Parsimonious reduction

In computational complexity theory and game complexity, a parsimonious reduction is a transformation from one problem to another (a reduction) that preserves the number of solutions. Informally, it is a bijection between the respective sets of solutions of two problems. A general reduction from problem ${\displaystyle A}$ to problem ${\displaystyle B}$ is a transformation that guarantees that whenever ${\displaystyle A}$ has a solution ${\displaystyle B}$ also has at least one solution and vice versa. A parsimonious reduction guarantees that for every solution of ${\displaystyle A}$, there exists a unique solution of ${\displaystyle B}$ and vice versa.

Parsimonious reductions are commonly used in computational complexity for proving the hardness of counting problems, for counting complexity classes such as #P. Additionally, they are used in game complexity, as a way to design hard puzzles that have a unique solution, as many types of puzzles require.