Abstract
The exploration of submodular optimization problems on the integer lattice offers a more precise approach to handling the dynamic interactions among repetitive elements in practical applications. In today’s data-driven world, the importance of efficient and reliable privacy-preserving algorithms has become paramount for safeguarding sensitive information. In this paper, we delve into the DR-submodular and lattice submodular maximization problems subject to cardinality constraints on the integer lattice, respectively. For DR-submodular functions, we devise a differential privacy algorithm that attains a \((1-1/e-\rho )\)-approximation guarantee with additive error \(O(r\sigma \ln |N|/\epsilon )\) for any \(\rho >0\), where N is the number of groundset, \(\epsilon \) is the privacy budget, r is the cardinality constraint, and \(\sigma \) is the sensitivity of a function. Our algorithm preserves \(O(\epsilon r^{2})\)-differential privacy. Meanwhile, for lattice submodular functions, we present a differential privacy algorithm that achieves a \((1-1/e-O(\rho ))\)-approximation guarantee with additive error \(O(r\sigma \ln |N|/\epsilon )\). We evaluate their effectiveness using instances of the combinatorial public projects problem and the budget allocation problem within the bipartite influence model.
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Acknowledgements
We are grateful to the guest editor and the anonymous referees for their helpful comments on an earlier version of this paper. The first two authors are supported by National Natural Science Foundation of China (No. 12131003). The third author is supported by National Natural Sciences and Engineering Research Council of Canada (NSERC) grant 06446, and National Natural Science Foundation of China (Nos. 11771386, 11728104). The fourth author is supported by the Province Natural Science Foundation of Shandong (No. ZR2022MA019).
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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. 59–67, 2021.
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Hu, J., Xu, D., Du, D. et al. Differentially private submodular maximization with a cardinality constraint over the integer lattice. J Comb Optim 47, 58 (2024). https://doi.org/10.1007/s10878-024-01158-2
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DOI: https://doi.org/10.1007/s10878-024-01158-2