Redukálhatóság

Redukálhatóság, often translated as reducibility, is a fundamental concept in various fields, particularly in theoretical computer science, logic, and mathematics. It describes a relationship between two problems or tasks, indicating that one problem can be transformed into another. Specifically, if problem A is reducible to problem B, it means that a solution to problem B can be used to solve problem A. This transformation typically involves an algorithm that converts an instance of problem A into an instance of problem B, and then uses the solution to the transformed problem B to obtain the solution for the original problem A.

The primary use of reducibility is in understanding the difficulty or complexity of problems. If problem A is reducible to problem B, and problem A is known to be difficult (e.g., NP-complete), then problem B must be at least as difficult as problem A. Conversely, if problem B is known to be easy (e.g., solvable in polynomial time), and problem A is reducible to it, then problem A must also be easy. This allows us to classify problems into complexity classes.

A common type of reducibility is polynomial-time reducibility, denoted by A ≤p B. This means that the transformation from problem A to problem B can be performed in polynomial time with respect to the size of the input to problem A. If A ≤p B and B is in a complexity class C, then A is also in C. This concept is crucial for proving that certain problems are NP-complete. If a known NP-complete problem can be reduced to a new problem in polynomial time, then the new problem is also NP-complete. Reducibility is a powerful tool for both proving hardness and demonstrating solvability.