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Return to NP, NP Complete, NP Hard.
It might be because of the name that many graduate students find it difficult to understand about NP problems. So, I thought of explaining them as easy as I can.
As the name says these problems can be solved in polynomial time, i.e.; <math>O(n), O(n^2) or O(n^k), </math>where <math>k</math> is a constant.
Some think <math>NP</math> as Non-polynomial. But actually its Non-deterministic Polynomial time. i.e.; these problems can be solved in polynomial time by a non-deterministic Turing machine and hence in exponential time by a deterministic Turing machine. In other words these problems can be verified (if a solution is given, say if its correct or wrong) in polynomial time. Examples include all P problems and one example of a problem not in <math>P</math> but in <math>NP</math> is Integer Factorization.
Over the years many problems in <math>NP</math> have been proved to be in <math>P</math> also (like Primality Testing). Still, there are many problems in <math>NP</math> still not proved to be in <math>P</math>. i.e.; the question still remains whether <math>P = NP</math>.
<math>NP</math> Complete Problems are an aid in solving this question. They are a subset of <math>NP</math> problems with the property that all other <math>NP</math> problems can be reduced to any <math>NP</math> Complete problem in polynomial time. So, they are the hardest problems in <math>NP</math> in terms of running time. Also, if anyone can show that an <math>NP</math> Complete Problem is in <math>P</math>, then all problems in <math>NP</math> will be in <math>P</math>, and hence <math>P = NP = NPC</math>.
All <math>NP</math> Complete problems are in <math>NP</math> because of the definition. Examples of <math>NP</math> Complete problems
These problems need not have any bound on their execution time. If any NP Complete Problem is polynomial time reducable to a problem, that problem belongs to NP Hard class. Hence, all NP Complete problems are also NP Hard. In other words if a NP Hard problem is non-deterministic polynomial time solvable, its a NP Complete problem. Example of a NP Hard problem that's not NP Complete is Halting Problem.
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