A New Method for Variable Elimination in Systems of Inequations

TL;DR

This paper introduces a faster variable elimination method for inequation systems using dual Diophantine problems and Hilbert bases, outperforming Fourier-Motzkin Elimination in efficiency and redundancy reduction.

Contribution

The paper presents a novel variable elimination technique leveraging dual Diophantine systems and Hilbert bases, improving speed and reducing redundancies over traditional methods.

Findings

The new method is significantly faster than FME.

It reduces redundant solutions compared to FME.

It can solve problems in a single step instead of multiple iterations.

Abstract

In this paper, we present a new method for variable elimination in systems of inequations which is much faster than the Fourier-Motzkin Elimination (FME) method. In our method, a linear Diophantine problem is introduced which is dual to our original problem. The new Diophantine system is then solved, and the final result is calculated by finding the dual inequations system. Our new method uses the algorithm Normaliz to find the Hilbert basis of the solution space of the given Diophantine problem. We introduce a problem in the interference channel with multiple nodes and solve it with our new method. Next, we generalize our method to all problems involving FME and in the end we compare our method with the previous method. We show that our method has many advantages in comparison to the previous method. It does not produce many of the redundant answers of the FME method. It also solves the whole problem in one step whereas the previous method uses a step by step approach in eliminating each auxiliary variable.