93.204.196.246 at 13:55, 2 September 2014

2014-09-02T13:55:59Z

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← Older revision		Revision as of 00:01, 5 January 2014
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	My name is ~~Adelaide Griffiths~~. ~~I life~~ in ~~Lublin~~ (~~Poland~~).<br><br>~~Have~~ a ~~look at my web site~~: [http://~~safedietplans~~.~~com diets~~]		In [[computational complexity theory]], a '''polynomial-time reduction''' is a method of solving one problem by means of a hypothetical [[subroutine]] for solving a different problem (that is, a [[Reduction (complexity)\|reduction]]), that uses [[polynomial time]] excluding the time within the subroutine. There are several different types of polynomial-time reduction, depending on the details of how the subroutine is used. Intuitively, a polynomial-time reduction proves that the first problem is no more difficult than the second one, because whenever an efficient [[algorithm]] exists for second problem, one exists for the first problem as well. Polynomial-time reductions are frequently used in complexity theory for defining both [[complexity class]]es and [[complete problem]]s for those classes.

			==Types of reduction==
			The three most common types of polynomial-time reduction, from the most to the least restrictive, are polynomial-time many-one reductions, truth-table reductions, and Turing reductions.
			*A polynomial-time [[many-one reduction]] from a problem ''A'' to a problem ''B'' (both of which are usually required to be [[decision problem]]s) is a polynomial-time algorithm for transforming inputs to problem ''A'' into inputs to problem ''B'', such that the transformed problem has the same output as the original problem. An instance ''x'' of problem ''A'' can be solved by applying this transformation to produce an instance ''y'' of problem ''B'', giving ''y'' as the input to an algorithm for problem ''B'', and returning its output. Polynomial-time many-one reductions may also be known as '''polynomial transformations''' or '''Karp reductions''', named after [[Richard Karp]]. A reduction of this type may be denoted by the expression <math>A \le_m^P B</math>.<ref name="goldreich">{{citation\|title=Computational Complexity: A Conceptual Perspective\|first=Oded\|last=Goldreich\|authorlink=Oded Goldreich\|publisher=Cambridge University Press\|year=2008\|isbn=9781139472746\|pages=59–60}}.</ref>
			*A polynomial-time [[truth-table reduction]] from a problem ''A'' to a problem ''B'' (both decision problems) is a polynomial time algorithm for transforming inputs to problem ''A'' into a fixed number of collection of inputs to problem ''B'', such that the output for the original problem can be expressed as a function of the outputs to the transformed problems. The function that maps outputs for ''B'' into the output for ''A'' must be the same for all inputs, so that it can be expressed by a [[truth table]]. A reduction of this type may be denoted by the expression <math>A \le_{tt}^P B</math>.<ref>{{citation
			\| last1 = Buss \| first1 = S.R. \| author1-link = Samuel Buss
			\| last2 = Hay \| first2 = L.
			\| contribution = On truth-table reducibility to SAT and the difference hierarchy over NP
			\| doi = 10.1109/SCT.1988.5282
			\| pages = 224–233
			\| title = Proceedings of Third Annual Structure in Complexity Theory Conference
			\| year = 1988}}.</ref>
			*A polynomial-time [[Turing reduction]] from a problem ''A'' to a problem ''B'' is an [[algorithm]] that solves problem ''A'' using a polynomial number of calls to a subroutine for problem ''B'', and polynomial time outside of those subroutine calls. Polynomial-time Turing reductions are also known as '''Cook reductions''', named after [[Stephen Cook]]. A reduction of this type may be denoted by the expression <math>A \le_T^P B</math>.<ref name="goldreich"/>
			The most frequently used of these are the many-one reductions, and in some cases the phrase "polynomial-time reduction" may be used to mean a polynomial-time many-one reduction.<ref>{{citation\|title=Complexity Theory: Exploring the Limits of Efficient Algorithms\|first=Ingo\|last=Wegener\|authorlink=Ingo Wegener\|page=60\|publisher=Springer\|year=2005\|isbn=9783540274773}}.</ref>

			==Completeness==
			A [[complete problem]] for a given complexity class '''C''' and reduction ≤ is a problem ''C'' that belongs to '''C''', such that every problem ''A'' in '''C''' has a reduction ''A'' ≤ ''C''.
			For instance, a problem is [[NP-complete\|'''NP'''-complete]] if it belongs to [[NP (complexity)\|'''NP''']] and all problems in '''NP''' have polynomial-time many-one reductions to it. A problem that belongs to '''NP''' can be proven to be '''NP'''-complete by finding a single polynomial-time many-one reduction to it, from a known '''NP'''-complete problem.<ref>{{citation\|title=Computers and Intractability: A Guide to the Theory of NP-Completeness\|first1=Michael R.\|last1=Garey\|author1-link=Michael Garey\|first2=D. S.\|last2=Johnson\|author2-link=David S. Johnson\|publisher=W. H. Freeman\|year=1979}}.</ref> Polynomial-time many-one reductions have been used to define complete problems for other complexity classes, including the [[PSPACE-complete\|'''PSPACE'''-complete]] languages and [[EXPTIME\|'''EXPTIME'''-complete]] languages.<ref>{{citation\|contribution=Complexity theory\|pages=241–267\|first=A. V.\|last=Aho\|authorlink=Alfred Aho\|title=Computer Science: The Hardware, Software and Heart of It\|doi=10.1007/978-1-4614-1168-0_12\|year=2011\|editor1-first=E. K.\|editor1-last=Blum\|editor2-first=A. V.\|editor2-last=Aho}}. See in particular p. 255.</ref>

			Every nontrivial decision problem in [[P (complexity)\|'''P''']] (the class of polynomial-time decision problems, where nontrivial means that not every input has the same output) may be reduced to every other nontrivial decision problem, by a polynomial-time many-one reduction. To transform an instance of problem ''A'' to ''B'', solve ''A'' in polynomial time, and then use the solution to choose one of two instances of problem ''B'' with different answers.
			Therefore, for complexity classes within '''P''' such as [[L (complexity)\|'''L''']], [[NL (complexity)\|'''NL''']], [[NC (complexity)\|'''NC''']], and '''P''' itself,
			polynomial-time reductions cannot be used to define complete languages: if they were used in this way, every nontrivial problem in '''P''' would be complete. Instead, weaker reductions such as [[log-space reduction]]s or [[NC (complexity)\|'''NC''']] reductions are used for defining classes of complete problems for these classes, such as the [[P-complete\|'''P'''-complete]] problems.<ref>{{citation\|last1=Greenlaw\|first1=Raymond\|first2=James\|last2=Hoover\|first3=Walter\|last3=Ruzzo\|year=1995\|title=Limits To Parallel computation; P-Completeness Theory\|isbn=0-19-508591-4}}. In particular, for the argument that every nontrivial problem in P has a polynomial-time many-one reduction to every other nontrivial problem, see p. 48.</ref>

			==Defining complexity classes==
			The definitions of the complexity classes '''NP''', '''PSPACE''', and '''EXPTIME''' do not involve reductions: reductions come into their study only in the definition of complete languages for these classes. However, in some cases a complexity class may be defined by reductions. If ''C'' is any [[decision problem]], then one can define a complexity class '''C''' consisting of the languages ''A'' for which <math>A \le_m^P B</math>. In this case, ''C'' will automatically be complete for '''C''', but '''C''' may have other complete problems as well.

			An example of this is the complexity class <math>\exists \mathbb{R}</math> defined from the [[existential theory of the reals]], a computational problem that is known to be [[NP-hard\|'''NP'''-hard]] and in [[PSPACE\|'''PSPACE''']], but is not known to be complete for '''NP''', '''PSPACE''', or any language in the [[polynomial hierarchy]]. <math>\exists \mathbb{R}</math> is the set of problems having a polynomial-time many-one reduction to the existential theory of the reals; it has several other complete problems such as determining the [[Crossing number (graph theory)\|rectilinear crossing number]] of an [[undirected graph]]. Each problem in <math>\exists \mathbb{R}</math> inherits the property of belonging to '''PSPACE''', and each <math>\exists \mathbb{R}</math>-complete problem is '''NP'''-hard.<ref>{{citation\|first=Marcus\|last=Schaefer\|contribution=Complexity of some geometric and topological problems\|url=http://ovid.cs.depaul.edu/documents/convex.pdf\|title=[[International Symposium on Graph Drawing\|Graph Drawing, 17th International Symposium, GS 2009, Chicago, IL, USA, September 2009, Revised Papers]]\|series=Lecture Notes in Computer Science\|publisher=Springer-Verlag\|volume=5849\|pages=334–344\|doi=10.1007/978-3-642-11805-0_32\|year=2010}}.</ref>

			Similarly, the complexity class [[GI (complexity)\|'''GI''']] consists of the problems that can be reduced to the [[graph isomorphism problem]]. Since graph isomorphism is known to belong both to '''NP''' and co-[[AM (complexity)\|'''AM''']], the same is true for every problem in this class. A problem is '''GI'''-complete if it is complete for this class; the graph isomorphism problem itself is '''GI'''-complete, as are several other related problems.<ref>{{citation
			\| first1 = Johannes \| last1 = Köbler
			\| first2 = Uwe \| last2 = Schöning \| author2-link = Uwe Schöning
			\| first3 = Jacobo \| last3 = Torán
			\| title = The Graph Isomorphism Problem: Its Structural Complexity
			\| publisher = Birkhäuser
			\| year = 1993
			\| isbn = 978-0-8176-3680-7
			\| oclc = 246882287}}.</ref>

			== References ==
			{{reflist}}

			{{DEFAULTSORT:Polynomial-Time Reduction}}
			[[Category:Computational complexity theory]]

			[[he:רדוקציה פולינומית]]

59.167.79.120: No edit, just adding note. Last couple of edits were trying to fix lexing errors (no idea WHAT was going on but it's good now)

2012-04-27T04:31:28Z

No edit, just adding note. Last couple of edits were trying to fix lexing errors (no idea WHAT was going on but it's good now)

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My name is Adelaide Griffiths. I life in Lublin (Poland).<br><br>Have a look at my web site: [http://safedietplans.com diets]

Cocktail sort - Revision history

93.204.196.246 at 13:55, 2 September 2014

66.25.206.181: /* Complexity */ +.

59.167.79.120: No edit, just adding note. Last couple of edits were trying to fix lexing errors (no idea WHAT was going on but it's good now)