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 Omegacombines 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 Omegais 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 Omegathe 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.
Highlights
In this article we present Polyhedral Omega, a new algorithm for computing a multivariate rational function expression for the set of all solutions to a linear Diophantine system
If Barvinok decompositions are used for the conversion of symbolic cones into rational functions, Polyhedral Omega runs in polynomial time in fixed dimension, i.e., it lies in the same complexity class as the fastest-known algorithms for the Rational Function Solution of Linear Diophantine System (rfsLDS) problem
Where each symbolic cone combination i αi f (Ci) is given in terms of a triple (V, o, q) where V is an integer matrix of generators, o is a vector indicating which faces of the cone are open and q is a rational vector giving the apex of the cone
Summary
In this article we present Polyhedral Omega, a new algorithm for computing a multivariate rational function expression for the set of all solutions to a linear Diophantine system. This algorithm connects two branches of research — partition analysis and polyhedral geometry — between which there has been little interaction in the past. We use this introduction to define the problem of computing rational function solutions to linear Diophantine systems and to give an overview of the algorithms developed in partition analysis and polyhedral geometry to solve it, before pointing out the benefits of Polyhedral Omega
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