Heuristics for the Mirrored Traveling Tournament Problem

1. Heuristics for the Mirrored Traveling Tournament Problem Celso C. RIBEIRO Sebastián URRUTIA

2. Summary • The Mirrored Traveling Tournament Problem • Constructive heuristic • Extended GRASP + ILS heuristic • Computational results

3. Motivation • The total distance traveled by teams in round robin tournaments is an important variable to be minimized, in order to reduce traveling costs and to give more time to the players for resting and training.

4. The Traveling Tournament Problem • The Traveling Tournament Problem (TTP) consists in generating an schedule for a tournament between n teams subject to: • The tournament is a time constrained double round-robin tournament: • There are exactly 2(n-1) rounds. • Each team plays against every other team twice, one at home and the other away. • No team can play more than three consecutive home or more than three consecutive away games. • No repeaters are allowed (A at B followed by B at A). • The goal is to minimize the total distance traveled by all teams during the tournament.

5. The Mirrored Traveling Tournament Problem (MTTP) has an additional constraint: The tournament is mirrored, i.e.: All teams face each other once in the first phase with n-1 rounds. In the second phase, with the last n-1 rounds, the teams play each other again in the same order, following an inverted home/away pattern. Common structure in Latin-American tournaments. The set of feasible solutions for the MTTP is a subset of the set of feasible solutions for the TTP. The Mirrored Traveling Tournament Problem

6. The Mirrored Traveling Tournament Problem • Some references: • Easton, Nemhauser, & Trick, “The traveling tournament problem: Description and benchmarks” (2001) • Trick, “Challenge traveling tournament instances”, web page: http://mat.gsia.cmu.edu/TOURN/ • Anagnostopoulos, Michel, Van Hentenryck, & Vergados, “A simulated annealing approach to the traveling tournament problem” (2003)

7. 1-Factorizations • Given a graph G=(V, E), a factor of G is a graph G’=(V,E’) with E’E. • G’ is a 1-factor if all its nodes have degree equal to one. • A factorization of G=(V,E) is a set of edge-disjoint factors G1=(V,E1), ..., Gp=(V,Ep), such that E1...Ep=E. • All factors in a 1-factorization of G are 1-factors.

8. 1-Factorizations 1 2 5 4 3 Example: 1-factorization of K6 6

9. 1-Factorizations 1 1 2 5 4 3 Example: 1-factorization of K6 6

10. 1-Factorizations 2 1 2 5 4 3 Example: 1-factorization of K6 6

11. 1-Factorizations 3 1 2 5 4 3 Example: 1-factorization of K6 6

12. 1-Factorizations 4 1 2 5 4 3 Example: 1-factorization of K6 6

13. 1-Factorizations 5 1 2 5 4 3 Example: 1-factorization of K6 6

14. 1-Factorizations • Mirrored tournament: games in the second phase are determined by those in the first. • If each edge of Kn represents a game, • each 1-factor of Kn represents a round and • each ordered 1-factorization of Kn represents a feasible schedule for n teams.

15. Constructive heuristic • Three steps: • Schedule games using abstract teams (structure of the draw). • Assign real teams to abstract teams. • Select stadium for each game (home/away pattern) in the first phase (mirrored tournament).

16. Constructive heuristic • Step 1: schedule games using abstract teams • This phase creates the structure of the tournament. • “Polygon method” is used. • Tournament structure is fixed and will not change in the other steps of the constructive heuristic.

17. Constructive heuristic

18. Constructive heuristic • Step 2: assign real teams to abstract teams • Build a matrix with the number of consecutive games for each pair of abstract teams: • For each pair of teams X and Y, an entry in this matrix contains the total number of times in which the other teams play consecutively with X and Y in any order.

19. Constructive heuristic

20. Constructive heuristic • Step 2: assign real teams to abstract teams • Greedily assign pairs of real teams with close home cities to pairs of abstract teams with large entries in the matrix with the number of consecutive games.

21. Constructive heuristic

22. Constructive heuristic • Step 3: select stadium for each game in the first phase of the tournament: • Two-part strategy: • Build a feasible assignment of stadiums, starting from a random assignment in the first round. • Improve the assignment of stadiums, performing a simple local search algorithm based on home-away swaps.

23. Constructive heuristic

24. Neighborhoods • Neighborhood “home-away swap”(HAS): select a game and exchange the stadium where it takes place. • Neighborhood “team swap” (TS): select two teams and swap their games; also swap the home-away assignment of their own game.

25. Neighborhoods • Neighborhood “partial round swap” (PRS): select two games AxB and CxD from round X and two games AxC and BxD from round Y, and swap their rounds (only for n8, not always possible).

26. Neighborhoods • Neighborhood “partial round swap” (PRS): select two games AxB and CxD from round X and two games AxC and BxD from round Y, and swap their rounds (only for n8, not always possible).

