Puneet Varma (Editor)

PTAS reduction

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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 .

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 + α ( ϵ ) times worse than the optimal solution, then the corresponding solution g ( x , y , ϵ ) to x (an instance of problem A) is at most 1 + ϵ times worse than the optimal solution.

    • Properties

    From the definition it is straightforward to show that:

  • A ≤ PTAS B and B ∈ PTAS ⟹ A ∈ PTAS
  • A ≤ PTAS B and A ∉ PTAS ⟹ B ∉ 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.

    References

    PTAS reduction Wikipedia
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