Polynomial-time approximation scheme

In computer science, 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, in polynomial time, 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] There exists also PTAS for the class of all dense constraint satisfaction problems (CSPs).[2]

The running time of a PTAS is required to be polynomial in n for every fixed ε but can be different for different ε. Thus an algorithm running in time O(n1/ε) or even O(nexp(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(nc) 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. 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. All problems in FPTAS are fixed-parameter tractable with respect to the standard parameterization. For example, the knapsack problem admits an FPTAS.[3]

Any strongly NP-hard optimization problem with a polynomially bounded objective function cannot have an FPTAS unless P=NP.[4] However, the converse fails: e.g. if P does not equal NP, knapsack with two constraints is not strongly NP-hard, but has no FPTAS even when the optimal objective is polynomially bounded.[5]

Unless P = NP, it holds that FPTAS ⊊ PTAS ⊊ APX.[6] 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 for each fixed .

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.[4]

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. [6]

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.

References
  1. Sanjeev Arora, Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems, Journal of the ACM 45(5) 753–782, 1998.
  2. Arora, S.; Karger, D.; Karpinski, M. (1999), "Polynomial Time Approximation Schemes for Dense Instances of NP-Hard Problems", Journal of Computer and System Sciences, 58 (1): 193–210, doi:10.1006/jcss.1998.1605
  3. Vazirani, Vijay (2001). Approximation algorithms. Berlin: Springer. pp. 69–70. ISBN 3540653678. OCLC 47097680.
  4. Vazirani, Vijay V. (2003). Approximation Algorithms. Berlin: Springer. pp. 294–295. ISBN 3-540-65367-8.
  5. H. Kellerer and U. Pferschy and D. Pisinger (2004). Knapsack Problems. Springer.CS1 maint: uses authors parameter (link)
  6. 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, Springer, pp. 5–28, doi:10.1007/BFb0053011, ISBN 9783540642015. See discussion following Definition 1.30 on p. 20.
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