Eur. Phys. J. B 65, 131-140 (2008)
https://doi.org/10.1140/epjb/e2008-00320-9

Analysis of the Karmarkar-Karp differencing algorithm

¹ Department of Physics, Emory University, Atlanta GA 30322-2430, USA
² Institut für Theoretische Physik, Otto-von-Guericke Universität, PF 4120, 39016 Magdeburg, Germany
³ Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA

Corresponding author: ^a sboettc@emory.edu

Received: 27 February 2008
Revised: 7 May 2008
Published online: 9 August 2008

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 , where c = 1/(2 ln 2)=0.7213.... Our calculations reveal subtle relations between the algorithm and Fibonacci-like sequences, and we establish an explicit identity to that effect.