#### Theory of P versus NP Problem, Complexity P & NP

The P versus NP Problem:

Assume that you are organizing housing accommodations for a group of 400 university students. Space is very limited and only 100 of the students will receive positions in the dormitory. To complicate matters, the Dean has given you with a list of pairs of incompatible students, and requested that no pair from this list seem in your final selection. This is an illustration of what computer scientists call an NP-problem, as it is simple to check when a given selection of 100 students proposed by the coworker is satisfactory (that is, no pair from taken from your coworker's list as well appears on the list from Dean's office), though the task of producing such a list from scratch appears to be too hard as to be fully impractical. Certainly, the total number of ways of selecting 100 students from the 400 applicants is bigger than the number of atoms in the known universe! Therefore no future civilization could ever hope to build a super-computer capable of solving the trouble by brute force; that is, by checking each and every possible combination of 100 students. Though, this apparent difficulty might just reflect the lack of ingenuity of your programmer. However, one of the outstanding troubles in computer science is finding out whether questions exist whose answer can be fast checked, however that need an impossibly long time to resolve by any direct procedure. Troubles like the one listed above certainly appear to be of this type, however so far no one has managed to verify that any of them really are too hard as they appear, that is, there really is no feasible way to create an answer with the help of a computer. Stephen Cook and Leonid Levin formulated the P (that is, simple to find) versus NP (that is, simple to check) problem independently in the year1971.

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