NP-completeness

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In computational complexity theory, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. NP is short for "nondeterministic polynomial-time" and a problem is NP-complete when: It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". Each input to the problem is associated with a collection of short (polynomial size) solutions, which might or might not validly solve the input. The output is "yes" when at least one of these solutions is valid, and "no" when none of them is. The validity of each solution can be verified quickly (namely, in polynomial time), and a brute-force search algorithm can find a valid solution (if one exists) by trying all possible solutions. The problem can be used to simulate every other problem for which we can verify quickly that a solution is valid. Hence, if we could find valid solutions of some NP-complete problem quickly (when they exist), we could quickly find valid...

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