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| Author(s): |
Fernando Fagundes Ferreira
Total Authors: 1
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| Document type: | Doctoral Thesis |
| Press: | São Carlos. |
| Institution: | Universidade de São Paulo (USP). Instituto de Física de São Carlos (IFSC/BT) |
| Defense date: | 2001-03-08 |
| Examining board members: |
Jose Fernando Fontanari;
Roland Koberle;
Domingos Humberto Urbano Marchetti;
Paulo Murilo Castro de Oliveira;
Roberto Nicolau Onody
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| Advisor: | Jose Fernando Fontanari |
| Abstract | |
In this thesis we present a statistical mechanics approach to a classical optimization problem called the number partitioning problem (NPP), which is stated as follows. Given a sequence of N positive real numbers {a1, a2, a3,....aN}, the number partitioning problem consists of partitioning them into two sets A and its complementary set Ac such that the absolute value of the difference of the sums of aj over the two sets is minimized. In each case in which the aj\'s are statistically independent random variables uniformly distributed in the unit interval, this NP-complete problem is equivalent to the problem of finding the ground state of an infinite range, random antiferromagnetic Ising model. Hence the probabilistic analysis of the NPP can be carried out within the framework of the standard statistical mechanics of disordered systems. In this vein we employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best-constrained and unconstrained partitions in the large N limit. Furthermore, we calculate analytically the fraction of metastable states, i.e. states that are stable against all single spin flips. We conclude the analysis of the so-called direct approach, in which the instances {ai} are fixed and the partitions are variable, with the analytical study of the linear programming relaxation of this NP-complete integer programming. In the second part of this thesis we propose and explore an inverse approach to the NPP, in which the optimal partitions are fixed and the instances are variable. Specifically, using the replica framework we study analytically the instance space of the number partitioning problem. We show that, regardless of the distribution of the instance entries, there is an upper bound αcN to the number of perfect random partitions (i.e. partitions for which that difference is zero). In particular, in the case where the two sets have the same cardinality (balanced partitions) we find αc =1/2. Moreover, in the case of unbalanced partitions, we show that perfect random partitions exist only if the difference between the cardinalities of the two sets scales like m N-1/2}. (AU) | |