In computer science, a polynomial-time approximation scheme is a type of approximation algorithm for 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. For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most L, with L being the length of the shortest tour. There exists also PTAS for the class of all dense constraint satisfaction problems. 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 or even O 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. 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 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 tractablewith respect to the standard parameterization. Both the knapsack problem and bin packing problem admit an FPTAS. Any strongly NP-hard optimization problem with a polynomially bounded objective function cannot have an FPTAS unless P=NP. 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. Unless P = NP, it holds that FPTAS ⊊ PTAS ⊊ APX. 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. 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.
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, 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.
