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A problem is complete for a complexity class if every problem in the class reduces to that problem, and it is also in the class itself. In this sense the problem represents the class, since any solution to it can, in combination with the reductions, be used to solve every problem in the class. However, in order to be useful, reductions must be easy.
For example, it's quite possible to reduce a difficult-to-solve NP-complete problem like the boolean satisfiability problem to a trivial problem, like determining if a number equals zero, by having the reduction machine solve the problem in exponential time and output zero only if there is a solution. However, this does not achieve much, because even though we can solve the new problem, performing the reduction is just as hard as solving the old problem.
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Likewise, a reduction computing a noncomputable function can reduce an undecidable problem to a decidable one. As Michael Sipser points out in Introduction to the Theory of Computation : "The reduction must be easy, relative to the complexity of typical problems in the class [ Therefore, the appropriate notion of reduction depends on the complexity class being studied.
When studying the complexity class NP and harder classes such as the polynomial hierarchy , polynomial-time reductions are used. Reductions are also used in computability theory to show whether problems are or are not solvable by machines at all; in this case, reductions are restricted only to computable functions. In case of optimization maximization or minimization problems, we often think in terms of approximation-preserving reduction.
Suppose we have two optimization problems such that instances of one problem can be mapped onto instances of the other, in a way that nearly optimal solutions to instances of the latter problem can be transformed back to yield nearly optimal solutions to the former. This way, if we have an optimization algorithm or approximation algorithm that finds near-optimal or optimal solutions to instances of problem B, and an efficient approximation-preserving reduction from problem A to problem B, by composition we obtain an optimization algorithm that yields near-optimal solutions to instances of problem A.
The following example shows how to use reduction from the halting problem to prove that a language is undecidable. Suppose H M , w is the problem of determining whether a given Turing machine M halts by accepting or rejecting on input string w. This language is known to be undecidable. Suppose E M is the problem of determining whether the language a given Turing machine M accepts is empty in other words, whether M accepts any strings at all.
We show that E is undecidable by a reduction from H. To obtain a contradiction, suppose R is a decider for E. We will use this to produce a decider S for H which we know does not exist. Given input M and w a Turing machine and some input string , define S M , w with the following behavior: S creates a Turing machine N that accepts only if the input string to N is w and M halts on input w , and does not halt otherwise. The decider S can now evaluate R N to check whether the language accepted by N is empty. If R accepts N , then the language accepted by N is empty, so in particular M does not halt on input w , so S can reject.
If R rejects N , then the language accepted by N is nonempty, so M does halt on input w , so S can accept.