Abstract
In this paper, we have proposed an algorithm based on min-to-min approach. In the proposed algorithm first the degree of each vertex of the graph is calculated. Next the vertex with minimum degree is selected, after which all the neighbors of the minimum degree are located. In the neighbors of the minimum degree vertex, again the vertex with the minimum degree is found and put into the set minimum vertex cover and deleted from the graph. Again, the degree of each vertex of the updated graph is calculated and again the same process is repeated until the graph becomes empty. In case of tie, all the neighbors of the minimum degree vertices are computed and then the minimum degree vertex in all of them is added to minimum vertex degree set. The same process is repeated until the graph becomes empty. The proposed algorithm is a very simple, efficient, and easy to understand and implement. The proposed min-to-min algorithm is evaluated on small as well as on large benchmark instances and the results indicate that the performance of the min-to-min algorithm is far better as compared to the other state-of-art algorithms in term of accuracy and computation complexity. We have also used the proposed method to solve the maximum independent set problem.
Highlights
A graph in field of computer science is the set of vertices, and collection of edges each of which connects a pair of vertices [1]
A graph is represented as G (V, E), where V denotes the set of vertices and E the collection of edges[2] .In graph theory a vertex cover is defined as the subset Vc ⸦ V, such that the vertices in the subset Vc covers all the edges E in the graph G(V, E)
Maximum independent set (MIS) problem can be converted to clique problem as well, minimum vertex cover can be converted into maximum independent set (MIS) problem in polynomial time, the above problems are interchangeable and are considered as NP-complete problems [10]
Summary
A graph in field of computer science is the set of vertices, and collection of edges each of which connects a pair of vertices [1]. Maximum independent set (MIS) problem can be converted to clique problem as well, minimum vertex cover can be converted into MIS problem in polynomial time, the above problems are interchangeable and are considered as NP-complete problems [10]. The exact algorithms will always provide a solution that is optimal, but the computation time increases exponentially with the size of the problem. The exact algorithms are best option for small size problems where an optimal solution is needed without any time constraints. The approximation algorithms are preferably the better option for researcher to come up with a solution to NP-complete problems [18,19]. Due to simplicity and intelligent selection of vertices in vertex cover set it save the time and improve the performance in term of optimality of the proposed algorithm.
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