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The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this sense it is the hardest of the problems to which solutions can be verified quickly so that if we could actually find solutions of some NP-Complete problem quickly, we could quickly find the solutions of every other problem to which a solution once given is easy to check.

the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct.

it is a problem for which the correctness of each solution can be verified quickly and a brute-force search algorithm can actually find a solution by trying all possible solutions.

In computational complexity theory, a problem is NP-complete when: Cook and Levin proved that each easy-to-verify problem can be solved as fast as SAT, which is hence NP-complete.

While it is easy to verify whether a given assignment renders the formula true, no essentially faster method to find a satisfying assignment is known than to try all assignments in succession.

The Boolean satisfiability problem (SAT) asks to determine if a propositional formula (example depicted) can be made true by an appropriate assignment of truth values to its variables.