Complexity Classes

To formalize the concepts of whether algorithms can be solved in polynomial time (or, whether a solution can be verified in polynomial time), we define the concept of complexity classes. Watch the following videos to learn more:

P vs. NP

The P vs. NP problem is one of the most fundamental and profound questions in computer science and mathematics, concerning the relationship between two classes of problems: those that can be solved in polynomial time (P) and those for which a solution can be verified in polynomial time (NP).

Specifically, the problem asks whether every problem whose solution can be verified quickly (in polynomial time) can also be solved quickly (in polynomial time), which would imply P = NP.

Despite significant efforts, no one has yet proven whether P equals NP or not. If P were equal to NP, it would mean that many complex problems in fields such as cryptography, optimization, and algorithm design could be solved efficiently, drastically altering our approach to computation.

Conversely, if P is not equal to NP, it would affirm the inherent difficulty of these problems, implying that some problems can be verified quickly but not solved quickly. This unresolved question continues to be a central focus of theoretical computer science, with far-reaching implications across numerous domains.

Watch the following videos to learn more about the P vs. NP problem: