Complexity Classes (P, NP)

| Problem Type | Verifiable in P time | Solvable in P time | Increasing Difficulty
___________________________________________________________|           |
| P            |        Yes           |        Yes         |           |
| NP           |        Yes           |     Yes or No *    |           |
| NP-Complete  |        Yes           |      Unknown       |           |
| NP-Hard      |     Yes or No **     |      Unknown ***   |           |
____________________________________________________________           V

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.


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