Differentially private submodular maximization with a cardinality constraint over the integer lattice

• Published in: Journal of combinatorial optimization 2024-05, Vol.47 (4), Article 58
• Main Authors: Hu, Jiaming, Xu, Dachuan, Du, Donglei, Miao, Cuixia
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Budgets; Combinatorial analysis; Combinatorics; Convex and Discrete Geometry; Integers; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Maximization; Operations Research/Decision Theory; Optimization; Privacy; Theory of Computation
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-024-01158-2

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 - ρ ) -approximation guarantee with additive error O ( r σ ln | N | / ϵ ) for any ρ > 0 , where N is the number of groundset, ϵ is the privacy budget, r is the cardinality constraint, and σ is the sensitivity of a function. Our algorithm preserves O ( ϵ r 2 ) -differential privacy. Meanwhile, for lattice submodular functions, we present a differential privacy algorithm that achieves a ( 1 - 1 / e - O ( ρ ) ) -approximation guarantee with additive error O ( r σ ln | N | / ϵ ) . We evaluate their effectiveness using instances of the combinatorial public projects problem and the budget allocation problem within the bipartite influence model.