An algebraic expression of the number partitioning problem

Abstract

In this paper we investigate the number partitioning problem, using the tropical semiring (max-plus algebra). We show that the problem is reduced to deciding whether one of integers is a solution of a tropical analogue of algebraic equations with coefficients composed of other integers. For n up to 6 we derive concretely and explicitly the equation and its solution set. The derivation requires only routine algebraic computations, so can be applied for n larger than 6. Our approach based on max-plus algebra reveals the mathematical structure of the problem and provides a new view point for the P versus NP problem, since the problem is well-known to be NP-complete.