如何证明问题不是NP完成的？

有什么通用的技术可以证明问题不是NP完全的吗？

我在考试中遇到这个问题，要求我证明是否有一些问题（参见下文）是NP-Complete。我想不出任何真正的解决方案，只是证明它在P中。显然，这不是一个真正的答案。

NP-Complete被定义为NP中的一组问题，所有NP问题都可以简化为它。因此，任何证据都应至少与这两个条件之一相矛盾。这个特定的问题确实存在于P中（因此也存在于NP中）。因此，我坚持证明NP中存在一些无法解决的问题。在地球上如何证明这一点？

这是我在考试中遇到的具体问题：

设为析取范式的弦集。令D N F S A T是来自D N F的字符串的语言，可以通过某些变量分配来满足该语言。显示是否D N F S A $DNF$$DNFSAT$$DNF$$DNFSAT$是否在NP-Complete中。

根据评论，您似乎想要一个无条件的答案。

但是，通过分配变量以满足第一个析取项，DNF-SAT位于L中。因此，如果它是NP完全的，则L ＝ NP。

另一方面，如果L = NP，那么在对数空间缩减下，DNF-SAT完全是NP完全的。（实际上，如果L = NP，那么在对数空间减少的情况下，NP中的每个问题都是NP完全的。）

因此，在对数空间缩减下，L = NP iff DNF-SAT是NP完全的。

因此，您目前不能像您想做的那样无条件地声明DNF-SAT不是NP完整的。不一定要假设P≠NP，但是答案确实必须以某些条件为条件，并且L≠NP是可以保证期望结果的最弱假设。

一个问题是NP完全如果它既是NP- 硬和中 NP。这意味着您需要反驳这两个之一。$Q$

1. 在P NP 的假设下，可以给出求解Q的多项式时间算法。稀疏地，假设图同构不是NP困难的，则可以证明Q对于图同构是可乘的。$\neq$$Q$$Q$
2. 您表明不在NP中。这很难，并且您通常必须使用其他假设，例如多项式层次结构的不崩溃，即NP ≠ coNP或表明对高于NP的其他某些类别来说较难，例如，通过证明其对NEXPTIME困难。$Q$$\neq$

Usually, the answer is to give a polynomial time algorithm, which would be the simplest for DNF-SAT, but this relies on the hypothesis that P $\neq$ NP. However, proving that DNF-SAT is not NP-complete without any assumptions implies, as Shaull points out, proving that P $\neq$ NP, so that is somewhat trickier.

By the nondeterministic time hierarchy, you could show that the problem is $\mathsf {NEXP}$-hard; as $\mathsf {NP} \ne \mathsf {NEXP}$, it is impossible to reduce the problem in polynomial time to any problem in $\mathsf {NP}$, so the problem will not be in $\mathsf {NP}$.

However, if your problem is not nearly that difficult, you may be hard-pressed to prove that it is not in $\mathsf {NP}$; and if it is in $\mathsf {NP}$, you'll be hard-pressed to show that $\mathsf {NP}$ is not Karp-reducible to your problem without assuming that $\mathsf P \ne \mathsf {NP}$.

As is the case with all proofs, there is no formula for proving a statement, you have to do some intelligent guesswork, trial and error and hopefully you'll be able to prove what you are trying to prove. To prove a problem is NOT NP-Complete, negate the definition (DeMorgran Law), that is to say prove the problem NOT in NP or prove the problem NOT NP-Hard.

I believe what the lecturer really wants is that you could distinguish problems that are in P from problems that are NP-complete in the meaning given language can you build an efficient algorithm ? if yes then it is suspected not to be NP-complete because we dont think that languages in P is NP-complete ! otherwise you still have to prove that the problem is NP-hard !

note that there exists some problems that we dont know there status such as graph isomorphism,factoring given number,... we think that these problems are not NP-complete but no one could prove that ! specifically we have evidences that graph isomorphism is not NP-complete! other problem is the unique game conjuncture that we suspect that unique game is NP-complete but no proof exists! so the approach you have describe is not helpful!