NP-Completeness

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...