Abstract
The large class of often important numerical problems, for which the computational effort grows exponentially with size, is called NP-complete.1 Simple examples are optimization problems, where the variables can take only discrete values. An important subclass is formed by the so-called combinatorial optimization problems. Here a cost function E has to be minimized, which depends on the order of a finite number of objects. The number of arrangements of N objects, and therefore the effort to find the minimum of E, grows exponentially with N.
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Notes
“NP” stands for “nondeterministic polynomial”, implying that no algorithm is known that would allow a solution on a deterministic computer with an effort that increases as a power of the size parameter. A proof for the nonexistence of such an algorithm is not known either (see e.g. [Ga79]).
Rapid cooling is commonly used to increase the strength of metals, such as steel.
The search for the “best” tour among 50 cities takes a few minutes on a modern PC; searching through all possible 49! combinations would take about 1053 years.
Our own attempts at solving the traveling-salesman problem with help of the Hopfield-Tank method also had little success. (Misquoting Arthur Miller, one may be tempted to speak of “the death of the traveling salesman”.)
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© 1995 Springer-Verlag Berlin Heidelberg
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Müller, B., Reinhardt, J., Strickland, M.T. (1995). Combinatorial Optimization. In: Neural Networks. Physics of Neural Networks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57760-4_11
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DOI: https://doi.org/10.1007/978-3-642-57760-4_11
Publisher Name: Springer, Berlin, Heidelberg
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