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Information, Physics, and Computation$
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Marc Mézard and Andrea Montanari

Print publication date: 2009

Print ISBN-13: 9780198570837

Published to Oxford Scholarship Online: September 2009

DOI: 10.1093/acprof:oso/9780198570837.001.0001

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Number partitioning

Number partitioning

(p.131) 7 Number partitioning
Information, Physics, and Computation

Marc Mézard

Andrea Montanari

Oxford University Press

Number partitioning is one of the most basic optimization problems. It is very easy to state: ‘Given the values of N assets, is there a fair partition of them into two sets?’ Nevertheless, it is very difficult to solve: it belongs to the NP-complete category, and the known heuristics are often not very good. It is also a problem with practical applications, for instance in multiprocessor scheduling. This chapter focuses on a particularly difficult case: the partitioning of a list of independent uniformly distributed random numbers. It discusses the phase transition occurring when the range of numbers varies, and shows that low cost configurations — the ones with a small unbalance between the two sets — can be seen as independent energy levels. Hence the model behaves analogously to the Random Energy Model.

Keywords:   number partitioning, phase transition, random numbers, Random Energy Model, NP-complete category, low cost configurations

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