1. A Linear Search Strategy Using Bounds Sharlee Climer and Weixiong Zhang

2. Sharlee ClimerWashington University in St. Louis2 Overview  Introduction  Example problem  Results for Traveling Salesman Problem  Future Work

3. Sharlee ClimerWashington University in St. Louis3 Introduction  Linear search strategy called cut-and-solve  At each step: A chunk of the solution space is cut away and solved, providing incumbent solutions A relaxed solution is found for remaining solution space  Iterate until relaxed solution is greater than or equal to incumbent  Cut-and-solve may not be useful for simple problem instances

4. Sharlee ClimerWashington University in St. Louis4 Introduction  Use cut-and-solve to solve Linear Programs (LPs)  LPs are useful for modeling: Traveling Salesman Problem Constraint Satisfaction Problem Minimum cost flow problem

5. Sharlee ClimerWashington University in St. Louis5 Introduction  Asymmetric Traveling Salesman Problem (ATSP) can be used to model: No-wait flowshop Stacker crane Tilted drilling machine Computer disk read head Robotic motion Pay phone coin collection

6. Sharlee ClimerWashington University in St. Louis6 Introduction Minimize Z = c ij x ij s.t.: x ij = 1 for j = 1,…,n x ij = 1 for i = 1,…,n x ij <= |W| - 1, for all proper non-empty subsets W of V x ij = {0,1}

7. Sharlee ClimerWashington University in St. Louis7 Introduction  NP-hard  Frequently solved using search trees Branch-and-bound Branch-and-cut  Search strategy Best-first search Depth-first search  Cut-and-solve has minimal memory requirements and no “wrong” subtrees

8. Sharlee ClimerWashington University in St. Louis8 Introduction  ATSP solution space is a high- dimensional convex polyhedron  See paper for algorithm  Simple 2D problem for example

9. Sharlee ClimerWashington University in St. Louis9 Example Minimize Z = y – 4/5 x s.t.: x >= 0 y <= 3 y + 13/6 x <= 9 y – 5/13 x >= 1/14 y + 3/5 x >= 6/5 x,y integers

10. Sharlee ClimerWashington University in St. Louis10 Example x >= 0 y <= 3 y + 13/6 x <= 9 y – 5/13 x >= 1/14 y + 3/5 x >= 6/5 x,y integers

11. Sharlee ClimerWashington University in St. Louis11 Example Minimize Z = y – 4/5 x x = 0 y = 3 Z = 3

12. Sharlee ClimerWashington University in St. Louis12 Example Minimize Z = y – 4/5 x x = 2 y = 1 Z = -0.6

13. Sharlee ClimerWashington University in St. Louis13 Example Minimize Z = y – 4/5 x x = 3.5 y = 1.4 Z = -1.4

14. Sharlee ClimerWashington University in St. Louis14 Example Add new constraint: y – 17/3 x >= -14

15. Sharlee ClimerWashington University in St. Louis15 Example Incumbent solution: x = 3 y = 2 Z = -0.4

16. Sharlee ClimerWashington University in St. Louis16 Example Minimize Z = y – 4/5 x x = 2.6 y = 1.0 Z = -1.1

17. Sharlee ClimerWashington University in St. Louis17 Example Add new constraint:

18. Sharlee ClimerWashington University in St. Louis18 Example New incumbent solution: x = 2 y = 1 Z = -0.6

19. Sharlee ClimerWashington University in St. Louis19 Example Minimize Z = y – 4/5 x x = 1.0 y = 0.6 Z = -0.2 Incumbent solution: Z = -0.6

20. Sharlee ClimerWashington University in St. Louis20 Example  Keep new constraints and incumbent solution from one iteration to the next  Incumbent yields pruning opportunities  No children to choose between  Need to decide size of cut

21. Sharlee ClimerWashington University in St. Louis21 ATSP results  Implemented using cplex with default settings  Compared with Concorde (Applegate, Bixby, Chvatal, & Cook)  Concorde designed for STSP  Used 2-node transformation

22. Sharlee ClimerWashington University in St. Louis22 ATSP results  Testbed: All 27 TSPLIB instances 6 each: rtilt, stilt, crane, disk, coin, shop, and super (Fischetti, Lodi, & Toth, 2002)  Cut-and-solve required 29.7% as much time as Concorde

23. Sharlee ClimerWashington University in St. Louis23 ATSP results rtilt class 100 cities 100 trials

24. Sharlee ClimerWashington University in St. Louis24 ATSP results stilt class

25. Sharlee ClimerWashington University in St. Louis25 ATSP results crane class

26. Sharlee ClimerWashington University in St. Louis26 ATSP results disk class

27. Sharlee ClimerWashington University in St. Louis27 ATSP results coin class

28. Sharlee ClimerWashington University in St. Louis28 ATSP results shop class

29. Sharlee ClimerWashington University in St. Louis29 Future work  Improve ATSP solver: Automatic determination of cut size Use specialized sparse problem solver  Clustering algorithm