Editing Computational complexity theory (section)

===Reduction===
{{main|Reduction (complexity)}}
Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at most as difficult as another problem. For instance, if a problem <math>X</math> can be solved using an algorithm for <math>Y</math>, <math>X</math> is no more difficult than <math>Y</math>, and we say that <math>X</math> ''reduces'' to <math>Y</math>. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as [[polynomial-time reduction]]s or [[log-space reduction]]s.

The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

This motivates the concept of a problem being hard for a complexity class. A problem <math>X</math> is ''hard'' for a class of problems <math>C</math> if every problem in <math>C</math> can be reduced to <math>X</math>. Thus no problem in <math>C</math> is harder than <math>X</math>, since an algorithm for <math>X</math> allows us to solve any problem in <math>C</math>. The notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of [[NP-hard]] problems.

If a problem <math>X</math> is in <math>C</math> and hard for <math>C</math>, then <math>X</math> is said to be ''[[complete (complexity)|complete]]'' for <math>C</math>. This means that <math>X</math> is the hardest problem in <math>C</math>. (Since many problems could be equally hard, one might say that <math>X</math> is one of the hardest problems in <math>C</math>.) Thus the class of [[NP-complete]] problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, <math>\Pi_2</math>, to another problem, <math>\Pi_1</math>, would indicate that there is no known polynomial-time solution for <math>\Pi_1</math>. This is because a polynomial-time solution to <math>\Pi_1</math> would yield a polynomial-time solution to <math>\Pi_2</math>. Similarly, because all NP problems can be reduced to the set, finding an [[NP-complete]] problem that can be solved in polynomial time would mean that P = NP.<ref name="Sipser2006"/>