PolynomialzeitNäherungsschemata

Polynomial time refers to a class of computational problems that can be solved by an algorithm whose running time is bounded by a polynomial function of the size of the input. This means that as the input size grows, the time required to solve the problem increases at a rate that is at most proportional to some power of the input size. For example, an algorithm with a running time of O(n^2) or O(n^3), where 'n' is the input size, is considered to be in polynomial time.

The concept of polynomial time is fundamental in complexity theory, a branch of computer science that classifies problems based on the resources (time and space) required to solve them. Problems that can be solved in polynomial time are generally considered to be efficiently solvable or tractable. This is in contrast to problems that require exponential time, where the running time grows much more rapidly with the input size, making them impractical to solve for even moderately large inputs.

The set of all decision problems solvable in polynomial time is known as P. A closely related class is NP, which stands for nondeterministic polynomial time. NP problems are those for which a proposed solution can be verified in polynomial time. The famous P versus NP problem, one of the most important unsolved problems in computer science, asks whether every problem in NP can also be solved in polynomial time (i.e., if P = NP). If P were equal to NP, it would have profound implications, suggesting that many problems currently considered hard could be solved efficiently.