An algorithm X of an undirected graph G returns, Vc. Vc is a set of vertices of G and every edge in G has at least one of its endpoints in Vc. In other words, the algorithm finds a set of the fewest vertices such that every edge includes at least one of the vertices of G in the set, Vc. The following is a pseudo-code of a proposed algorithm for the vertex cover problem using the degree of each vertex. Vc = { } //Vc is a set of vertices covering edges V' = All the vertices in G E' = All the edges in G While E' is not empty Find a vertex v of the highest possible degree in E' Add v to Vc. Remove vertex v from V' and remove all the edges that hit v from E' return Vc. Vc needs to cover all of the edges in G. This proposed algorithm is based on the idea that selecting a vertex including the most edges at each step of the while loop will result in covering the most edges. With that in mind, please answer the following three questions: What type (approach) of the algorithm is the pseudo code above? Does it result in finding the smallest vertex cover in the end? If so, briefly describe how it finds them along with its time complexity in your own words. If not, give a counterexample along with your best-guessed complexity of solving this problem in your own words.

Computer Networking: A Top-Down Approach (7th Edition)

7th Edition

ISBN:9780133594140

Author:James Kurose, Keith Ross

Publisher:James Kurose, Keith Ross

Chapter1: Computer Networks And The Internet

Section: Chapter Questions

Problem R1RQ: What is the difference between a host and an end system? List several different types of end...

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An algorithm X of an undirected graph G returns, Vc. Vc is a set of vertices of G and every edge in G has at least one of its endpoints in Vc. In other words, the algorithm finds a set of the fewest vertices such that every edge includes at least one of the vertices of G in the set, Vc.

The following is a pseudo-code of a proposed algorithm for the vertex cover problem using the degree of each vertex.

Vc = { } //Vc is a set of vertices covering edges

V' = All the vertices in G

E' = All the edges in G

While E' is not empty

Find a vertex v of the highest possible degree in E'

Add v to Vc.

Remove vertex v from V' and remove all the edges that hit v from E'

return Vc.

Vc needs to cover all of the edges in G. This proposed algorithm is based on the idea that selecting a vertex including the most edges at each step of the while loop will result in covering the most edges.

With that in mind, please answer the following three questions:

What type (approach) of the algorithm is the pseudo code above?

Does it result in finding the smallest vertex cover in the end? If so, briefly describe how it finds them along with its time complexity in your own words. If not, give a counterexample along with your best-guessed complexity of solving this problem in your own words.