Solving Tall Dense Linear Programs in Nearly Linear Time

In this paper we provide an \(O_~(nd+d_3)\) 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 \(O_~(\sqrt{d–√})\)-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.