Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 A grid implementation of a GRASP-ILS heuristic for the mirrored traveling tournament problem Aletéia ARAÚJO Vinod REBELLO Celso RIBEIRO Sebastián URRUTIA

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Summary Motivation The Mirrored Traveling Tournament Problem Extended GRASP + ILS heuristic –Construction phase –Neighborhoods Parallel implementations of GRASP- ILS –PAR-MP Computational results EsporteMax: optimization in sports management and scheduling

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Motivation Game scheduling is a difficult task, involving different types of constraints, logistic issues, multiple objectives, and several decision makers. Total distance traveled is an important variable to be minimized, to reduce traveling costs and to give more time to the players for resting and training. Timetabling is the major area of applications of OR in sports.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Formulation Traveling Tournament Problem (TTP) –n (even) teams take part in a tournament. –Each team has its own stadium at its home city. –Distances between the stadiums are known. –A team playing two consecutive away games goes directly from one city to the other, without returning to its home city.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Formulation –Tournament is a strict double round- robin tournament: There are 2(n-1) rounds, each one with n/2 games. Each team plays against every other team twice, one at home and the other away. –No team can play more than three games in a home stand (home games) or in a road trip (away games). Goal: minimize the total distance traveled by all teams.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Formulation Mirrored Traveling Tournament Problem (MTTP): –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. Set of feasible solutions for the MTTP is a subset of the feasible solutions for the TTP.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 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 G 1 =(V,E 1 ),..., G p =(V,E p ), such that E 1 ...  E p =E. All factors in a 1-factorization of G are 1-factors. Oriented 1-factorization: assign orientations to the edges of a 1- factorization

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Mirrored tournament: games in the second phase are determined by those in the first. –Each edge of K n represents a game. –Each 1-factor of K n represents a round. –Each ordered oriented 1-factorization of K n represents a feasible schedule for n teams. Example: K 6 1-Factorizations

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 1-Factorizations

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Three steps: 1.Schedule games using abstract teams: polygon method defines the structure of the tournament 2.Assign real teams to abstract teams: greedy heuristic to QAP (number of travels between stadiums of the abstract teams x distances between the stadiums of the real teams) 3.Select stadium for each game (home/away pattern) in the first phase (mirrored tournament): 1.Build a feasible assignment of stadiums, starting from a random assignment of stadiums in the first round. 2.Improve this assignment of stadiums, using a simple local search algorithm based on home- away swaps. Constructive heuristic

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Constructive heuristic Example: “polygon method” for n=6 1 st round

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Constructive heuristic Example: “polygon method” for n=6 2 nd round

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Constructive heuristic Example: “polygon method” for n=6 3 rd round

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Constructive heuristic Example: “polygon method” for n=6 4 th round

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Constructive heuristic Example: “polygon method” for n=6 5 th round

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Constructive heuristic Abstract teams (n=6) RoundABCDEF 1/6FEDCBA 2/7DCBAFE 3/8BAEFCD 4/9EDFBAC 5/10CFAEDB

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 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. –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: QAP heuristic

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Constructive heuristic ABCDEF A B C D E F443430

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Constructive heuristic Real teams (n=6) RoundFLUSA N FLAGREPALPAY 1/6PAYPALGR E FLASA N FLU 2/7GR E FLASA N FLUPAYPAL 3/8SA N FLUPALPAYFLAGRE 4/9PALGREPAYSA N FLUFLA 5/10FLAPAYFLUPALGR E SA N

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 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.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Constructive heuristic Real teams (n=6) RoundFLUSANFLAGREPALPAY L A U A U L N L E Y U FLA A L N

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Neighborhood home-away swap (HAS)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Neighborhood home-away swap (HAS)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Neighborhood team-swap (TS)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Neighborhood team-swap (TS)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Neighborhood partial round swap (PRS)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Neighborhood partial round swap (PRS)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Ejection chain: game rotation (GR) 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.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Ejection chain: game rotation (GR)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Ejection chain: game rotation (GR) Enforce game (1,3) to be played in round 2

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Ejection chain: game rotation (GR) Enforce game (1,3) to be played in round 2

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Ejection chain: game rotation (GR)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Ejection chain: game rotation (GR)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Ejection chain: game rotation (GR)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Ejection chain: game rotation (GR)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Ejection chain: game rotation (GR)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Ejection chain: game rotation (GR)

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Ejection chain: game rotation (GR) Ejection chain moves are able to find solutions unreachable with other neighborhoods.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Neighborhoods Only movements in neighborhoods PRS and GR may change the structure of the initial schedule. However, PRS moves not always exist, due to the structure of the solutions built by polygon method e.g. for 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: escape from local optima

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 GRASP + ILS heuristic Hybrid improvement heuristic for the MTTP: –Combination of GRASP and ILS metaheuristics. –Initial solutions: randomized version of the constructive heuristic. –Local search with first improving move: use TS, HAS, PRS, and HAS cyclically in this order until a local optimum for all neighborhoods is found. –Perturbation: random movement in GR neighborhood. –Detailed algorithm to appear in EJOR.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 GRASP + ILS heuristic while.not.StoppingCriterion S  BuildGreedyRandomizedSolution() S, S  LocalSearch(S) repeat S’  Perturbation(S) S’  LocalSearch(S’) S  AceptanceCriterion(S,S’) S*  UpdateGlobalBestSolution(S,S*) S  UpdateIterationBestSolution(S,S) until ReinitializationCriterion end GRASP construction phase ILS phase

