 # Prove using the steps from the class that n/2-Clique problem is NP-...

## Question

Prove using the steps from the class that n/2-Clique problem is NP-Hard. You can use without proof that n/3-Clique is NP-Complete problem.

## Solution Preview

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We prove that n/2-clique is NP-Complete in two steps, as the lecture shows.
1). n/2-clique is in NP class
2). n/2-clique is in NP-Hard
For the step 1), we make a “guess” of a sequence of vertices for a graph provided as input and we verify whether the size of the sequence is n/2 and whether there is a clique determined by these n/2 vertices. Since each of these can be done in polynomial time (assuming that n is finite, then n/2 is also finite and gives the input size since the number of edges in this case is (n/2)*(n/2 -1)), it follows that our problem is in NP.
For step 2), we need to build a reduction in polynomial time from another known NP-Complete problem. We use n/3-clique problem in this case....

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