Analysis of the Karmarkar-Karp Differencing Algorithm

TL;DR

This paper provides an in-depth analysis of the Karmarkar-Karp differencing algorithm for number partitioning, revealing finite size effects, asymptotic behavior, and connections to Fibonacci-like sequences through a nonlinear rate equation.

Contribution

It introduces a novel analysis method using a nonlinear rate equation to understand the algorithm's performance and asymptotics, highlighting subtle mathematical relations.

Findings

Finite size effects significantly influence algorithm performance.

Asymptotic scaling follows a $n^{-c\\ln n}$ pattern with $c=1/(2\ln2)$.

Explicit relations between the algorithm and Fibonacci-like sequences are established.

Abstract

The Karmarkar-Karp differencing algorithm is the best known polynomial time heuristic for the number partitioning problem, fundamental in both theoretical computer science and statistical physics. We analyze the performance of the differencing algorithm on random instances by mapping it to a nonlinear rate equation. Our analysis reveals strong finite size effects that explain why the precise asymptotics of the differencing solution is hard to establish by simulations. The asymptotic series emerging from the rate equation satisfies all known bounds on the Karmarkar-Karp algorithm and projects a scaling $n^{-c\ln n}$, where $c=1/(2\ln2)=0.7213...$. Our calculations reveal subtle relations between the algorithm and Fibonacci-like sequences, and we establish an explicit identity to that effect.