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NP-Completeness

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Encyclopedia of Machine Learning

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Definition

A decision problem consists in identifying symbol strings, presented as inputs, that have some specified property. The output consists in a yes/no or 0/1 answer. A decision problem belongs to the class P if there exists an algorithm, that is, a deterministic procedure, for deciding any instance of the problem in a length of time bounded by a polynomial function of the length of the input.

A decision problem is in the class NP if it is possible for every yes-instance of the problem to verify in polynomial time, after having been supplied with a polynomial-length witness, that the instance is indeed of the desired property.

An example is the problem to answer the question for two given numbers n and m whether n has a divisor d strictly between m and n. This problem is in NP: if the answer is positive, then such a divisor d will be a witness, since it can be easily checked that d lies between the required bounds, and that n is indeed divisible by d. However, it is not known...

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Recommended Reading

  • Stephen Cook (1971). The complexity of theorem proving procedures. Proceedings of the third annual ACM symposium on theory of computing, 151–158.

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  • Leonid Levin (1973). Universal’nye pereborne zadachi. Problemy Peredachi Informatsii 9(3): 265–266.

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  • English translation, Universal Search Problems, in B. A. Trakhtenbrot (1984). A Survey of Russian Approaches to Perebor (Brute-Force Searches) Algorithms. Annals of the History of Computing 6(4): 384–400.

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Editor information

Editors and Affiliations

  1. School of Computer Science and Engineering, University of New South Wales, Sydney, Australia, 2052

    Claude Sammut

  2. Faculty of Information Technology, Clayton School of Information Technology, Monash University, P.O. Box 63, Victoria, Australia, 3800

    Geoffrey I. Webb

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© 2011 Springer Science+Business Media, LLC

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(2011). NP-Completeness. In: Sammut, C., Webb, G.I. (eds) Encyclopedia of Machine Learning. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30164-8_603

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