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5 Prove that the Traveling Salesman Problem, (finding shortest tour

5 Prove that the Traveling Salesman Problem, (finding shortest tour

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5. Prove that the Traveling Salesman Problem, (finding

shortest tour visiting n points in the plane) is in class NP

a) Optimization formulation

b) Decision formulation

c) Polynomial-size certificate

d) Polynomial time verification algorithm

Solution

Prove that TSP is NP-complete

Let first try to prove that TSP belongs to NP.

If we check the tour contains each vertex one for any trip

in the TSP problem.

Thus we sum up total cost of edges and we will finally find

the minimum cost.

This can be done in polynomial time and thus TSP is in NP

Next we will prove that iTSp is NP hard. We can do it

either by proving that hamilton cycle TSP.

Assume G =(V,E) to be in instance of hamiltionian cycle.

We will create a complete graph CG = (V’,E’)

Total cost will be given as

t(I,j) = 0 if (I,j) belongs to E

t(I,j) = 1 if(I,j) doesnt belong to E

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5. Prove that the Traveling Salesman Problem, (finding shortest tour visiting n points in the plane) is in class NP a) Optimization formulation b) Decision formulation c) Polynomial-size certificate d) Polynomial time verification algorithm Solution Prove that TSP is NP-complete Let first try to prove that TSP belongs to NP. If we check the tour contains each vertex one for any trip in the TSP problem. Thus we sum up total cost of edges and we will finally find the minimum cost. This can be done in polynomial time and thus TSP is in NP Next we will prove that iTSp is NP hard. We can do it ei ...
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