How do you find the maximum independent set?
A maximum independent line set of ‘G’ with maximum number of edges is called a maximum independent line set of ‘G’. L3 is the maximum independent line set of G with maximum edges which are not the adjacent edges in graph and is denoted by β1 = 3. Line independent number (Matching number) = β1 = [n/2] α1 + β1 = n.
What is approximation algorithm explain with an example?
A simple example of an approximation algorithm is one for the minimum vertex cover problem, where the goal is to choose the smallest set of vertices such that every edge in the input graph contains at least one chosen vertex.
What is maximum independent set problem?
The Maximum Independent Set (MIS) problem in graph theory is the task of finding the largest independent set in a graph, where an independent set is a set of vertices such that no two vertices are adjacent. There is currently no known efficient algorithm to find maximum independent sets.
What is 2 approximation algorithm?
An algorithm with approximation ratio k is called a k-approximation algorithm; both algorithms above would be called 2-approximation algorithms. When the approximation ratio is close to 1, it is often more useful to look at the approximation error, which is defined as the approximation ratio minus 1.
Is maximum independent set NP-complete?
Maximum independent sets and maximum cliques The independent set decision problem is NP-complete, and hence it is not believed that there is an efficient algorithm for solving it. The maximum independent set problem is NP-hard and it is also hard to approximate.
Which one is better approximation algorithm?
Therefore, a 1-approximation algorithm gives an optimal solution. Some problems have polynomial-time approximation algorithm with small constant approximate ratios, while others have best-known polynomial time approximation algorithms whose approximate ratios grow with n.
Why do you need approximation algorithms give using any one example and its algorithm how the use of approximation improves solution for NP hard problems?
The goal of the approximation algorithm is to come close as much as possible to the optimal solution in polynomial time. Features of Approximation Algorithm : An approximation algorithm guarantees to run in polynomial time though it does not guarantee the most effective solution.
Is maximum independent set NP?
The maximum independent set problem is NP-hard. However, it can be solved more efficiently than the O(n2 2n) time that would be given by a naive brute force algorithm that examines every vertex subset and checks whether it is an independent set.
What is the maximum size of an independent set in the following tree?
1 Answer. The maximum size of the independent set in this tree is 10. This can be obtained by the following dynamic programming over tree: for each vertex, we will calculate the maximum independent set of a subtree of this vertex with this vertex included and without.
What is Alpha approximation algorithm?
An algorithm is a factor α approximation (α-approximation algorithm) for a problem iff for every instance of the problem it can find a solution within a factor α of the optimum solution. If the problem is a maximization, α < 1 and this definition guarantees that the approximate solution is at least α times the optimum.
What is epsilon approximation?
– An epsilon-approximate algorithm is an f(n)-approximate with f(n) <= epsilon for some constant epsilon. scheme is a polynomial time approx. scheme iff for every (fixed) e > 0, the computing time of A(e) is polynomial in the problem size.