The P versus NP problem is a major unsolved problem in theoretical computer science. Informally, it asks whether every decision problem for which a proposed positive answer can be quickly verified can also be quickly solved.
Here, "quickly" means an algorithm exists that solves the task and runs in polynomial time (as opposed to, say, exponential time), meaning the task completion time is bounded above by a polynomial function on the size of the input to the algorithm. The general class of questions that some algorithm can answer in polynomial time is "P" or "class P". For some questions, there is no known way to find an answer quickly, but if provided with an answer, it can be verified quickly. The class of questions where an answer can be verified in polynomial time is "NP", standing for "nondeterministic polynomial time".Nondeterminism is not randomness. A nondeterministic Turing machine accepts an input if at least one of its possible computation paths accepts. Equivalently, for NP problems the paths can represent possible polynomial-size certificates, and the machine accepts when at least one certificate passes polynomial-time verification.
An answer to the P versus NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If P ≠ NP, which is widely believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be verified in polynomial time.
The problem has been called the most important open problem in computer science. Aside from being an important problem in computational theory, a proof either way would have profound implications for mathematics, cryptography, algorithm research, artificial intelligence, game theory, multimedia processing, philosophy, economics and many other fields. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute, each of which carries a US$1,000,000 prize for the first correct solution.
Example Consider the following yes/no problem: given an incomplete Sudoku grid of size , is there at least one legal solution where every row, column, and square contains the integers 1 through ? It is straightforward to verify "yes" instances of this generalized Sudoku problem, given a candidate solution, by just checking the contents of the rows, columns and squares. However, it is not known whether there is a polynomial-time algorithm that can correctly answer "yes" or "no" to all instances of this problem. Therefore, generalized Sudoku is in NP (quickly verifiable), but may or may not be in P (quickly solvable). (It is necessary to consider a generalized version of Sudoku, as any fixed size Sudoku has only a finite number of possible grids. In this case the problem is in P, as the answer can be found by table lookup.)
History
The precise statement of the P versus NP problem was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures", and independently by Leonid Levin in 1973.
Although the P versus NP problem was formally defined in 1971, there were previous inklings of the underlying problems involved. In 1955, mathematician John Nash wrote a letter to the National Security Agency, speculating that the time required to crack a sufficiently complex code would increase exponentially with the length of the key. If proved,Nash expressed skepticism that the conjecture would be proven: "The nature of this conjecture is that I cannot prove it, even for a simple type of cipher. Nor do I expect it to be proven." this would imply what is now called P ≠ NP, since a proposed key can be verified in polynomial time. In a 1956 letter written by Kurt Gödel to John von Neumann, Gödel asked whether theorem-proving (now known to be co-NP-complete) could be solved in quadratic or linear time, and posited that if so, then the discovery of mathematical proofs could be automated.
Context The relation between the complexity classes P and NP is studied in computational complexity theory, the part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps it takes to solve a problem) and space (how much memory it takes to solve a problem).
In such analysis, a model of the computer for which time must be analyzed is required. Typically such models assume that the computer is deterministic (given the computer's present state and any inputs, there is only one possible action that the computer might take) and sequential (it performs actions one after the other).
In this theory, the class P consists of all decision problems (defined below) solvable on a deterministic sequential machine in a duration polynomial in the size of the input; the class NP consists of all decision problems whose positive solutions are verifiable in polynomial time given the right information, or equivalently, whose solution can be found in polynomial time on a nondeterministic machine. Equivalently, a language is in NP if there are a relation decidable in polynomial time and a polynomial such that
In this formulation, P versus NP asks whether polynomial-time decidable predicates are closed under polynomially bounded existential quantification.Sipser, Michael: Introduction to the Theory of Computation, Second Edition, International Edition, page 270. Thomson Course Technology, 2006. Definition 7.19 and Theorem 7.20. Clearly, P ⊆ NP. Arguably, the biggest open question in theoretical computer science concerns the relationship between those two classes: Is P equal to NP?
Since 2001, William Gasarch has conducted three polls of researchers concerning P ≠ NP and related questions. The share of responders believing that P ≠ NP was 61% in 2001, 83% in 2011, and 88% in 2018, with 100, 151, and 124 responses, respectively. When restricted to experts, 99% of 2018 responders believed P ≠ NP. These polls do not imply whether P = NP; as Gasarch stated: "This does not bring us any closer to solving P=?NP or to knowing when it will be solved, but it attempts to be an objective report on the subjective opinion of this era."
