Polynomial-time approximation scheme
{{short description|Type of approximation algorithm}}
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 {{math|ε > 0}} and produces a solution that is within a factor {{math|1 + ε}} of being optimal (or {{math|1 – ε}} for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most {{math|(1 + ε)L}}, with {{mvar|L}} being the length of the shortest tour.Sanjeev Arora, Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems, Journal of the ACM 45(5) 753–782, 1998.
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 {{math|O(n{{sup|1/ε}})}} or even {{math|O(n{{sup|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 {{math|O(n{{sup|(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 {{math|O(n{{sup|c}})}} for a constant {{mvar|c}} independent of {{math|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 {{mvar|n}} and {{math|1/ε}}.
Unless P = NP, it holds that {{nowrap|FPTAS ⊊ PTAS ⊊ APX}}.{{citation|first=Thomas|last=Jansen|contribution=Introduction to the Theory of Complexity and Approximation Algorithms|pages=5–28|title=Lectures on Proof Verification and Approximation Algorithms|series=Lecture Notes in Computer Science |editor1-first=Ernst W.|editor1-last=Mayr|editor2-first=Hans Jürgen|editor2-last=Prömel|editor3-first=Angelika|editor3-last=Steger|editor3-link=Angelika Steger|publisher=Springer|year=1998|volume=1367 |isbn=9783540642015|doi=10.1007/BFb0053011}}. See discussion following Definition 1.30 on [https://books.google.com/books?id=_C8Ly1ya4cgC&pg=PA20 p. 20]. 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 {{math|n{{sup|polylog(n)}}}} for each fixed {{math|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 {{math|ε > 0}} and, in polynomial time, produces a solution that has a high probability of being within a factor {{math|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 {{mvar|n}}, but not necessarily in {{math|ε}}; with further restrictions on the running time in {{math|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.{{cite book
| last = Vazirani
| first = Vijay V.
| title = Approximation Algorithms
| publisher = Springer
| year = 2003
| pages = 294–295
| location = Berlin
| isbn = 3-540-65367-8
}}
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.
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.
See also
- Parameterized approximation scheme, an approximation scheme that runs in FPT time
References
External links
- Complexity Zoo: [https://complexityzoo.net/Complexity_Zoo:P#ptas PTAS], [https://complexityzoo.net/Complexity_Zoo:E#eptas EPTAS].
- Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson, Marek Karpinski, and Gerhard Woeginger, [https://www.csc.kth.se/tcs/compendium/ A compendium of NP optimization problems] – list which NP optimization problems have PTAS.
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