Number partitioning as random energy model

TL;DR

This paper models the number partitioning problem as a random energy model, revealing that energies and configurations are uncorrelated, which impacts the understanding of its optimization landscape.

Contribution

It introduces a local random energy perspective for number partitioning, supported by simulations and heuristic reasoning, highlighting the lack of exploitable structure.

Findings

Energies behave like uncorrelated random variables

Adjacent configurations are uncorrelated in energy

No exploitable relation between configuration geometry and energy

Abstract

Number partitioning is a classical problem from combinatorial optimisation. In physical terms it corresponds to a long range anti-ferromagnetic Ising spin glass. It has been rigorously proven that the low lying energies of number partitioning behave like uncorrelated random variables. We claim that neighbouring energy levels are uncorrelated almost everywhere on the energy axis, and that energetically adjacent configurations are uncorrelated, too. Apparently there is no relation between geometry (configuration) and energy that could be exploited by an optimization algorithm. This ``local random energy'' picture of number partitioning is corroborated by numerical simulations and heuristic arguments.