Reduktionsäquivalente

Reduktionsäquivalente is a term used in theoretical computer science and complexity theory to describe a problem that can be transformed into another problem in such a way that the solvability of the first problem implies the solvability of the second, and vice-versa. More precisely, if problem A is a reductionsäquivalente to problem B, it means that an algorithm for solving B can be used to solve A, and an algorithm for solving A can be used to solve B. This concept is fundamental to understanding the relationships between different computational problems and is crucial for proving the difficulty of problems.

The process of showing that problem A is a reductionsäquivalente to problem B is typically done by constructing a polynomial-time reduction from A to B. This reduction is a function that takes an instance of problem A and transforms it into an instance of problem B such that the solution to the instance of B can be used to derive the solution to the original instance of A. If such a polynomial-time reduction exists, and if problem B is known to be hard (e.g., NP-hard), then problem A must also be at least as hard as B.

Conversely, if problem B can be reduced to problem A in polynomial time, and problem A is known to be easy (e.g., solvable in polynomial time), then problem B must also be easy. Reduktionsäquivalenz is a symmetric relationship, meaning if A is a reductionsäquivalente to B, then B is also a reductionsäquivalente to A. This equivalence is a powerful tool for classifying problems into complexity classes and for establishing lower bounds on the computational resources required to solve them.