Analysis of the Karmarkar-Karp differencing algorithm |
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Authors: | S Boettcher S Mertens |
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Institution: | 1.Department of Physics,Emory University,Atlanta,USA;2.Institut für Theoretische Physik,Otto-von-Guericke Universit?t,Magdeburg,Germany;3.Santa Fe Institute,Santa Fe,USA |
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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 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. |
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Keywords: | PACS" target="_blank">PACS 02 60 Pn Numerical optimization 89 75 Da Systems obeying scaling laws 89 75 Fb Structures and organization in complex systems |
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