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. There exists also PTAS for the class of all dense CSP 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(n1/ε) or even O(nexp(1/ε)) counts as a PTAS.
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. An example of a problem that has an FPTAS is the knapsack problem.
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.
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