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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 \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) 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.

Properties

From the definition it is straightforward to show that:

  • \text{A} \leq_{\text{PTAS}} \text{B} and \text{B} \in \text{PTAS} \implies \text{A} \in \text{PTAS}
  • \text{A} \leq_{\text{PTAS}} \text{B} and \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.{{cite book|last1=Crescenzi|first1=Pierluigi|title=Proceedings of Computational Complexity. Twelfth Annual IEEE Conference |chapter=A short guide to approximation preserving reductions |date=1997|pages=262–273|chapter-url=https://scholar.archive.org/work/lksautq5erevpkljbei674cjla|publisher=IEEE Computer Society|location=Washington, D.C.|doi=10.1109/CCC.1997.612321 |isbn=0-8186-7907-7 |s2cid=18911241 }}

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

See also

References

{{Reflist}}

  • Ingo Wegener. Complexity Theory: Exploring the Limits of Efficient Algorithms. {{isbn|3-540-21045-8}}. Chapter 8, pp. 110–111. [https://books.google.com/books?id=1fo7_KoFUPsC Google Books preview]

Category:Reduction (complexity)

Category:Approximation algorithms