Using the Johnson-Lindenstrauss lemma in linear and integer programming

TL;DR

This paper explores the novel application of the Johnson-Lindenstrauss lemma to feasibility problems in linear and integer programming, extending its use beyond traditional Euclidean distance-based algorithms.

Contribution

It introduces the first approach to applying the Johnson-Lindenstrauss lemma in linear and integer programming feasibility problems.

Findings

Demonstrates potential for dimension reduction in optimization problems

Extends the application scope of the Johnson-Lindenstrauss lemma

Provides initial insights into feasibility problem solutions

Abstract

The Johnson-Lindenstrauss lemma allows dimension reduction on real vectors with low distortion on their pairwise Euclidean distances. This result is often used in algorithms such as $k$-means or $k$ nearest neighbours since they only use Euclidean distances, and has sometimes been used in optimization algorithms involving the minimization of Euclidean distances. In this paper we introduce a first attempt at using this lemma in the context of feasibility problems in linear and integer programming, which cannot be expressed only in function of Euclidean distances.