SCIENCE 13 May 1983, Volume 220, Number 4598 with N, so that in practice exact solu- tions can be attempted only on problems involving a few hundred cities or less. The traveling salesman belongs to the large class of NP-complete (nondeter- Optimization by ministic polynomial time complete) problems, which has received extensive Simulated Annealing study in the past 10 years (3). N o method for exact solution with a computing ef- fort bounded by a power of N has been S. Kirkpatrick, C. D. Gelatt, Jr., M. P. Vecchi found for any of these problems, but if such a solution were found, it could be mapped into a procedure for solving all members of the class. It is not known what features of the individual problems In this article we briefly review the sure of the "goodness" of some complex system. The cost function depends on in the NP-complete class are the cause of central constructs in combinatorial opti- mization and in statistical mechanics and the detailed configuration of the many their difficulty. parts of that system. We are most famil- Since the NP-complete class of prob- then develop the similarities between the two fields. We show how the Metropolis iar with optimization problems occurring lems contains many situations of practi- algorithm for approximate numerical in the physical design of computers, so cal interest, heuristic methods have been simulation of the behavior of a many- examples used below are drawn from developed with computational require- body system at a finite temperature pro- vides a natural tool for bringing the tech- Summary. There is a deep and useful connection between statistical mechanics niques of statistical mechanics to bear on (the behavior of systems with many degrees of freedom in thermal equilibrium at a optirn~ization. finite temperature) and multivariate or combinatorial optimization (finding the mini- We have applied this point of view to a mum of a given function depending on many parameters). A detailed analogy with number of problems arising in optimal annealing in solids provides a framework for optimization of the properties of very design of computers. Applications to large and complex systems. This connection to statistical mechanics exposes new partitioning, component placement, and information and provides an unfamiliar perspective on traditional optimization prob- wiring of electronic systems are de- lems and methods. scribed in this article. In each context, we introduce the problem and discuss the improvements available from optimi- zation. that context. The number of variables ments proportional to small powers of N. Of classic optimization problems, the involved may range up into the tens of Heuristics are rather problem-specific: thousands. travel~~ng salesman problem has received there is no guarantee that a heuristic The classic example, because it is so the most intensive study. To test the procedure for finding near-optimal solu- simply stated, of a combinatorial optimi- tions for one NP-complete problem will power of simulated annealing, we used zation problem is the traveling salesman be effective for another. the algorithm on traveling salesman problem. Given a list of N cities and a There are two basic strategies for problems with as many as several thou- means of calculating the cost of traveling heuristics: "divide-and-conquer" and it- sand cities. This work is described in a final section, followed by our conclu- between any two cities, one must plan erative improvement. In the first, one sions. the salesman's route, which will pass divides the problem into subproblems of through each city once and return finally manageable size, then solves the sub- to the starting point, minimizing the total problems. The solutions to the subprob- Combinatorial Optimization cost. Problems with this flavor arise in lems must then be patched back togeth- all areas of scheduling and design. Two er. For this method to produce very good subsidiary problems are of general inter- solutions, the subproblems must be natu- The subject of combinatorial optimiza- tion (1) consists of a set of problems that est: predicting the expected cost of the rally disjoint, and the division made must are central to the disciplines of computer salesman's optimal route, averaged over be an appropriate one, so that errors science and engineering. Research in this some class of typical arrangements of made in patching do not offset the gains area aims at developing efficient tech- cities, and estimating or obtaining bounds for the computing effort neces- niques for finding minimum or maximum S. Kirkpatrick and C. D. Gelatt, Jr., are research sary to determine that route. staff members and M. P. Vecchi was a visiting values of a function of very many inde- scientist at IBM Thomas J . Watson Research Cen- All exact methods known for deter- pendent variables (2). This function, usu- ter, Yorktown Heights, New York 10598. M. P. Vecchi's present address is Instituto Venezolano de mining an optimal route require a com- ally called the cost function or objective lnvestigaciones Cientificas, Caracas 1010A. Vene- puting effort that increases exponentially function, represents a quantitative mea- znela. 13 MAY I983
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