grid implementation of a grasp-ils heuristic for the mirrored ttp mic, august 2005 1/60 a grid...

MIC, August 2005 1/60 Grid implementation of a GRASP-ILS heuristic for the mirrored TTP

A grid implementation of a GRASP-ILSheuristic for the mirrored traveling

tournament problem

Aletéia ARAÚJOVinod REBELLOCelso 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


• 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

1-Factorizations

4 3

2

1

5

6


1-Factorizations

4 3

21

5

64 3

2

1

5

64 3

2

1

5

6

4 3

2

1

5

64 3

2

1

5

6


• Three steps:1. Schedule games using abstract teams: polygon

method defines the structure of the tournament2. 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



4 3

2

1

5

6

Example: “polygon method” for n=6

1st round



3 2

1

5

4

6


2nd round



2 1

5

4

3

6


3rd round



1 5

4

3

2

6


4th round



5 4

3

2

1

6


5th round



Abstract teams (n=6)

Round

A B C D E F

1/6 F E D C B A

2/7 D C B A F E

3/8 B A E F C D

4/9 E D F B A C

5/10 C F A E D B


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



A B C D E F

A 0 1 6 5 2 4

B 1 0 2 5 6 4

C 6 2 0 2 5 3

D 5 5 2 0 2 4

E 2 6 5 2 0 3

F 4 4 3 4 3 0


Constructive heuristic Real teams (n=6)

Round

FLU SAN

FLA GRE

PAL PAY

1/6 PAY PAL GRE

FLA SAN

FLU

2/7 GRE

FLA SAN

FLU PAY PAL

3/8 SAN

FLU PAL PAY FLA GRE

4/9 PAL GRE

PAY SAN

FLU FLA

5/10 FLA PAY FLU PAL GRE

SAN


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.


Constructive heuristic Real teams (n=6)

Round FLU SAN FLA GRE PAL PAY

1/6 PAY@PA

LGRE

@FLA

SAN@FL

U

2/7 GRE@FL

ASAN

@FLU

PAY@PA

L

3/8@SA

NFLU

@PAL

PAY FLA@GR

E

4/9 PAL@GR

E@PA

YSAN

@FLU

FLA

5/10@FL

APAY FLU

@PAL

GRE@SA

N


Neighborhood home-away swap (HAS)

4 3

21

5

64 3

2

1

5

64 3

2

1

5

6

4 3

2

1

5

64 3

2

1

5

6


Neighborhood team-swap (TS)

4 3

21

5

64 3

2

1

5

64 3

2

1

5

6

4 3

2

1

5

64 3

2

1

5

6


Neighborhood team-swap (TS)

4 3

12

5

64 3

1

2

5

64 3

1

2

5

6

4 3

1

2

5

64 3

1

2

5

6


Neighborhood partial round swap (PRS)

7 4

3

1

8

6

2

5

7 4

3

1

8

6

2

5


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.



4 3

21

5

64 3

2

1

5

64 3

2

1

5

6

4 3

2

1

5

64 3

2

1

5

6



4 3

21

5

64 3

2

1

5

64 3

2

1

5

6

4 3

2

1

5

64 3

2

1

5

6 Enforce game (1,3) tobe played in round 2



4 3

21

5

64 3

2

1

5

64 3

2

1

5

6

4 3

2

1

5

64 3

2

1

5

6



4 3

21

5

64 3

2

1

5

64 3

2

1

5

6

4 3

2

1

5

64 3

2

1

5

6Ejection 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 heuristicwhile .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 ReinitializationCriterionend

GRASP constructio

n 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 (quality)Instance

Sequential

PAR-I/O/1P

PAR-MP Improvement (%)

circ8 140 140 140 -

circ10 276 272 272 1.45

circ12 456 456 456 -

circ14 714 714 714 -

circ16 1004 984 980 2.39

circ18 1364 1308 1306 4.25

circ20 1882 1882 1882 -

nl8 41928 41928 41928 -

nl10 63832 63832 63832 -

nl12 120655 120655 120655 -

nl14 208086 208086 208086 -

nl16 285614 280174 279618 2.09

br24 506433 503158 503158 0.65

Independent or less-

cooperative strategies


Computational results (quality)Instance

Sequential

PAR-I/O/1P

PAR-MP Improvement (%)

circ8 140 140 140 -

circ10 276 272 272 1.45

circ12 456 456 456 -

circ14 714 714 714 -

circ16 1004 984 980 2.39

circ18 1364 1308 1306 4.25

circ20 1882 1882 1882 -

nl8 41928 41928 41928 -

nl10 63832 63832 63832 -

nl12 120655 120655 120655 -

nl14 208086 208086 208086 -

nl16 285614 280174 279618 2.09

br24 506433 503158 503158 0.65

Best known solution for (unmirrored) TTP


Computation times (seconds)

Instance

Seq.

PAR–I PAR-O PAR-1P PAR-MP

1 proc.

8 proc

s.

24 proc

s.

8 proc

s.

24 proc

s.

8 proc

s.

24 proc

s.

8 proc

s.

24 procs

.circ12 8.5 1.33 0.55 1.32 0.54 1.33 0.35 1.33 0.34

circ14 1.1 0.87 0.73 0.86 0.71 0.65 0.52 0.65 0.51

circ16115.

356.54

24.78

56.58 24.77 55.30 24.45 55.31 24.50

circ18284.

243.19

13.62

43.24 13.62 24.73 13.60 24.70 13.62

circ20578.

3177.2

843.4

7176.6

743.47

106.34

43.47106.3

043.46

nl12 24.0 4.23 1.34 4.23 1.33 3.91 1.33 3.91 1.33

nl14 69.9 5.32 4.34 5.32 4.35 5.85 4.35 5.84 4.34

nl16514.

2474.1

273.6

2474.2

673.58

323.33

73.59322.2

773.54


Computational results (speedups)

Instance

PAR–I PAR-O PAR-1P PAR-MP

8 proc

s.

24 proc

s.

8 proc

s.

24 proc

s.

8 proc

s.

24 proc

s.

8 proc

s.

24 proc

s.circ20

3.2613.3

03.27 13.31 5.43 13.30 5.44

13.32

nl16 1.08 6.98 1.09 6.99 1.59 6.99 1.60 7.01

br246.87

22.16

6.89 22.18 6.93 22.79 6.9622.8

3


Computational results

Instance Target PAR-I PAR-O PAR-1P PAR-MP

circ10 2725,590.5

34,703.6

88,391.1

81,802.4

2

circ16 9845,454.4

71,787.1

82,490.3

1678.31

circ18 1,3088,445.0

627,649.

509,952.2

0269.23

nl16 280,1741,311.4

7793.87

8,188.81

2,205.72

br24 503,1587,250.4

97,082.7

23,082.3

04,460.1

8

Times (seconds) to medium targets on 10 processors:



Instance New best solution value

Improvement (%)

Times (s)

circ16 980 0.40 783.14

circ18 1306 0.15 25,828.60

nl16 279618 0.20 20,512.50

Best solutions found exclusively by PAR-MP:



Instance 2 x t(s) Target PAR-I PAR-O PAR-1P

circ16 2,000 980 990 984 998

circ18 50,000 1306 1308 1308 1384

nl16 40,000 279618 280174 280174 298357

Best solutions found by the other strategies in twice the time used 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

grid implementation of a grasp-ils heuristic for the mirrored ttp mic, august 2005 1/60 a grid...

Documents