NP-Problem

A problem is assigned to the NP (nondeterministic polynomial time) class if it is solvable in polynomial time by a nondeterministic Turing machine.

A P-problem (whose solution time is bounded by a polynomial) is always also NP. If a problem is known to be NP, and a solution to the problem is somehow known, then demonstrating the correctness of the solution can always be reduced to a single P (polynomial time) verification. If P and NP are not equivalent, then the solution of NP-problems requires (in the worst case) an exhaustive search.

Linear programming, long known to be NP and thought not to be P, was shown to be P by L. Khachian in 1979. It is an important unsolved problem to determine if all apparently NP problems are actually P.

A problem is said to be NP-hard if an algorithm for solving it can be translated into one for solving any other NP-problem. It is much easier to show that a problem is NP than to show that it is NP-hard. A problem which is both NP and NP-hard is called an NP-complete problem.

 

 

NP-Hard Problem  http://mathworld.wolfram.com/NP-HardProblem.html

 

A problem is NP-hard if an algorithm for solving it can be translated into one for solving any NP-problem (nondeterministic polynomial time) problem. NP-hard therefore means "at least as hard as any NP-problem," although it might, in fact, be harder.

 

NP-Complete Problem   http://mathworld.wolfram.com/NP-CompleteProblem.html

 

A problem which is both NP (verifiable in nondeterministic polynomial time) and NP-hard (any NP-problem can be translated into this problem). Examples of NP-hard problems include the Hamiltonian cycle and traveling salesman problems.

In a landmark paper, Karp (1972) showed that 21 intractable combinatorial computational problems are all NP-complete.

 

 

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 克雷数学研究所（Clay Mathematics Institute，简称CMI）七个千禧年大奖难题：

http://www.claymath.org/millennium/

 

P vs NP Problem

 

If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit (by car), how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily (given the methods I know) find a solution.