Solving Tall Dense Linear Programs in Nearly Linear Time

In this paper we provide an O~(nd+d3) time randomized algorithm for solving linear programs with d variables and n constraints with high probability. To obtain this result we provide a robust, primal-dual O~(d−−√)-iteration interior point method inspired by the methods of Lee and Sidford (2014, 2019) and show how to efficiently implement this method using new data-structures based on heavy-hitters, the Johnson-Lindenstrauss lemma, and inverse maintenance. Interestingly, we obtain this running time without using fast matrix multiplication and consequently, barring a major advance in linear system solving, our running time is near optimal for solving dense linear programs among algorithms that do not use fast matrix multiplication.