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Parallel implementations of metaheuristics Robustness Granularity: coarse grain implementation suitable to grid environments (communication) Master-slave Single-walk vs. multiple-walk Cooperative vs. independent Cost and frequency of communication: few communication steps Nature of the information to be shared

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Parallel strategy PAR-I Parallel strategy with independent processes. PAR-I is equivalent to running the sequential algorithm simultaneously on multiple machines. After receiving the seed, each process computes a new solution. Then, each process runs an ILS local search phase until the reinitialization criterion is met. Procedure stops when a solution at least as good as a given target is found.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Parallel strategy PAR-O Parallel strategy with one-off cooperation. Identical to PAR-I, except for the first iteration of the main loop. After each process executes the first GRASP construction phase, the initial solution found by each of them is sent to the master. The master selects and broadcats the best initial solution to all procesors.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Parallel strategy PAR-O All workers run the ILS local search phase of the first iteration using the same initial solution. The following iterations are executed independently. Processors stop ILS phase after 50 steps deteriorating solution quality. This strategy is called one-off cooperation because exchange only occurs at the first iteration.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Parallel strategy PAR-1P Master manages the exchange of information collected along the trajectories investigated by each worker. It keeps the best solution found by any worker. Each time the best solution is improved, the master broadcasts its cost to all workers. The idea is to use this information not only to converge faster to a target solution, but also to find better solutions than the independent search strategies.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Parallel strategy PAR-1P Each time a worker completes the ILS phase, it will compare the cost of the solution found with that of the best solution held by the master. If it is better, the worker sends its solution to the master, otherwise the solution is discarded. Then, the worker chooses between two possibilities: –It requests the best solution held by the master to start the ILS local search phase with this solution; or –The worker restarts from the GRASP construction phase Workers indirectly exchange elite solutions found along their search trajectories.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Parallel strategy PAR-MP Master handles a centralized pool of elite solutions, collecting and distributing them upon request. Slaves start their searches from different initial solutions. Slaves exchange and share elite solutions found along their search trajectories. Master updates the pool of elite solutions with a newly received solution according to some criteria based on the quality and diversity of the solutions already in the pool.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Parallel strategy PAR-MP When a slave completes an iteration (ILS phase), it can either request an elite solution from the pool or construct a new initial solution randomly. To guarantee diversity within the pool, the insertion of a new solution depends on the state of the pool and on how this solution was generated. When a slave requests an elite solution from the master, a solution is selected at random from the pool and sent back to it.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Computational results Circular instances with n = 12,..., 20 teams. MLB instances with n = 12,..., 16 teams. –All available from –Largest unmirrored instances exactly solved to date: n=6 (sequential), n=8 (parallel) Random number generator: Mersenne Twister of Matsumoto and Nishimura. Algorithms implemented using C++ and MPI-LAM (version 7.0.6).

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Computational results Numerical results on a dedicated cluster of 1.7 GHz Pentium IV, with 256 Mbytes of RAM. Parameter M (size of the pool) set to the number of slave processes used in the parallel execution. Probability of choosing a solution from the pool fixed at 10%.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Computational results (quality) Instan ce Sequenti al PAR- I/O/1P PAR-MPImprovement (%) circ circ circ circ circ circ circ nl nl nl nl nl br Independent or less- cooperative strategies

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Computational results (quality) Instan ce Sequenti al PAR- I/O/1P PAR-MPImprovement (%) circ circ circ circ circ circ circ nl nl nl nl nl br Best known solution for (unmirrored) TTP

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Computation times (seconds) Instan ce Seq. PAR–IPAR-OPAR-1PPAR-MP 1 pro c. 8 procs. 24 proc s. 8 procs. 24 procs. 8 procs. 24 procs. 8 procs. 24 procs. circ circ circ circ circ nl nl nl

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Computational results (speedups) Instan ce PAR–IPAR-OPAR-1P PAR-MP 8 procs. 24 proc s. 8 procs. 24 procs. 8 procs. 24 procs. 8 procs. 24 proc s. circ nl br

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Computational results InstanceTargetPAR-IPAR-OPAR-1PPAR-MP circ , , , , circ , , , circ181,308 8, , , nl16280,1741, , , br24503,158 7, , , , Times (seconds) to medium targets on 10 processors:

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Computational results InstanceNew best solution value Improvem ent (%) Times (s) circ circ , nl , Best solutions found exclusively by PAR-MP:

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Computational results Instance2 x t(s)TargetPAR-IPAR-OPAR-1P circ162, circ1850, nl1640, Best solutions found by the other strategies in twice the time used by PAR-MP:

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Concluding remarks (1/2) Parallel metaheuristics compared to their sequential counterparts demand more programming and design effort. Cooperative strategies lead to more robust implementations (e.g. PAR-MP) and are able to find better solutions in smaller computation times. Experiments on a “true” grid environment with hundreds of processors.

Grid implementation of a GRASP-ILS heuristic for the mirrored TTP MIC, August /60 Concluding remarks (2/2) EsporteMax: OR applications in sports timetabling and management –FutMax: Number of points a team has to win to be qualified for the play-offs (Brazilian press and TV) –Schedule of the Brazilian national tournaments of basketball (men/women) –Collaboration in the scheduling of the Chilean national soccer tournament –Referee assignment