Reduction from the partition problem: Dynamic lot sizing problem with polynomial complexity

TL;DR

This paper demonstrates a polynomial reduction from the partition problem to a dynamic lot sizing problem, showing that solving the latter can efficiently solve the former, revealing insights into the complexity of these problems.

Contribution

It introduces a polynomial reduction from the partition problem to the dynamic lot sizing problem and analyzes the complexity of solving the latter.

Findings

Partition problem can be solved in pseudo-polynomial time via dynamic programming.

Numerical experiments validate the reduction and solution approach.

Discussion on potential polynomial time solutions for the partition problem.

Abstract

In this note, we polynomially reduce an instance of the partition problem to a dynamic lot sizing problem, and show that solving the latter problem solves the former problem. By solving the dynamic program formulation of the dynamic lot sizing problem, we show that the instance of the partition problem can be solved with pseudo-polynomial time complexity. Numerical results on solving instances of the partition problem are also provided using an implementation of the algorithm that solves the dynamic program. We conclude by discussing polynomial time solvability of the partition problem through further observation on the dynamic program formulation of the dynamic lot sizing problem.