{"title":"Differentially private submodular maximization with a cardinality constraint over the integer lattice","authors":"Jiaming Hu, Dachuan Xu, Donglei Du, Cuixia Miao","doi":"10.1007/s10878-024-01158-2","DOIUrl":null,"url":null,"abstract":"<p>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 <span>\\((1-1/e-\\rho )\\)</span>-approximation guarantee with additive error <span>\\(O(r\\sigma \\ln |N|/\\epsilon )\\)</span> for any <span>\\(\\rho >0\\)</span>, where <i>N</i> is the number of groundset, <span>\\(\\epsilon \\)</span> is the privacy budget, <i>r</i> is the cardinality constraint, and <span>\\(\\sigma \\)</span> is the sensitivity of a function. Our algorithm preserves <span>\\(O(\\epsilon r^{2})\\)</span>-differential privacy. Meanwhile, for lattice submodular functions, we present a differential privacy algorithm that achieves a <span>\\((1-1/e-O(\\rho ))\\)</span>-approximation guarantee with additive error <span>\\(O(r\\sigma \\ln |N|/\\epsilon )\\)</span>. We evaluate their effectiveness using instances of the combinatorial public projects problem and the budget allocation problem within the bipartite influence model.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-024-01158-2","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.