PTAS reduction

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 ${\displaystyle {\text{A}}\leq _{\text{PTAS}}{\text{B}}}$.

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.

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 ${\displaystyle f(x)}$ 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 ${\displaystyle f(x)}$ (an instance of problem B) is at most ${\displaystyle 1+\alpha (\epsilon )}$ times worse than the optimal solution, then the corresponding solution ${\displaystyle g(x,y,\epsilon )}$ to x (an instance of problem A) is at most ${\displaystyle 1+\epsilon }$ times worse than the optimal solution.

Properties

From the definition it is straightforward to show that:

• ${\displaystyle {\text{A}}\leq _{\text{PTAS}}{\text{B}}}$ and ${\displaystyle {\text{B}}\in {\text{PTAS}}\implies {\text{A}}\in {\text{PTAS}}}$
• ${\displaystyle {\text{A}}\leq _{\text{PTAS}}{\text{B}}}$ and ${\displaystyle {\text{A}}\not \in {\text{PTAS}}\implies {\text{B}}\not \in {\text{PTAS}}}$

L-reductions imply PTAS reductions. As a result, one may show the existence of a PTAS reduction via a L-reduction instead.[1]

PTAS reductions are used to define completeness in APX, the class of optimization problems with constant-factor approximation algorithms.