What's "P=NP?", and why is it such a famous question?
P stands for polynomial time. NP stands for non-deterministic polynomial time.
Definitions:
Polynomial time means that the complexity of the algorithm is O(n^k), where n is the size of your data (e. g. number of elements in a list to be sorted), and k is a constant.
Complexity is time measured in the number of operations it would take, as a function of the number of data items.
Operation is whatever makes sense as a basic operation for a particular task. For sorting, the basic operation is a comparison. For matrix multiplication, the basic operation is multiplication of two numbers.
Now the question is, what does deterministic vs. non-deterministic mean? There is an abstract computational model, an imaginary computer called a Turing machine (TM). This machine has a finite number of states, and an infinite tape, which has discrete cells into which a finite set of symbols can be written and read. At any given time, the TM is in one of its states, and it is looking at a particular cell on the tape. Depending on what it reads from that cell, it can write a new symbol into that cell, move the tape one cell forward or backward, and go into a different state. This is called a state transition. Amazingly enough, by carefully constructing states and transitions, you can design a TM, which is equivalent to any computer program that can be written. This is why it is used as a theoretical model for proving things about what computers can and cannot do.
There are two kinds of TM's that concern us here: deterministic and non-deterministic. A deterministic TM only has one transition from each state for each symbol that it is reading off the tape. A non-deterministic TM may have several such transition, i. e. it is able to check several possibilities simultaneously. This is sort of like spawning multiple threads. The difference is that a non-deterministic TM can spawn as many such "threads" as it wants, while on a real computer only a specific number of threads can be executed at a time (equal to the number of CPUs). In reality, computers are basically deterministic TMs with finite tapes. On the other hand, a non-deterministic TM cannot be physically realized, except maybe with a quantum computer.
It has been proven that any problem that can be solved by a non-deterministic TM can be solved by a deterministic TM. However, it is not clear how much time it will take. The statement P=NP means that if a problem takes polynomial time on a non-deterministic TM, then one can build a deterministic TM which would solve the same problem also in polynomial time. So far nobody has been able to show that it can be done, but nobody has been able to prove that it cannot be done, either.
NP-complete problem means an NP problem X, such that any NP problem Y can be reduced to X by a polynomial reduction. That implies that if anyone ever comes up with a polynomial-time solution to an NP-complete problem, that will also give a polynomial-time solution to any NP problem. Thus that would prove that P=NP. Conversely, if anyone were to prove that P!=NP, then we would be certain that there is no way to solve an NP problem in polynomial time on a conventional computer.
An example of an NP-complete problem is the problem of finding a truth assignment that would make a boolean expression containing n variables true.
For the moment in practice any problem that takes polynomial time on the non-deterministic TM can only be done in exponential time on a deterministic TM or on a conventional computer.
For example, the only way to solve the truth assignment problem is to try 2^n possibilities.
- A yes-or-no problem is in P (Polynomial time) if the answer can be computed in polynomial time.
- A yes-or-no problem is in NP (Non-deterministic Polynomial time) if a yes answer can be verified in polynomial time.
Intuitively, we can see that if a problem is in P, then it is in NP. Given a potential answer for a problem in P, we can verify the answer by simply recalculating the answer.
Less obvious, and much more difficult to answer, is whether all problems in NP are in P. Does the fact that we can verify an answer in polynomial time mean that we can compute that answer in polynomial time?
There are a large number of important problems that are known to be NP-complete (basically, if any these problems are proven to be in P, then all NP problems are proven to be in P). If P = NP, then all of these problems will be proven to have an efficient (polynomial time) solution.
Most scientists believe that P!=NP. However, no proof has yet been established for either P = NP or P!=NP. If anyone provides a proof for either conjecture, they will win US $1 million.
To give the simplest answer I can think of:
Suppose we have a problem that takes a certain number of inputs, and has various potential solutions, which may or may not solve the problem for given inputs. A logic puzzle in a puzzle magazine would be a good example: the inputs are the conditions ("George doesn't live in the blue or green house"), and the potential solution is a list of statements ("George lives in the yellow house, grows peas, and owns the dog"). A famous example is the Traveling Salesman problem: given a list of cities, and the times to get from any city to any other, and a time limit, a potential solution would be a list of cities in the order the salesman visits them, and it would work if the sum of the travel times was less than the time limit.
Such a problem is in NP if we can efficiently check a potential solution to see if it works. For example, given a list of cities for the salesman to visit in order, we can add up the times for each trip between cities, and easily see if it's under the time limit. A problem is in P if we can efficiently find a solution if one exists.
(Efficiently, here, has a precise mathematical meaning. Practically, it means that large problems aren't unreasonably difficult to solve. When searching for a possible solution, an inefficient way would be to list all possible potential solutions, or something close to that, while an efficient way would require searching a much more limited set.)
Therefore, the P=NP problem can be expressed this way: If you can verify a solution for a problem of the sort described above efficiently, can you find a solution (or prove there is none) efficiently? The obvious answer is "Why should you be able to?", and that's pretty much where the matter stands today. Nobody has been able to prove it one way or another, and that bothers a lot of mathematicians and computer scientists. That's why anybody who can prove the solution is up for a million dollars from the Claypool Foundation.
We generally assume that P does not equal NP, that there is no general way to find solutions. If it turned out that P=NP, a lot of things would change. For example, cryptography would become impossible, and with it any sort of privacy or verifiability on the Internet. After all, we can efficiently take the encrypted text and the key and produce the original text, so if P=NP we could efficiently find the key without knowing it beforehand. Password cracking would become trivial. On the other hand, there's whole classes of planning problems and resource allocation problems that we could solve effectively.
You may have heard the description NP-complete. An NP-complete problem is one that is NP (of course), and has this interesting property: if it is in P, every NP problem is, and so P=NP. If you could find a way to efficiently solve the Traveling Salesman problem, or logic puzzles from puzzle magazines, you could efficiently solve anything in NP. An NP-complete problem is, in a way, the hardest sort of NP problem.
So, if you can find an efficient general solution technique for any NP-complete problem, or prove that no such exists, fame and fortune are yours.