# Complexity Classes (P, NP)

Notes on `Yes`

or `No`

entries:

* An NP problem that is also P is solvable in P time.

** An NP-Hard problem that is also NP-Complete is verifiable in P time.

*** NP-Complete problems (all of which form a subset of NP-hard) might be. The rest of NP hard is not.

### Complexity Classes for Decision Problems

Decision problem: A problem with a yes or no answer.

Note that the following definitions and discussions are centered about the decision problem (or if a solution is verifiable), and do not mention if you can find the solution (or if the problem is solvable).

P is a complexity class that represents the set of all decision problems that can be solved in polynomial time.

That is, given an instance of the problem, the answer yes or no can be decided in polynomial time.

NP is a complexity class that represents the set of all decision problems for which the instances where the answer is "yes" have proofs that can be verified in polynomial time.

This means that if someone gives us an instance of the problem and a certificate (sometimes called a witness) to the answer being yes, we can check that it is correct in polynomial time.

Note that NP is for Nondeterministic Polynomial time; not non-polynomial time.

NP-Complete: NP-Complete is a complexity class which represents the set of all problems

`X`

in NP for which it is possible to reduce any other NP problem`Y`

to`X`

in polynomial time.Intuitively this means that we can solve

`Y`

quickly if we know how to solve`X`

quickly.Precisely,

`Y`

is reducible to`X`

, if there is a polynomial time algorithm`f`

to transform instances`y`

of`Y`

to instances`x = f(y)`

of`X`

in polynomial time, with the property that the answer to`y`

is yes, if and only if the answer to`f(y)`

is yes.

NP-Hard: Intuitively, these are the problems that are at least as hard as the NP-complete problems.

Note that NP-hard problems do not have to be in NP, and they do not have to be decision problems.

The precise definition here is that a problem X is NP-hard, if there is an NP-complete problem Y, such that Y is reducible to X in polynomial time.

References: https://stackoverflow.com/questions/1857244/what-are-the-differences-between-np-np-complete-and-np-hard

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