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，重点关注包装、覆盖及混合包装-覆盖这三类基础问题。研究聚焦于Hsu-Roth-Roughgarden-Ullman（ICALP 2014）提出的高灵敏度约束隐私场景：相邻实例可能仅因单个任意约束的差异而不同，因此在隐私保护条件下无法期望近似满足每个约束。我们提出的隐私求解器在仅违反可控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证，同时给出了仅依赖维度的全新数据无关边界。我们的技术手段包括：基于正则化对偶视角开发的稠密乘性权重更新方法，通过利用正线性规划特有的结构特性对其展开分析。