Solving Positive Linear Programs with Differential Privacy

2026-04-29

AI Summary

This work addresses the challenge of solving non-negative linear programs—including packing, covering, and mixed variants—under differential privacy constraints, particularly in high-sensitivity regimes with stringent privacy requirements where satisfying all constraints is infeasible. The authors propose a dense multiplicative weights update algorithm grounded in a regularized dual perspective, integrating differential privacy mechanisms with structural properties of the problem for refined dual analysis. Their key contributions include the first data-independent error bound that depends solely on the problem dimension, alongside an improved instance-dependent error bound. The method yields high-quality approximate solutions under strong privacy guarantees, violating only a small and controllable number of constraints, and significantly outperforms existing approaches in balancing approximation accuracy and constraint violation control.

Abstract

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.

Problem

• differential privacy
• positive linear programs
• packing
• covering
• constraint violation

Innovation

• differential privacy
• positive linear programs
• multiplicative weights update
• constraint violation
• data-independent bounds