*Marc Mézard and Andrea Montanari*

- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780198570837
- eISBN:
- 9780191718755
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570837.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

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 ...
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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.Less

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.

*Hidetoshi Nishimori*

- Published in print:
- 2001
- Published Online:
- January 2010
- ISBN:
- 9780198509417
- eISBN:
- 9780191709081
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509417.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

A decision-making problem is often formulated as the minimization or maximization of a multivariable function, an optimization problem. This chapter shows that the methods of statistical mechanics ...
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A decision-making problem is often formulated as the minimization or maximization of a multivariable function, an optimization problem. This chapter shows that the methods of statistical mechanics are useful to study some types of optimization problems including the number partitioning, the graph partitioning, the knapsack problem, and the satisfiability problem. All these problems are shown to be formulated and solved using the theory of spin glasses, in particular the replica method. Then, discussions are continued on the mathematical properties of simulated annealing, an approximate numerical method for generic optimization problems.Less

A decision-making problem is often formulated as the minimization or maximization of a multivariable function, an optimization problem. This chapter shows that the methods of statistical mechanics are useful to study some types of optimization problems including the number partitioning, the graph partitioning, the knapsack problem, and the satisfiability problem. All these problems are shown to be formulated and solved using the theory of spin glasses, in particular the replica method. Then, discussions are continued on the mathematical properties of simulated annealing, an approximate numerical method for generic optimization problems.