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Lesson 43, Topic 1
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# Basic Concepts

##### Akash
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The theory of NP-completeness which we present here does not provide a
method of obtaining polynomial time algorithms for problems in the second
group. Nor does it say that algorithms of this complexity do not exist.
Instead,what we do is show that many of the problems for which there are no known polynomial time algorithms are computationally related. In fact,
we establish two classes of problems. These are given the names NP-hard
and NP-complete. A problem that is NP-complete has the property that
it can be solved in polynomial time if and only if all other NP-complete
problems can also be solved in polynomial time. If an NP-hard problem
can be solved in polynomial time,then all NP-complete problems can be
solved in polynomial time. All NP-complete problems are NP-hard, but
some NP-hard problems are not known to be NP-complete.

Although one can define many distinct problem classes having the
properties stated above for the NP -hard and NP -complete classes,the classes we study are related to nondeterministic computations(to be defined later). The relationship of these classes to nondeterministic computations together with the apparent power of nondeterminism leads to the intuitive (though as yet unproved)conclusion that no NP-complete or NP-hard problem is polynomially solvable.

We see that the class of NP-hard problems(and the subclass of NV complete problems)is very rich as it contains many interesting problems from
a wide variety of disciplines. First,we formalize the preceding discussion of
the classes.