Instance Space of the Number Partitioning Problem

TL;DR

This paper analytically explores the instance space of the number partitioning problem using the replica framework, revealing bounds on perfect partitions and linking the problem to models of replicators with Hebb rule interactions.

Contribution

It introduces an analytical study of the number partitioning problem's instance space, identifying bounds on perfect partitions and connecting it to models of replicators.

Findings

Upper bound $rac{1}{2}N$ on perfect partitions

Characterization of instance properties with perfect solutions

Connection to replicator models with Hebb interactions

Abstract

Within the replica framework we study analytically the instance space of the number partitioning problem. This classic integer programming problem consists of partitioning a sequence of N positive real numbers $\{a_1, a_2,..., a_N}$ (the instance) into two sets such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. We show that there is an upper bound $\alpha_c N$ to the number of perfect partitions (i.e. partitions for which that difference is zero) and characterize the statistical properties of the instances for which those partitions exist. In particular, in the case that the two sets have the same cardinality (balanced partitions) we find $\alpha_c=1/2$. Moreover, we show that the disordered model resulting from hte instance space approach can be viewed as a model of replicators where the random interactions are given by the Hebb rule.