Why are NP problems called that way (and NP-hard and NP-complete)?

Question

Really.. I\'m having the last test for graduation this Tuesday, and that\'s one of the things I just never could understand. I realize that a solution for NP problem can be

Answer 1

Let's begin with NP: in NP, N is for "nondeterministic" and P is for "polynomial time". It is the class of problems that can be solved in polynomial time if you have a nondeterministic Turing machine that can branch at each cycle to explore possibilities in parallel (the "verify the solution" alternative definition has become popular recently but it does not make clear what "N" means). The nondeterministic machine can be compared to a parallel computer with an infinite number of processors, and the ability to fork() at each instruction.

Saying that a problem Q is "NP-hard" means that any problem in NP can be reduced to problem Q (in polynomial time). Since the relation "can be reduced to" between problems is an order relation, you can think of "NP-hard" as meaning "at least as hard as all NP problems".

An "NP-complete" problem is simply one of the problems in NP that is NP-hard. I guess that class of problems needed a name, but I'm not sure how to explain the choice of the word "complete".