The differencing algorithm LDM for partitioning: a proof of a conjecture of Karmarkar and Karp
Mathematics of Operations Research
Exponentially small bounds on the expected optimum of the partition and subset sum problems
Random Structures & Algorithms
Phase transition and finite-size scaling for the integer partitioning problem
Random Structures & Algorithms - Special issue on analysis of algorithms dedicated to Don Knuth on the occasion of his (100)8th birthday
Proof of the local REM conjecture for number partitioning. I: Constant energy scales
Random Structures & Algorithms
Proof of the local REM conjecture for number partitioning. I: Constant energy scales
Random Structures & Algorithms
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We continue our analysis of the number partitioning problem with n weights chosen i.i.d. from some fixed probability distribution with density ρ. In Part I of this work, we established the so-called local REM conjecture of Bauke, Franz and Mertens. Namely, we showed that, as n → ∞, the suitably rescaled energy spectrum above some fixed scale α tends to a Poisson process with density one, and the partitions corresponding to these energies become asymptotically uncorrelated. In this part, we analyze the number partitioning problem for energy scales αn that grow with n, and show that the local REM conjecture holds as long as n-1-4αn → 0, and fails if αn grows like κn1-4 with κ 0. We also consider the SK-spin glass model, and show that it has an analogous threshold: the local REM conjecture holds for energies of order o(n), and fails if the energies grow like κn with κ 0. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009 This work was done during the period when C.N. was a graduate student at Stanford and later a post-doc at Microsoft Research.