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Fix the driver behind crashes, sound loss and screen glitchesFind Drivers →Repair Windows errors before they cause bigger problemsFix Now →P vs NP asks whether every problem whose answer can be checked efficiently can also be solved efficiently. Computer scientists know that P is contained in NP, but they have not proved whether P and NP are equal. The Clay Mathematics Institute currently lists the problem as unsolved.
What do P and NP mean?
P and NP are classes of decision problems: questions whose answers are yes or no. In the standard model, input size is measured by the length of its encoding, and “polynomial time” means the running time is bounded by a polynomial in that length.
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P: problems that can be solved in polynomial time
P contains decision problems that a deterministic algorithm can decide in polynomial time. Informally, the algorithm works through the input and returns yes or no within a number of steps that grows polynomially as the input gets larger.
NP: problems with efficiently checkable certificates
NP is the class decidable by a nondeterministic polynomial-time algorithm. An equivalent way to understand it is that, for a yes answer, there is a certificate—a proposed solution of polynomial length—that a deterministic algorithm can verify in polynomial time. NP does not mean “not polynomial,” and it does not mean that every problem in the class is known to be difficult.
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This formal framing depends on a computational model, usually a Turing machine, and on measuring time relative to input length. Stephen Cook’s official problem description sets out these definitions.
Why ask whether checking is easier than solving?
A proposed solution may be easy to verify even when finding one seems harder. The Clay Mathematics Institute illustrates this with a Hamiltonian path: given a proposed route through a graph’s cities, it is straightforward to check whether it visits every city exactly once. Finding such a route may appear much more challenging. That contrast motivates the question; it does not prove that Hamiltonian path is outside P.
“Efficient” here is a mathematical, asymptotic standard, not a promise that an algorithm will feel fast on every input. A polynomial-time algorithm can still be impractical for real workloads if its polynomial has a large exponent, its constants are high, or the inputs are already large.
What is known about P vs NP?
P is a subset of NP: if an algorithm can solve a problem in polynomial time, its answer can also be checked in polynomial time. The unanswered part is whether every problem in NP is also in P—that is, whether P = NP.
The Clay Mathematics Institute currently lists P vs NP as unsolved and includes it among the Millennium Prize Problems. The Institute says Stephen Cook and Leonid Levin formulated the question independently in 1971. Its P vs NP overview presents the central issue as whether a solution that is easy to check is also easy to find.
What would each possible answer mean?
If P = NP
Every decision problem in NP would have a polynomial-time algorithm. That would be a major result about the relationship between finding and checking solutions, but it would not automatically make every such algorithm useful in practice. Polynomial growth can still be too steep for realistic input sizes, and the claim concerns worst-case asymptotic running time.
If P ≠ NP
At least one problem whose solutions can be verified in polynomial time would have no polynomial-time decision algorithm. This would establish a limit on efficient solving, although it would not by itself identify which real-world instances are hard or rule out useful methods for particular cases.
Why does the question matter beyond theory?
The answer would clarify fundamental limits on efficient algorithms and the power of verification. It also matters to fields such as optimization and cryptography, but practical consequences depend on the exact algorithms, assumptions, and reductions involved. A proof that P = NP would not, on its own, show that all encryption becomes breakable or that every difficult optimization task becomes practical to solve.
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Work around the question continues. A 2025 Clay Mathematics Institute workshop, P vs NP and Complexity Lower Bounds, covered directions including circuit lower bounds, meta-complexity, algebraic and geometric complexity theory, hardness of approximation, and proof complexity. These are research areas related to understanding computational limits, not evidence that the central question has been settled.
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