Du

  3. School of Mathematical Sciences, University of Chinese Academy Sciences, Beijing, 100049, People’s Republic of China

    Ruiqi Yang

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  1. Min Cui
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  2. Donglei Du
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  3. Dachuan Xu
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  4. Ruiqi Yang
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Correspondence to Ruiqi Yang.

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A preliminary version of this paper appeared in Proceedings of the 10th International Conference on Computational Data and Social Networks, pp. 50–58, 2021.

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Cui, M., Du, D., Xu, D. et al. Two approximation algorithms for maximizing nonnegative weakly monotonic set functions. J Comb Optim 45, 54 (2023). https://doi.org/10.1007/s10878-022-00978-4

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