P versus NP problem presentation

1. P VERSUS NP
By Farid El Hajj
2012

2. OVERVIEW OF P VERSUS NP PROBLEM
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The P versus NP problem belongs to complexity
theory. P denotes the class of computer algorithms
which can be executed in a fast way in order to solve a
decision or a search problem. Whereas
NP corresponds to the problems
NP
which can be checked for solution
correctness in a fast way
whenever a solution is
provided.
Every P problem belongs
P
to NP set (P is included in NP)
The opposite question
whether NP is identical to P
is a major unsolved question.
We don’t know if there exists
a fast algorithm which can find
the solution to a NP problem
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3. FORMAL DEFINITION OF P VERSUS NP
PROBLEM
Formally, P is the class of problems solved by a
deterministic Turing machine in a polynomial
time. NP is the class of problems which are
solved by a non-deterministic Turing machine in
a polynomial time, which means it can be solved
by brute force algorithm in an exponential time.
 We know that any P problem is also an NP one.
In fact the fast algorithm that solves the P
problem can be its fast verification algorithm.
The question is could we find an NP problem
which doesn’t have definitely a fast algorithm to
solve it. If so then NP is different from P. If no
then NP is identical to P.
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4. DESCRIPTION OF P VERSUS NP PROBLEM
Most famous open problem in theoretical
computer science.
 Its solution can deepen our understanding of
algorithm complexity or maybe it will give us a
mathematical tool for solving hard problems in
an efficient way.
 Great work has been spent on providing a
solution but no one has ever been able to solve it.
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5. HISTORY OF THE PROBLEM
Described in its current form by Cook in 1971 in
a paper where he proved the first NP-Complete
problem: the Boolean satisfiability problem. Also
he proved that any NP problem could be reduced
to an NP-Complete one using a Turing machine
in a polynomial time.
 In 1973 Karp gave another 21 NP-Complete
problems.
 Since then tremendous effort has been deployed
to find an algorithm that will reduce any NP
problem to P, or on the contrary to prove that
these two classes of problems are different.
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6. TECHNICAL IMPACT
The P versus NP problem has a large impact
which makes it a fundamental open problem.
Imagine one could derive a polynomial time
algorithm for solving an NP-Complete problem,
then any NP problem with exponential time
could be reduced and solved in a polynomial time
in an efficient way.
 No such algorithm has been found, nor the
impossibility to find such an algorithm has been
proved.
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7. LARGER IMPACT
If someone could derive such an algorithm the
result will have an impact on several domains:
theorem proving could be done in an efficient and
tractable way. Optimization and resource
management will be an easy matter. The impact
will be tremendous on many and unthought-of
domains such as evolutional biology and
economics and much more.
 If P equals NP then current cryptographic
schemes will be broken in a reasonable time and
power. The one way functions assumed to be hard
could be broken like a piece of toy.
 But many experts think that P is not equal to
NP.
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