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 A ≤ PTAS B {\displaystyle {\text{A}}\leq _{\text{PTAS}}{\text{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.
Definition 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 ) {\displaystyle 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 ) {\displaystyle f(x)} (an instance of problem B) is at most 1 + α ( ϵ ) {\displaystyle 1+\alpha (\epsilon )} times worse than the optimal solution, then the corresponding solution g ( x , y , ϵ ) {\displaystyle g(x,y,\epsilon )} to x (an instance of problem A) is at most 1 + ϵ {\displaystyle 1+\epsilon } times worse than the optimal solution.
Properties From the definition it is straightforward to show that:
A ≤ PTAS B {\displaystyle {\text{A}}\leq _{\text{PTAS}}{\text{B}}} and B ∈ PTAS ⟹ A ∈ PTAS {\displaystyle {\text{B}}\in {\text{PTAS}}\implies {\text{A}}\in {\text{PTAS}}}
A ≤ PTAS B {\displaystyle {\text{A}}\leq _{\text{PTAS}}{\text{B}}} and A ∉ PTAS ⟹ B ∉ PTAS {\displaystyle {\text{A}}\not \in {\text{PTAS}}\implies {\text{B}}\not \in {\text{PTAS}}}
L-reductions imply PTAS reductions. As a result, one may show the existence of a PTAS reduction via a L-reduction instead. PTAS reductions are used to define completeness in APX, the class of optimization problems with constant-factor approximation algorithms.
See also Approximation-preserving reduction L-reduction APX
References
Ingo Wegener. Complexity Theory: Exploring the Limits of Efficient Algorithms. ISBN 3-540-21045-8. Chapter 8, pp. 110–111. Google Books preview
