- A binary relation P(x,y) is in TFNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(x,y) holds given both x and y, and for every x, there exists a y such that P(x,y) holds.
The job of an TFNP algorithm is to establish, given an x give one possible value for a y such that P(x,y) holds.
In contrast to FNP, there is no known recursive enumeration of machines for problems in TFNP. It seems that such classes, and therefore TFNP, do not have complete problems.
- Megiddo and Papadimitriou. A Note on Total Functions, Existence Theorems and Computational Complexity (1989).