From Elliott-MacMahon to an Algorithm for General Linear Constraints on Naturals

We describe a new algorithm for solving a conjunction of linear diophantine equations, inequations and disequations in natural numbers. We derive our algorithm from one proposed by Elliott in 1903 for solving a single homogeneous equation. This algorithm was then extended to solve homogeneous systems of equations by MacMahon. We show how it further extends to an algorithm which solves general linear constraints in nonnegative integers and allows a parallel implementation. This algorithm provides a parametric representation of the solutions from which minimal solutions may be extracted immediately. Moreover, it may be easily implemented in parallel. It has however one drawback: it is redundant which means that the same minimal solution is usually generated many times. We show how this redundancy may be eliminated at the cost of an increase in the space complexity.