NP-completeness
To attack the P = NP question, the concept of NP-completeness is very useful. NP-complete problems are problems that any other NP problem is reducible to in polynomial time and whose solution is still verifiable in polynomial time. That is, any NP problem can be transformed into any NP-complete problem. Informally, an NP-complete problem is an NP problem that is at least as "tough" as any other problem in NP.
NP-hard problems are those at least as hard as NP problems; i.e., all NP problems can be reduced (in polynomial time) to them. NP-hard problems need not be in NP; i.e., they need not have solutions verifiable in polynomial time.
For instance, the Boolean satisfiability problem is NP-complete by the Cook–Levin theorem, so any instance of any problem in NP can be transformed mechanically into a Boolean satisfiability problem in polynomial time. The Boolean satisfiability problem is one of many NP-complete problems. If any NP-complete problem is in P, then it would follow that P = NP. However, many important problems are NP-complete, and no fast algorithm for any of them is known.
From the definition alone it is unintuitive that NP-complete problems exist; however, a trivial NP-complete problem can be formulated as follows: given a Turing machine M guaranteed to halt in polynomial time, does a polynomial-size input that M will accept exist? It is in NP because (given an input) it is simple to check whether M accepts the input by simulating M; it is NP-complete because the verifier for any particular instance of a problem in NP can be encoded as a polynomial-time machine M that takes the solution to be verified as input. Then the question of whether the instance is a yes or no instance is determined by whether a valid input exists.
The first natural problem proven to be NP-complete was the Boolean satisfiability problem, also known as SAT. As noted above, this is the Cook–Levin theorem; its proof that satisfiability is NP-complete contains technical details about Turing machines as they relate to the definition of NP. However, after this problem was proved to be NP-complete, proof by reduction provided a simpler way to show that many other problems are also NP-complete, including the game Sudoku discussed earlier. In this case, the proof shows that a solution of Sudoku in polynomial time could also be used to complete Latin squares in polynomial time. This in turn gives a solution to the problem of partitioning tri-partite graphs into triangles, which could then be used to find solutions for the special case of SAT known as 3-SAT, which then provides a solution for general Boolean satisfiability. So a polynomial-time solution to Sudoku leads, by a series of mechanical transformations, to a polynomial time solution of satisfiability, which in turn can be used to solve any other NP-problem in polynomial time. Using transformations like this, a vast class of seemingly unrelated problems are all reducible to one another, and are in a sense "the same problem".
Harder problems
Although it is unknown whether P = NP, problems outside of P are known. Just as the class P is defined in terms of polynomial running time, the class EXPTIME is the set of all decision problems that have exponential running time. In other words, any problem in EXPTIME is solvable by a deterministic Turing machine in O(2p(n)) time, where p(n) is a polynomial function of n. A decision problem is EXPTIME-complete if it is in EXPTIME, and every problem in EXPTIME has a polynomial-time many-one reduction to it. A number of problems are known to be EXPTIME-complete. Because it can be shown that P ≠ EXPTIME, these problems are outside P, and so require more than polynomial time. In fact, by the time hierarchy theorem, they cannot be solved in significantly less than exponential time. Examples include finding a perfect strategy for chess positions on an N × N board and similar problems for other board games.
The problem of deciding the truth of a statement in Presburger arithmetic requires even more time. Fischer and Rabin proved in 1974 that every algorithm that decides the truth of Presburger statements of length n has a runtime of at least for some constant c. Hence, the problem is known to need more than exponential run time. Even more difficult are the undecidable problems, such as the halting problem. They cannot be completely solved by any algorithm, in the sense that for any particular algorithm there is at least one input for which that algorithm will not produce the right answer; it will either produce the wrong answer, finish without giving a conclusive answer, or otherwise run forever without producing any answer at all.
It is also possible to consider questions other than decision problems. One such class, consisting of counting problems, is called #P: whereas an NP problem asks "Are there any solutions?", the corresponding #P problem asks "How many solutions are there?". Clearly, a #P problem must be at least as hard as the corresponding NP problem, since a count of solutions immediately tells if at least one solution exists, if the count is greater than zero. Surprisingly, some #P problems that are believed to be difficult correspond to easy (for example linear-time) P problems. For these problems, it is very easy to tell whether solutions exist, but thought to be very hard to tell how many. Many of these problems are #P-complete, and hence among the hardest problems in #P, since a polynomial time solution to any of them would allow a polynomial time solution to all other #P problems.
Problems in NP not known to be in P or NP-complete