polynomial-time bounded algorithm for Minimum Vertex Cover

Introduction#

This is a polynomial algorithm for getting the minimum vertex cover of connected undirected graph. The time complexity of this algorithm is O(n2)

Parameters#

Remarks#

The first thing you have to do in this algorithm to get all of the vertices of the graph sorted in descending order according to its degree.

After that you have iterate on them and add each one to final vertices set which don’t have any adjacent vertex in this set.

In the final stage iterate on the final vertices set and remove all of the vertices which have one of its adjacent vertices in this set.

Algorithm Pseudo Code

Algorithm PMinVertexCover (graph G)

Input connected graph G

Output Minimum Vertex Cover Set C

Set C <- new Set<Vertex>() 

Set X <- new Set<Vertex>() 

X <- G.getAllVerticiesArrangedDescendinglyByDegree()

for v in X do
    List<Vertex> adjacentVertices1 <- G.getAdjacent(v)

    if !C contains any of adjacentVertices1 then
        
        C.add(v)

for vertex in C do

    List<vertex> adjacentVertices2 <- G.adjacentVertecies(vertex)

    if C contains any of adjacentVertices2 then
        
        C.remove(vertex)

        
return C

C is the minimum vertex cover of graph G

we can use bucket sort for sorting the vertices according to its degree because the maximum value of degrees is (n-1) where n is the number of vertices then the time complexity of the sorting will be O(n)