The Easiest Hard Problem: Number Partitioning

TL;DR

This paper reviews the classical NP-hard number partitioning problem, detailing its phase transition between easy and hard instances, and discusses the implications for algorithms and applications like encryption and scheduling.

Contribution

It provides a simple derivation of the phase transition in random number partitioning and discusses its algorithmic implications, with detailed and rigorous results.

Findings

Identifies a clear easy-hard phase transition in number partitioning.

Provides a simple derivation of the phase transition.

Discusses algorithmic implications of the phase transition.

Abstract

Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an "easy-hard" phase transition similar to the phase transitions observed in other combinatorial problems like $k$-SAT. In contrast to most other problems, number partitioning is simple enough to obtain detailled and rigorous results on the "hard" and "easy" phase and the transition that separates them. We review the known results on random integer partitioning, give a very simple derivation of the phase transition and discuss the algorithmic implications of both phases.