From: lexfridman
The P vs NP problem is one of the most fundamental and enduring questions in theoretical computer science. It explores the relationship between two classes of problems: those that can be solved efficiently (in polynomial time), known as P, and those for which a proposed solution can be checked quickly, known as NP.
Understanding P and NP
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P (Polynomial Time): This is the class of all decision problems for which there exists a deterministic algorithm that can solve the problem in time that is polynomial with respect to the size of the input. For example, algorithmic operations like sorting or basic arithmetic can generally be performed in polynomial time [01:12:06].
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NP (Non-deterministic Polynomial Time): This is the class of decision problems for which, given a “yes” answer, there exists a proof of this answer (called a “witness”) that can be checked quickly (in polynomial time) by a deterministic algorithm. A classic example of an NP problem is factoring a large integer into its prime components; verifying that given factors are correct can be done efficiently, but finding the factors might not be [01:14:45].
The Core Question
The central question of the P vs NP problem is: Is P equal to NP? In other words, if every problem for which a solution can be verified quickly (in NP) can also be solved quickly (in P).
Currently, the consensus in the theoretical computer science community is that P is not equal to NP, although there is no formal proof to confirm this. This question remains unproven and is considered one of the greatest unsolved problems in computer science [01:15:57].
Implications of P = NP
If it were proven that P = NP, it would imply that every problem for which a solution can be quickly checked can also be quickly solved. This would revolutionize fields that rely on solving complex problems, such as cryptography, optimization, and automated theorem proving [01:18:00].
Practical Perspective
Despite the theoretical uncertainty, most researchers operate under the assumption that P does not equal NP, as many problems in NP seem to lack efficient solving algorithms. This assumption guides much of the practical work surrounding algorithms and computational complexity.
Conclusion
The P vs NP problem touches on deep philosophical and practical aspects of computation, touching on everything from the feasibility of certain kinds of scientific progress to the security of encrypted communications. Its resolution, either confirming or denying that P equals NP, would have profound implications for both the fields of computation and philosophy [01:24:22].
Further Exploration
For more in-depth insight into complexity classes and computational resources, the Complexity Zoo is a comprehensive online resource detailing various complexity classes [01:09:52].