In computational complexity theory, a PTAS reduction is an approximation-preserving 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 .
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 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 (an instance of problem B) is at most times worse than the optimal solution, then the corresponding solution to x (an instance of problem A) is at most times worse than the optimal solution.
From the definition it is straightforward to show that:
PTAS reductions are used to define completeness in APX, the class of optimization problems with constant-factor approximation algorithms.
- Crescenzi, Pierluigi (1997). "A Short Guide To Approximation Preserving Reductions". Proceedings of the 12th Annual IEEE Conference on Computational Complexity (Washington, D.C.: IEEE Computer Society): 262–.
- Ingo Wegener. Complexity Theory: Exploring the Limits of Efficient Algorithms. ISBN 3-540-21045-8. Chapter 8, pp. 110–111. Google Books preview
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