27. Neighborhoods • Neigborhood “game rotation” (GR) (ejection chain): • Enforce a game to be played at some round: add a new edge to a 1-factor of the 1-factorization associated with the current schedule. • Use an ejection chain to recover a 1-factorization.

28. Neighborhoods 2 1 2 5 4 3 6 Enforce game 1vs. 3 at round (factor) 2.

29. Neighborhoods 2 1 2 5 4 3 6 Teams 1 and 3 are now playing twice in this round.

30. Neighborhoods 2 1 2 5 4 3 6 Eliminate the other games played by teams 1 and 3 in this round.

31. Neighborhoods 2 1 2 5 4 3 6 Enforce the former oponents of teams 1 and 3 to play each other in this round: new game 2 vs. 4 in this round.

32. Neighborhoods 4 1 2 5 4 3 6 Consider the factor where game 2 vs. 4 was scheduled.

33. Neighborhoods 4 1 2 5 4 3 6 Enforce game 1 vs. 4 (eliminated from round 2) to be played in this round.

34. Neighborhoods • Continue with the applications of these steps, until the game enforced in the beginning is removed from the round where it was played in the original schedule. • Only movements in neighborhoods PRS and GR are able to change the structure of the schedule of the initial solution built by the “polygon method”. • However, PRS cannot always be used, due to the structure of the solutions built by “polygon method” for several values of n. • n = 6, 8, 12, 14, 16, 20, 24 • PRS moves may appear after an ejection chain move is made. • The ejection chain move is able to find solutions that are not reachable through other neighborhoods.

35. GRASP + ILS heuristic • The constructive heuristic and the neighborhoods were used to develop a hybrid improvement heuristic for the MTTP: • This heuristic is based on the GRASP and ILS metaheuristics. • Initial solutions: randomized version of the constructive heuristic. • Local search: use TS, HAS, PRS and HAS cyclically in this order until a local optimum for all neighborhoods is found. (do not search in GR!!!) • Perturbation: random movement in GR neighborhood. • Algorithm fully described in the paper.

36. GRASP + ILS heuristic while .not.StoppingCriterion S  GenerateRandomizedInitialSolution() S  LocalSearch(S) repeat S’  Perturbation(S,history) S’  LocalSearch(S’) S  AceptanceCriterion(S,S’,history) S*  UpdateBestSolution(S,S*) until ReinitializationCriterion end

37. Computational results • Benchmark circular instances with n = 12, 14, 16, 18, and 20 teams. • Harder benchmark MLB instances with n = 12, 14, and 16 teams. • All available from http://mat.gsia.cmu.edu/TOURN/ • 2003 edition of the Brazilian national soccer championship with 24 teams.

38. Computational results • All numerical results on a Pentium IV 2.0 MHz machine. • Comparisons with best known approximate solutions for the corresponding less constrainednot necessarily mirrored instances.

39. Computational results • Constructive heuristic: • Very fast • Instance MLB16: 1000 runs in approximately 1 second • Average gap is 17.1% • Better solutions than those found after several days of computations by some metaheuristic aproachs to the not necessarily mirrrored version of the problem

40. Computational results • GRASP + ILS heuristic: time limit is 10 minutes only • Largest gap with respect to the best known solution for the less constrained not necessarily mirrored problem was 9,5%. (before this work, times were measured in days!)

41. Computational results

42. Computational results

43. Computational results

44. Computational results • New heuristic improved by 3.9% and 1.4% the best known solutions for the corresponding less constrained unmirrored instances circ18 and circ20. • Computation times are smaller than computation time of other heuristics, e.g. for instance MLB14: • Anagnostopoulos et al. (2003): approximately five days of computation time • GRASP + ILS: 10 minutes

45. Computational results • Total distance traveled for the 2003 edition of the Brazilian soccer championship with 24 teams (instance br24) in 15 min. (Pentium IV 2.0 MHz): Our solution: 506,433 kms Realized (official draw): 1, 048,134 kms (52% reduction) • Approximate corresponding potential savings in airfares:US$ 1,700,000

46. Concluding Remarks • Constructive heuristic is very fast and effective. • GRASP + ILS heuristic found very good solutions to benchmark instances: • Very fast (10 minutes) • Solutions found for some instances are even better than those available for the corresponding less constrained not necessarily mirrored instances. • Optimal solution for MLB and circ instances with n = 4 and 6 • For a new class of easier instances the heuristic found the optimal solution for n = 4, 6, 8, 10,12 and 16. • Urrutia & Ribeiro, “Minimizing travels by maximizing breaks in round robin tournament schedules” (2004)

47. Concluding Remarks • Effectiveness of the ejection chain neighborhood. • Mirrored schedules are good schedules. • Significant savings in airfare costs and traveled distance in the real instance.