On an optimal constraint aggregation method for integer programming and on an analytic expression of the number of integer points in a polytope

TL;DR

This paper introduces a polynomial-time method for optimally aggregating linear Diophantine equations and provides an explicit formula for counting solutions when the system can be condensed into a single equation.

Contribution

It presents a new aggregation framework for linear Diophantine equations and derives an explicit solution count formula for aggregated systems.

Findings

Aggregated system of minimum size can be constructed in polynomial time.

An explicit formula for the number of solutions in aggregated systems.

Improved understanding of solution enumeration for integer systems.

Abstract

In this paper we give a new aggregation framework for linear Diophantine equations. In particular, we prove that an aggregated system of minimum size can be built in polynomial time. We also derive an analytic formula that gives the number of solutions of the system when it is possible to aggregate the system into one equation.