We study differentially private approximation algorithms for positive linear programs (LPs with nonnegative coefficients and variables), focusing on the fundamental families of packing, covering, and mixed packing-covering formulations. We focus on the high-sensitivity, constraint-private regime of Hsu-Roth-Roughgarden-Ullman (ICALP 2014), where neighboring instances may differ by an arbitrary single constraint, so one cannot hope to approximately satisfy every constraint under privacy. We give private solvers that return approximate solutions while violating only a controlled number of constraints. Our algorithms improve the prior instance-dependent guarantees, and also yield new data-independent bounds that depend only on the dimension. Our techniques involve a dense multiplicative weights update method developed from a regularized dual viewpoint, which we analyze in a way that exploits structure specific to positive LPs.

翻译：我们研究针对正线性规划（系数和变量均为非负的线性规划）的差分隐私近似算法，重点关注覆盖、打包及混合打包-覆盖这三类基本公式。我们聚焦于Hsu-Roth-Roughgarden-Ullman（ICALP 2014）提出的高灵敏度、约束隐私场景，其中相邻实例可能仅相差任意单个约束，因此无法在隐私保护下近似满足每个约束。我们给出的隐私求解器能够在仅违反有限数量约束的前提下返回近似解。本文算法改进了先前的实例相关保障，并提出了仅依赖维度的全新数据无关界。我们的技术方法基于从正则化对偶视角发展出的密集乘性权重更新法，并通过挖掘正线性规划特有结构的方式进行分析。