1. 1 State of the art for TSP TSP instances of thousand of cities can be consistently solved to optimality. Instances of up to 25000 cities have been solved: All cities in Sweden! Technique: Branch-and-Bound combined with cutting plane algorithms: Branch- and-cut.

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4. 4 x 2 · 2 is a valid inequality and a cutting plane

5. 5 Valid inequality for an ILP Given integer linear program, a new inequality is called valid if it does not change the set of integer solutions to the program. It is a cutting plane if it removes the (non- integer) optimum solution to the relaxed linear program.

6. 6 Cutting plane algorithm for ILP input ILP instance P P’ := LP-relaxation of P while(optimal solution to P’ is not integer){ add valid cutting plane to P P’ := LP-relaxation of P } return optimal solution to P’ How to find valid cutting plane?

7. 7 Gomory Cutting Plane Algorithm Assume that all initial coefficients in (standard form) ILP instance are integer. Solve LP-instance using simplex method. Suppose the simplex method terminates with a non-integer optimal solution.

8. 8 Some line in final dictionary reads x i = b i +  j a j x j where b i is not integer. Let b’ i = b i - b b i c and a’ j = a j - b a j c. x i - b b i c -  j b a j c x j = b’ i +  j a j ’ x j For every feasible integer solution: Left hand side is integer. Right hand side is strictly bigger than zero. x i - b b i c -  j b a j c x j ¸ 1 is a valid cut.

9. 9 Gomory Cutting Plane Algorithm For ILP Partial correctness is obvious. Termination is not guaranteed. Can be guaranteed if simplex algorithm uses special pivot rule and line of final dictionary is chosen carefully (non-trivial!) Not very practical.

10. 10 Cutting planes for TSP (and other problems) Use ILP formulation of TSP. Use specially tailored cutting planes.

11. 11 TSP as ILP

12. 12 Symmetric TSP Distance from i to j is equal to distance from j to i. Let y ij = x ij + x ji

13. 13 The TSP tour has to pass every cut at least twice. S T  i 2 S, j 2 T y ij ¸ 2 is a valid inequality.

14. 14 Cutting plane algorithm for TSP Find optimal solution x * of LP-relaxation of Let y * ij = x * ij + x * ji Check if some S,T has  i 2 S, j 2 T y ij < 2. This can be efficiently done by Max flow algorithm. Add inequality  i 2 S, j 2 T y ij ¸ 2.

15. 15 Cutting plane algorithm for TSP At some point we will fail to find new cuts. Still may not have integer solution. If not, add other kinds of specially tailored inequalities (big literature) or BRANCH.

16. 16 Branch-and-cut for TSP Branch-and-bound with relaxation being LP-relaxation + set of valid inequalities. When to stop adding inequalities and start branching is a matter of heuristics and experiments. Yields state-of-the art solver. Many non- trivial implementation issues.