論文裏面有一個NP問題，不知答辯的時候會不會問到這個問題，今天把看到的內容寫到這裏。
一個NP－完全的問題具有如下性質：它可以在多項式時間內求解，當且僅當所有的其他的NP－完全問題也可以在多項式時間內求解。
P是所有可在多項式時間內用確定算法求解的判定問題的集合。NP是所有可在多項式時間內用不確定算法求解的判定問題的集合。
令L1和L2是兩個問題，如果有一確定的多項式時間算法求解L1，而這個算法使用了一個在多項式時間內求解L2的確定算法，則稱L1約化爲L2。
如果可滿足性約化爲一個問題L，則稱L問題是NP-難度的。如果L是NP難度的且L（－NP，則稱L是NP－完全的。
NP-hard Problems
Example of difficult problems, which cannot be solved int "traditional" way, are NP problems.
There are many tasks for which we know fast (polynomial) algorithms. There are also some problems that are not possible to be solved algorithmicaly. For some problems was proved that they are not solvable in polynomial time.
But there are many important tasks, for which it is very difficult to find a solution, but once we have it, it is easy to check the solution. This fact led to NP-complete problems. NP stands for nondeterministic polynomial and it means that it is possible to "guess" the solution (by some nondeterministic algorithm) and then check it, both in polynomial time. If we had a machine that can guess, we would be able to find a solution in some reasonable time.
Studying of NP-complete problems is for simplicity restricted to the problems, where the answer can be yes or no. Because there are tasks with complicated outputs, a class of problems called NP-hard problems has been introduced. This class is not as limited as class of NP-complete problems.
For NP-problems is characteristic that some simple algorithm to find a solution is obvious at a first sight - just trying all possible solutions. But this algorithm is very slow (usually O(2^n)) and even for a bit bigger instances of the problems it is not usable at all.
Today nobody knows if some faster exact algorithm exists. Proving or disproving this remains as a big task for new researchers (and maybe you! :-)). Today many people think, that such an algorithm does not exist and so they are looking for some alternative methods - example of these methods are genetic algorithms.
Examples of the NP problems are satisfiability problem, travelling salesman problem or knapsack problem.