Solving tall dense linear programs in nearly linear time
Jan van den Brand, Yin Tat Lee, Aaron Sidford, Zhao Song
Abstract
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(√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.