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The travelling salesman problem

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.

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The travelling salesman problem

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  1. 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.

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

  3. 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.

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

  5. 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.

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

  7. 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.

  8. 3 B A 2 4 2 5 5 5 C 4 6 6 E D The travelling salesman problem Finding a lower bound Example Now, delete D and its arcs.

  9. 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.

  10. 3 B A 2 4 2 5 5 5 C 4 6 6 E D The travelling salesman problem Finding a lower bound Example Finally, delete E and its arcs.

  11. 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.

  12. The travelling salesperson problem Finding a lower bound The greatest lower bound is 17, by deleting vertex E. So the lower bound for the travelling salesperson problem is 17.

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