Statistical Mechanics Analysis of the Continuous Number Partitioning Problem

TL;DR

This paper applies statistical mechanics methods to analyze the linear programming relaxation of the NP-complete number partitioning problem, revealing that the difference in set sizes is not self-averaging.

Contribution

It provides an analytical study of the LP relaxation of number partitioning using statistical mechanics, a novel approach in this context.

Findings

Calculated the probability distribution of set size differences.

Showed the difference is not self-averaging.

Provided analytical insights into the LP relaxation of the problem.

Abstract

The number partitioning problem consists of partitioning a sequence of positive numbers ${a_1,a_2,..., a_N}$ into two disjoint sets, ${\cal A}$ and ${\cal B}$, such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. We use statistical mechanics tools to study analytically the Linear Programming relaxation of this NP-complete integer programming. In particular, we calculate the probability distribution of the difference between the cardinalities of ${\cal A}$ and ${\cal B}$ and show that this difference is not self-averaging.