Unbounded knapsack problem and double partitions

TL;DR

This paper explores the connection between the unbounded knapsack problem and double partition problems, offering a geometric perspective on classical variable elimination methods for solving these systems.

Contribution

It provides a geometric interpretation of Sylvester and Cayley's variable elimination approach, applying it to the unbounded knapsack problem.

Findings

Geometric interpretation of double partition solutions

Application of variable elimination to knapsack problem

Reduction of double partition to scalar partitions

Abstract

The unbounded knapsack problem can be considered as a particular case of the double partition problem that asks for a number of nonnegative integer solutions to a system of two linear Diophantine equations with integer coefficients. In the middle of 19th century Sylvester and Cayley suggested an approach based on the variable elimination allowing a reduction of a double partition to a sum of scalar partitions. This manuscript discusses a geometric interpretation of this method and its application to the knapsack problem.