Polynomial-time approximation scheme

Last updated December 20, 2024

In computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems).

A PTAS is an algorithm which takes an instance of an optimization problem and a parameter $ε > 0$ and produces a solution that is within a factor $1 + ε$ of being optimal (or $1 - ε$ for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most $(1 + ε) L$ , with $L$ being the length of the shortest tour.^[1]

The running time of a PTAS is required to be polynomial in the problem size for every fixed ε, but can be different for different ε. Thus an algorithm running in time $O (n 1/ε)$ or even $O (n exp(1/ε))$ counts as a PTAS.

Variants

Deterministic

A practical problem with PTAS algorithms is that the exponent of the polynomial could increase dramatically as ε shrinks, for example if the runtime is $O (n (1/ε)!)$ . One way of addressing this is to define the efficient polynomial-time approximation scheme or EPTAS, in which the running time is required to be $O (n c)$ for a constant $c$ independent of $ε$ . This ensures that an increase in problem size has the same relative effect on runtime regardless of what ε is being used; however, the constant under the big-O can still depend on ε arbitrarily. In other words, an EPTAS runs in FPT time where the parameter is ε.

Even more restrictive, and useful in practice, is the fully polynomial-time approximation scheme or FPTAS, which requires the algorithm to be polynomial in both the problem size $n$ and $1/ε$ .

Unless P = NP, it holds that FPTAS ⊊ PTAS ⊊ APX .^[2] Consequently, under this assumption, APX-hard problems do not have PTASs.

Another deterministic variant of the PTAS is the quasi-polynomial-time approximation scheme or QPTAS. A QPTAS has time complexity $n polylog (n)$ for each fixed $ε > 0$ . Furthermore, a PTAS can run in FPT time for some parameterization of the problem, which leads to a parameterized approximation scheme.

Randomized

Some problems which do not have a PTAS may admit a randomized algorithm with similar properties, a polynomial-time randomized approximation scheme or PRAS. A PRAS is an algorithm which takes an instance of an optimization or counting problem and a parameter $ε > 0$ and, in polynomial time, produces a solution that has a high probability of being within a factor $ε$ of optimal. Conventionally, "high probability" means probability greater than 3/4, though as with most probabilistic complexity classes the definition is robust to variations in this exact value (the bare minimum requirement is generally greater than 1/2). Like a PTAS, a PRAS must have running time polynomial in $n$ , but not necessarily in $ε$ ; with further restrictions on the running time in $ε$ , one can define an efficient polynomial-time randomized approximation scheme or EPRAS similar to the EPTAS, and a fully polynomial-time randomized approximation scheme or FPRAS similar to the FPTAS.^[3]

As a complexity class

The term PTAS may also be used to refer to the class of optimization problems that have a PTAS. PTAS is a subset of APX, and unless P = NP, it is a strict subset. ^[2]

Membership in PTAS can be shown using a PTAS reduction, L-reduction, or P-reduction, all of which preserve PTAS membership, and these may also be used to demonstrate PTAS-completeness. On the other hand, showing non-membership in PTAS (namely, the nonexistence of a PTAS), may be done by showing that the problem is APX-hard, after which the existence of a PTAS would show P = NP. APX-hardness is commonly shown via PTAS reduction or AP-reduction.

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References

↑ Sanjeev Arora, Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems, Journal of the ACM 45(5) 753–782, 1998.
1 2 Jansen, Thomas (1998), "Introduction to the Theory of Complexity and Approximation Algorithms", in Mayr, Ernst W.; Prömel, Hans Jürgen; Steger, Angelika (eds.), Lectures on Proof Verification and Approximation Algorithms, Lecture Notes in Computer Science, vol. 1367, Springer, pp. 5–28, doi:10.1007/BFb0053011, ISBN 9783540642015 . See discussion following Definition 1.30 on p. 20.
↑ Vazirani, Vijay V. (2003). Approximation Algorithms. Berlin: Springer. pp. 294–295. ISBN 3-540-65367-8.

External links

Complexity Zoo: PTAS, EPTAS.
Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski, and Gerhard Woeginger, A compendium of NP optimization problems – list which NP optimization problems have PTAS.

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[1] Sanjeev Arora, Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems, Journal of the ACM 45(5) 753–782, 1998.

[Jansen-2] 1 2 Jansen, Thomas (1998), "Introduction to the Theory of Complexity and Approximation Algorithms", in Mayr, Ernst W.; Prömel, Hans Jürgen; Steger, Angelika (eds.), Lectures on Proof Verification and Approximation Algorithms, Lecture Notes in Computer Science, vol. 1367, Springer, pp. 5–28, doi:10.1007/BFb0053011, ISBN 9783540642015 . See discussion following Definition 1.30 on p. 20.

[vvv-3] Vazirani, Vijay V. (2003). Approximation Algorithms. Berlin: Springer. pp. 294–295. ISBN 3-540-65367-8.

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