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A grid implementation of a GRASP-ILS heuristic for the mirrored traveling tournament problem

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A grid implementation of a GRASP-ILS

heuristic for the mirrored traveling

tournament problem

Aletéia ARAÚJO

Vinod REBELLO

Celso RIBEIRO

Sebastián URRUTIA

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

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.

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.

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.

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.

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.
- Oriented 1-factorization: assign orientations to the edges of a 1-factorization

1-Factorizations

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- Mirrored tournament: games in the second phase are determined by those in the first.
- Each edge of Kn represents a game.
- Each 1-factor of Kn represents a round.
- Each ordered oriented 1-factorization of Kn represents a feasible schedule for n teams.
- Example: K6

Constructive heuristic

- Three steps:
- Schedule games using abstract teams: polygon method defines the structure of the tournament
- 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)
- Select stadium for each game (home/away pattern) in the first phase (mirrored tournament):
- Build a feasible assignment of stadiums, starting from a random assignment of stadiums in the first round.
- Improve this assignment of stadiums, using a simple local search algorithm based on home-away swaps.

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

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.

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.

Ejection chain: game rotation (GR)

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Enforce game (1,3) to

be played in round 2

Ejection chain: game rotation (GR)

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Enforce game (1,3) to

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Ejection chain: game rotation (GR)

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Ejection chain moves are able to find solutions unreachable with other neighborhoods.

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

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.

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

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

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.

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.

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.

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.

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.

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.

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.

Computational results

- Circular instances with n = 12, ..., 20 teams.
- MLB instances with n = 12, ..., 16 teams.
- All available from http://mat.gsia.cmu.edu/TOURN/
- 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).

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%.

Computational results

Times (seconds) to medium targets on 10 processors:

Computational results

Best solutions found exclusively by PAR-MP:

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.

Concluding remarks (2/2)

- EsporteMax: OR applications in sports timetabling and management
- FutMax: http://www.futmax.org

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

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