The travelling salesman problem

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Presentation transcript:

The travelling salesman problem Finding a lower bound To find a lower bound for the weight of the minimum Hamiltonian cycle: Choose an arbitrary node. Delete that node and all its arcs. Find the length of the minimum connector for the remaining arcs. Add the weights of the two least weight arcs from the deleted node to the weight of the minimum connector.

The travelling salesman problem Finding a lower bound Example A B C D E 3 5 4 2 6 First, delete A and its arcs.

The travelling salesman problem Finding a lower bound Weight of minimum connector = 11 Example Lower bound = 11 + 2 + 3 = 16 3 B A 2 4 2 5 5 5 C 4 6 6 E D First, delete A and its arcs. Find the weight of the minimum connector for the remaining network. Add the two least weight arcs from A to give a lower bound.

The travelling salesman problem Finding a lower bound Example A B C D E 3 5 4 2 6 Now, delete B and its arcs.

The travelling salesman problem Finding a lower bound Weight of minimum connector = 11 Example Lower bound = 11 + 2 + 3 = 16 3 B A 2 4 2 5 5 5 C 4 6 6 E D Now, delete B and its arcs. Find the weight of the minimum connector for the remaining network. Add the two least weight arcs from B to give a lower bound.

The travelling salesman problem Finding a lower bound Example A B C D E 3 5 4 2 6 Now, delete C and its arcs.

The travelling salesman problem Finding a lower bound Weight of minimum connector = 10 Example Lower bound = 10 + 2 + 4 = 16 3 B A 2 4 2 5 5 5 C 4 6 6 E D Now, delete C and its arcs. Find the weight of the minimum connector for the remaining network. Add the two least weight arcs from C to give a lower bound.

The travelling salesman problem Finding a lower bound Example A B C D E 3 5 4 2 6 Now, delete D and its arcs.

The travelling salesman problem Finding a lower bound Weight of minimum connector = 10 Example Lower bound = 10 + 2 + 4 = 16 3 B A 2 4 2 5 5 5 C 4 6 6 E D Now, delete D and its arcs. Find the weight of the minimum connector for the remaining network. Add the two least weight arcs from D to give a lower bound.

The travelling salesman problem Finding a lower bound Example A B C D E 3 5 4 2 6 Finally, delete E and its arcs.

The travelling salesman problem Finding a lower bound Weight of minimum connector = 7 Example Lower bound = 7 + 5 + 5 = 17 3 B A 2 4 2 5 5 5 C 4 6 6 E D Finally, delete E and its arcs. Find the weight of the minimum connector for the remaining network. Add the two least weight arcs from E to give a lower bound.

The travelling salesperson problem Finding a lower bound Vertex deleted Lower bound A 16 B C D E 17 The greatest lower bound is 17, by deleting vertex E. So the lower bound for the travelling salesperson problem is 17.