A Note on Fourier-Motzkin Elimination with Three Eliminating Variables

TL;DR

This paper examines the complexity of Fourier-Motzkin elimination with three variables, highlighting its exponential growth in inequalities and discussing implications for information-theoretic security proofs.

Contribution

It demonstrates the difficulty of brute-force Fourier-Motzkin elimination in simple cases and emphasizes the need for more general proof strategies.

Findings

Inequalities grow doubly exponentially with the number of variables.

Most generated inequalities are redundant.

Direct elimination becomes unmanageable for more than a few variables.

Abstract

In this note, we show how difficult the brute-force Fourier-Motzkin elimination is, even in a simple case with three eliminating variables. Specifically, we first give a theorem, which plays quite an important role in the study of information-theoretic security for a multiple access wiretap (MAC-WT) channel, and then prove it for the case with three users by directly using the Fourier-Motzkin procedure. It is shown that large amounts of inequalities are generated in the elimination procedure while most of them are redundant. Actually, the number of generated inequalities grows doubly exponentially with the number of users or eliminating variables. It thus becomes unmanageable to directly prove the theorem in this brute-force way. Besides the great complexity, another disadvantage of the direct strategy is that it works only if the number of users is given. Obviously, this makes the strategy inappropriate for the proof of the theorem since it is a general result for any number of users. It is thus urgent and challenging to generally prove the theorem.