Using Optimization to Solve Positive LPs Faster in Parallel

Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operations research, and optimization. Despite the consistent efforts on this problem, all known nearly-linear-time algorithms require $\tilde{O}(\varepsilon^{-4})$ iterations to converge to $1\pm \varepsilon$ approximate solutions. This $\varepsilon^{-4}$ dependence has not been improved since 1993, and limits the performance of parallel implementations for such algorithms. Moreover, previous algorithms and their analyses rely on update steps and convergence arguments that are combinatorial in nature and do not seem to arise naturally from an optimization viewpoint. In this paper, we leverage new insights from optimization theory to construct a novel algorithm that breaks the longstanding $\varepsilon^{-4}$ barrier. Our algorithm has a simple analysis and a clear motivation. Our work introduces a number of novel techniques, such as the combined application of gradient descent and mirror descent, and a truncated, smoothed version of the standard multiplicative weight update, which may be of independent interest.