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

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