Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems

TL;DR

Polyhedral Omega is a novel, efficient algorithm that unifies partition analysis and polyhedral geometry to solve linear Diophantine systems, offering simplicity and competitive performance.

Contribution

It introduces a new algorithm combining partition analysis and polyhedral geometry, unifying two research areas with symbolic cones for solving LDS.

Findings

Polyhedral Omega outperforms previous partition analysis-based solvers.

It is competitive with state-of-the-art geometric LDS solvers.

The algorithm is the simplest available for solving linear Diophantine systems.

Abstract

Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega the simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.