CIENCE 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). No 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 In this article we briefly review the sure of the "goodness" of sorr ne complex what features of the individual problems central constructs in combinatorial opti- system. The cost function d lepends on in the NP-complete class are the cause of mization and in statistical mechanics and the detailed configuration of the many their difficulty. then develop the similarities between the parts of that system. We are r most famil- Since the NP-complete class of prob- two fields. We show how the Metropolis iar with optimization problemr s occurring lems contains many situations of practi- algorithm for approximate numerical in the physical design of con cal interest, heuristic methods have been nputers, so simulation of the behavior of a many- examples used below are di rawn 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 optimization. We have applied this point of view to a finite temperature) and multivariate or combinatorial optimization (finding the mini- 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 partitioning, component placement, and large and complex systems. This connection to statistical mechanics exposes new information and provides an unfamiliar perspective on traditional optimization prob- wiring of electronic systems are de- scribed in this article. In each context, lems and methods. 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: traveling salesman problem has received thousands. there is no guarantee that a heuristic the most intensive study. To test the The classic example, because it is so procedure for finding near-optimal solu- power of simulated annealing, we used simply stated, of a combinatorial optimi- tions for one NP-complete problem will the algorithm traveling salesman zation problem is the traveling salesman be effective for another. on problems with as many as several thou- problem. Given a list of N cities and a There are two basic strategies for means of calculating the cost of traveling sand cities. This work is described in a heuristics: "divide-and-conquer" and it- final section, followed by our conclu- erative improvement. In the first, one between any two cities, one must plan the salesman's route, which will pass divides the problem into subproblems of sions. 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 The subject of combinatorial optimiza- subsidiary problems are of general inter- solutions, the subproblems must be natu- tion (I) 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 niques for finding minimum or maximum bounds for the computing effort neces- S. Kirkpatrick and C. D. Gelatt, Jr., are research values of a function of very many inde- sary to determine that route. staff members and M. P. Vecchi was a visiting scientist at IBM Thomas J. Watson Research Cen- pendent variables (2). This function, usu- All exact methods known for deter- ter, Yorktown Heights, New York 10598. M. P. mining an optimal route require a com- Vecchi's present address is Instituto Venezolano de ally called the cost function or objective Investigaciones Cientificas, Caracas IOIOA, Vene- function, represents a quantitative mea- puting effort that increases exponentially zuela. 13 MAY 1983 671
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