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Some examples of NP problems.
Every problem in P :
We showed earlier that P problems are in NP ,
so every P problem is an example of an NP problem.
Examples include:
This is, to some extent, the nub of the
problem. We don't know that there are any
problems in NP that aren't also in P, because
we don't know that NP is strictly bigger.
That is the P vs NP question, and now you
can see it arising naturally from just thinking
about these things. 

But that's not interesting. What we want are examples
of problems that are strictly harder
(See problem A is harder than problem B)  problems
which are in NP, but for which no Polynomial Time
algorithm is know.
Some strictly harder problems
There are currently no known algorithms for solving
these problems in Polynomial Time:
 Factoring Integers
 Travelling Salesman Problem
 Three Colouring Graphs
 The Knapsack Problem
It's important to note that not every instance of these problems
takes a long time to solve. It's conjectured that the vast majority
of instances can be solved in polynomial time, and that it's only
a vanishingly small proportion that take longer.
Modern cryptography relies on finding hard instances of problems such
as these. The most commonly known  the RSA publickey system  relies
on being able to find integers that have a factorisation that's hard to
find.
Why are these hard(er)?
Broadly speaking, in each case there are an exponential number
of possibilities, and no apparent way to exploit a concept of
"closeness". There are n! possible orderings of the cities
to visit, but having path that visits all but one city doesn't
(necessarily!) help at all. It may be completely wrong in a
nonlocal way.
Similarly with the threecolouring question. The colour of
one node may be forced by choices made of colours that are, in
some sense, a long way away.
We think that in some very real sense Integer Factorisation (IF)
is easier than Graph ThreeColouring (G3C) or the
Travelling Salesman Problem (TSP).


Consider integer factorisation. Knowing that 1111111=239*4649
doesn't help at all in trying to find the factorisation
of 1111109, even though these numbers only differ by 2.
Intuition about why a given problem is "hard" in this sense is
not easy to come by, and once obtained, is not easy to express.
At best we can say that the search space is "large" and "brittle."
It's not enough simply that there are a lot of possible solutions.
The question of finding a minimal spanning tree for a graph has
a huge search space, but there is a polynomial time algorithm
because the solution space is, in some sense, "nice."
For a large list of NP problems that are in some sense "hard"
you can check this page on wikipedia:
The sense in which these are known to be "hard" is the subject
of the next section.
Prev: Problem A Is Harder Than Problem B / Next: What Is NPComplete?
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