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The CLIQUE problem -- problem of finding the maximum clique in a graph -- is NP-complete. That is, CLIQUE is

  1. in NP and
  2. there is an NP complete problem, 3-SAT for one, that reduces to CLIQUE in polynomial time.

Part 2. above is fine -- all over in every resource and very well explained. For Part 1., from what I know, we need to have the following: Given a specific solution instance, we need to show that it can be verified, in polynomial time, that that solution is an answer to this problem. So for instance, given a specific graph and a subgraph of it, we should be able to check whether that subgraph is a clique of maximum size in that graph.

The resources I've read so far are phrasing this Part 1. here as "easy, straightforward, etc" or "it can be shown in $O(n^2)$ time that the given subgraph is a clique/not". However, the verification here is not just whether it's a clique, but also is whether it is a maximum clique in the graph. How can this be decided in polynomial time?

What am I missing here?

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What you're missing is the difference between decision problems and optimization problems. "Finding the maximum clique" is an optimization problem. The corresponding decision problem is "Does there exist a clique of size $\ge k$?". The latter problem is in NP. NP is a class of decision problems, so it doesn't make sense to ask whether an optimization problem is in NP; that's only meaningful for decision problems. Any textbook or standard resource on NP should explain the difference between decision problems vs. optimization problems in more depth.

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  • $\begingroup$ "Does there exist a clique of size ≥k?" is also the decision problem of the following problem (call it CLIQUE-Z) "Is there a clique of size k or greater in the graph". So you`re saying, the decision problem of CLIQUE is the same as that of CLIQUE-Z above. in that case, how would i know what the "right" decision is to show my problem is NP? $\endgroup$ – Roam Oct 3 '13 at 0:30
  • $\begingroup$ "Is there a clique of size k or greater in the graph" is telling me the specific solution instance CAN BE a solution but not whether/not it is. Do i have the optimal solution (a maximum clique in the graph) at hand while i'm verifying that solution instance, is that how? $\endgroup$ – Roam Oct 3 '13 at 0:32
  • $\begingroup$ @Roam, I'll repeat the advice in my question: read a textbook or standard resource on NP. Any good textbook should have a discussion of decision problems, optimization problems, and their relationship. Go do your homework. Then if you still don't understand, you can come back and ask another question -- but if you want to understand this topic, you're going to need to put in some effort on your own. $\endgroup$ – D.W. Oct 3 '13 at 5:08
  • $\begingroup$ @Roam You comment is barely legible, but it seems as if your were trying to be rude; please be nice instead! D.W. is quite right -- you don't seem to have basic definitions straight. Please check out our reference question; in particular, the posts linked in the comments may be of interest to you. $\endgroup$ – Raphael Oct 6 '13 at 15:03

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