33 Table 52 Shows the comparison of all presented algorithms on graph G

# 33 table 52 shows the comparison of all presented

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33 Table 5.2: Shows the comparison of all presented algorithms on graph (G) Algorith m type Size of vertex- cover Solution set Complexity Remarks Branch and bound 5 {a,d,g,h,k} It grows exponentially fast with problem size for all values of c. 1. BB is complete algorithm that is ensured to find the minimum vertex cover. 2. If no vertex cover of the desired size is found, some covering marks have to be removed and be placed elsewhere, i.e. the algorithm has to backtrack. Approxi mation 10 6 6 {a,b,c,d,e,g,h, i,j,k} {b,c,g,h,i,k} {b,c,e,f,i,j} O(V+E) 1. This gives different solutions but all solutions near to optimal. 2. This is a polynomial-time 2- approximation algorithm means that the solution returned by algorithm is at most twice the size of an optimal. Greedy Clever greedy 7 5 {a,b,c,g,h,i,k} {a,d,g,h,k} O( V+E) O (logV) 1. It is easy to find in some situations where this algorithm fails to yield a optimal solution. 2. Greedy algorithm is not a 2- approximation 3. Clever greedy algorithm always gives solutions better than simple greedy. Genetic 6 {a,c,d,g,h,k} Time complexity measured by the overall number of candidate solutions examined until the optimum is found. 1. GA is an optimization technique based on the natural evolution. 2. GA fails to obtain consistent results for specific type of regular graphs. 3. For large problems, the growth of the number of evaluations required by GA becomes faster. 4. It gives better results when it is combined in to local optimization technique.
34 5 .2 Alom’s extended algorithm for Vertex Cover Problem Alom‘s algorithm is extended in order to give the all possible vertex cover for undirected graph. The reason behind that for some larger graphs this algorithm may be fails to give exact optimal solution. This algorithm gives all minimal vertex covers and minimum vertex covers. This paper presents a formal description of the algorithm. Given a simple graph G with |V|= n vertices and |E|= m edges, this algorithm finds every possible minimal vertex cover. This is followed by a small example illustrating the steps of the algorithm. OPTIMAL-VERTEX-COVER (V, E) 1. For i=1, 2, 3……n in turn 2. G ' = V {v i } and E {e E: v i e} 3. Apply the Algo1 on G ' 4. VC=V ' {v i }. 5. Return VC Primal- dual O(V log V+E) 1. It reduces the degree of infeasibility of the primal one at the same time. 2. The algorithm terminates as soon as the primal solution is feasible. 3. The final dual solution is used as a lower-bound for the optimum solution value by means of weak duality. Alom‘s 5 {a,d,g,h,k} O(E) 1. It gives always optimal solution to the given graph. 2. Complexity is same as with approximation algorithm. 3. For larger graphs, may be this algorithm lost to give an optimal solution.
35 Algo1(V ' , E ' ) { 1. V ' ø 2. E ' E [ G ' ] 3. While E ' ø { 4. Count incident edges of all vertices of graph G ' 5. Vm Choose a vertex which has maximum degree in the current graph; 6. If (More than one vertex have maximum number of edges) then 7. V m Choose that vertex which has at least one edge that is not covered by others which have maximum number of edges. Otherwise choose an arbitrary edge. 8. V ' = V ' V m . 9. Remove the all incident edges of vertex V m } 10. Return V ' } Figure 5.2: