PTAS reduction

In computational complexity theory, a PTAS reduction is a reduction that is often used to perform reductions between solutions to optimization problems. It preserves the property that a problem has a polynomial time approximation scheme (PTAS) and is used to define completeness for certain classes of optimization problems such as APX. Notationally, if there is a PTAS reduction from a problem A to a problem B, we write $ext\{A\}\; leq\_\{\; ext\{PTAS\; ext\{B\}$.

With ordinary polynomial-time many-one reductions, if we can describe a reduction from a problem A to a problem B, then any polynomial-time solution for B can be composed with that reduction to obtain a polynomial-time solution for the problem A. Similarly, our goal in defining PTAS reductions is so that given a PTAS reduction from an optimization problem A to a problem B, a PTAS for B can be composed with the reduction to obtain a PTAS for the problem A.

Formally, we define a PTAS reduction from A to B using three polynomial-time computable functions, "f", "g", and "α", with the following properties:

* "f" maps instances of problem A to instances of problem B.
* "g" takes an instance "x" of problem A, an approximate solution to the corresponding problem $f(x)$ in B, and an error parameter ε and produces an approximate solution to "x".
* "α" maps error parameters for solutions to instances of problem A to error parameters for solutions to problem B.
* If the solution "y" to $f(x)$ (an instance of problem B) is at most $1\; +\; alpha(epsilon)$ times worse than the optimal solution, then the corresponding solution $g(x,y,epsilon)$ to "x" (an instance of problem A) is at most $1\; +\; epsilon$ times worse than the optimal solution.

From this definition it is straightforward to show that:
* $ext\{A\}\; leq\_\{\; ext\{PTAS\; ext\{B\}$ and $ext\{B\}\; in\; ext\{PTAS\}\; implies\; ext\{A\}\; in\; ext\{PTAS\}$
* $ext\{A\}\; leq\_\{\; ext\{PTAS\; ext\{B\}$ and $ext\{A\}\; otin\; ext\{PTAS\}\; implies\; ext\{B\}\; otin\; ext\{PTAS\}$
* $ext\{A\}\; leq\_\{\; ext\{PTAS\; ext\{B\}$ and $ext\{B\}\; in\; ext\{APX\}\; implies\; ext\{A\}\; in\; ext\{APX\}$

References

* Ingo Wegener. Complexity Theory: Exploring the Limits of Efficient Algorithms. ISBN 3540210458. Chapter 8, pp.110–111. [http://books.google.com/books?vid=ISBN3540210458 Google Books preview]