NP Hardness boundary

I have two problems.

开发者_Python百科

1. Is there a clique of size k in a graph? NP Hard

2. Is there a clique of size 50 in a graph? - Can be found out in polynomial time O(n^50)

Why is the second problem not NP hard where as the first one is?

EDIT: Assuming P!=NP

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n^k is exponential whereas n^50 is polynomial.

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The first problem is NP-hard because an arbitrary NP-complete problem (say, 3-SAT) can be reduced to it in polynomial time. (by the definition of NP-hardness)

The second problem is not NP-hard, because an arbitrary NP-complete problem cannot be reduced to it (say, 3-SAT, with >50 clauses).

In fact, the second problem is in P, because O(n^50) belongs to P. But that isn't the reason why it is not NP-hard (specifically, NP doesn't mean non-polynomial).