Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
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The property of being “NP-complete” and the “P &equil;? NP” question are recognized as important theoretical concepts with which a practicing computer scientist should be acquainted. This tutorial will briefly introduce a simple model in which the significance of “NP-completeness” is discussed and will illustrate what makes a problem NP-complete with several examples. The second part of the tutorial discusses some of the ways of dealing with NP-complete problems. The computational models used in this presentation are deliberately extremely simple. The very simplicity of the model allows it to easily capture an important property of NP-complete problems: finding a solution may be difficult, but if one can guess (or be given) a solution, the validity of this solution can easily be tested. This is the first part of the definition of NP-completeness and puts a (very high) upper limit on how difficult an NP-complete problem can be. The second part of the definition gives a lower limit with a requirement that a problem be as hard as a well-known class of difficult questions. The tutorial will illustrate how relative difficulty is established and will discuss the expected but unproved absolute difficulty of NP-complete problems. The second half of the tutorial will be devoted to examining the implications of NP-completeness. It will be seen that NP-completeness is not always as serious limitation as it might appear to be. For example, sometimes it is possible to compute an approximation in a reasonable amount of time, with a guarantee that the approximation is not too far from the true optimum value. Sometimes it is possible to do a probabilistic analysis which shows that an algorithm does well on the average. Thus the notion of NP-completeness can help people redirect their efforts effectively by giving them some idea of what are realistic goals.