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arXiv · 2002.02304

Solving Tall Dense Linear Programs in Nearly Linear Time

Jan van den Brand·Yin Tat Lee·Aaron Sidford·Zhao Song
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Abstract

In this paper we provide an $\tilde{O}(nd+d^{3})$ 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 $\tilde{O}(\sqrt{d})$-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.

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Jan van den Brand, Yin Tat Lee, Aaron Sidford, Zhao Song. 2020-02-06. Solving Tall Dense Linear Programs in Nearly Linear Time. https://arxiv.org/abs/2002.02304

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