Clique np problem

Hi Fellow EE Experts

Looking for a quick proof here on the following. one showing it can be done and an example showing it cant.

Looking at the following NP Problem.
http://en.wikipedia.org/wiki/Clique_problem

How would I show that for one instance an example can be solved in a finite amount of time and the other an exponential?

Any Help would be grealty appreciated.

Many Thanks
Steve
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Stephen MandersonSoftware EngineerAsked:
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ozoConnect With a Mentor Commented:
The time to solve is generally a function of the number of vertices.
If there was an algorithm that could solve any instance of the problem in a time that was a polynomial function of the number of vertices, then there would be an algorithm to solve any NP-C problem in polynomial time.
Since no one knows how to do that, no one knows how to write a deterministic algorithm that can solve any instance of clique in polynomial time.
This does not prevent there from being an algorithm that can quickly solve a particular instance of clique.

But it doesn't really make sense to say that a particular problem can be solved in exponential time,
Exponential time and polynomial time describes how the time to solve varies as function of the size of the problem.
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ozoCommented:
You can show it is in  NP-Complete  by reducing it to Independent Set
http://en.wikipedia.org/wiki/Independent_set_problem
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ozoCommented:
Any given instance of the problem can be solved in P time by an algorithm that knows the answer to that given instance.
But for any given algorithm there exist instances of the problem that cannot be solved in polynomial time, unless every other NP-Complete problem can also be solved in Polynomial time,
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Stephen MandersonSoftware EngineerAuthor Commented:
Thanks for the links Ozo tryiong to understand the way to prove for one instance the graph could be P time.

You say reducing it to an independent set, but isnt an independant set the opposite of a clique ?

Hmm im pretty lost when it comes to this, just to help me along the right lines would this be correct in trying to prove a clique problem in p time?

6
  \
   4-----5----
   I       I     1
   3-----2----

The clique is formed from vertices 1,2 and 5 which could be solved in p time

However

If there was an exponential amount of vertices it would be NP to calculate if a clique is present?
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Stephen MandersonSoftware EngineerAuthor Commented:
Sorry i meant that as the number of vertices increases so does the time taken to solve the problem.
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