What do you mean by polynomial time?
(definition) Definition: When the execution time of a computation, m(n), is no more than a polynomial function of the problem size, n. More formally m(n) = O(nk) where k is a constant.
What is a polynomial time approximation scheme PTAS for an NP hard problem?
Polynomial Time Approximation Scheme (PTAS) is a type of approximate algorithms that provide user to control over accuracy which is a desirable feature. In FPTAS, algorithm need to polynomial in both the problem size n and 1/ε. Example (0-1 knapsack problem): We know that 0-1 knapsack is NP Complete.
Do approximation algorithms run in polynomial time?
An approximation algorithm guarantees to run in polynomial time though it does not guarantee the most effective solution.
What is a polynomial time reduction?
In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. If both the time required to transform the first problem to the second, and the number of times the subroutine is called is polynomial, then the first problem is polynomial-time reducible to the second.
What is meant by approximation algorithm?
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solution to the optimal one.
What is approximation ratio in DAA?
Intuitively, the approximation ratio measures how bad the approximate solution is distinguished with the optimal solution. A large (small) approximation ratio measures the solution is much worse than (more or less the same as) an optimal solution.
What do you mean by polynomial time approximation algorithm?
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). …
What is polynomial time and exponential time?
Polynomial time. A polynomial is a sum of terms that look like Constant * x^k Exponential means something like Constant * k^x. (in both cases, k is a constant and x is a variable). The execution time of exponential algorithms grows much faster than that of polynomial ones.
Which is the most restrictive polynomial time approximation scheme?
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.
Is the running time of a PRAS polynomial?
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.
Are there any problems that cannot be solved in polynomial time?
\\(NP\\)-hard problems are a vast family of problems that, to the best of our knowledge, cannot be solved in polynomial time.1When presented with a \\(NP\\)-hard problem, we can take one of three possible strategies: Run a super-polynomial algorithm anyway.
Which is an example of an approximation algorithm?
The above definition of an approximation algorithm gives no measure of how good the algorithm actually is. For instance, a valid approximation algorithm for the Graph Coloring problem is to color each vertex with its own color. Thus, we need a way to differentiate approximation algorithms based on how “good” they actually are.
