P,NP,NPC问题

论文里面有一个NP问题,不知答辩的时候会不会问到这个问题,今天把看到的内容写到这里。《P,NP,NPC问题》

 

一个NP-完全的问题具有如下性质:它可以在多项式时间内求解,当且仅当所有的其他的NP-完全问题也可以在多项式时间内求解。

P是所有可在多项式时间内用确定算法求解的判定问题的集合。NP是所有可在多项式时间内用不确定算法求解的判定问题的集合。

令L1和L2是两个问题,如果有一确定的多项式时间算法求解L1,而这个算法使用了一个在多项式时间内求解L2的确定算法,则称L1约化为L2。

如果可满足性约化为一个问题L,则称L问题是NP-难度的。如果L是NP难度的且L(-NP,则称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.

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