Augoust 2004 1/47

Heuristics for the Mirrored Traveling

Tournament Problem

Celso C. RIBEIROSebastián URRUTIA

Augoust 2004 2/47

Summary

• The Mirrored Traveling Tournament Problem

• Constructive heuristic• Extended GRASP + ILS heuristic• Computational results

Augoust 2004 3/47

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.

Augoust 2004 4/47

• 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.

The Traveling Tournament Problem

Augoust 2004 5/47

• 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

Augoust 2004 6/47

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)

Augoust 2004 7/47

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.

Augoust 2004 8/47

4 3

2

1

5

6

1-Factorizations

Example: 1-factorization of K6

Augoust 2004 9/47

4 3

2

1

5

6

1

1-Factorizations

Example: 1-factorization of K6

Augoust 2004 10/47

4 3

2

1

5

6

2

1-Factorizations

Example: 1-factorization of K6

Augoust 2004 11/47

4 3

2

1

5

6

3

1-Factorizations

Example: 1-factorization of K6

Augoust 2004 12/47

4 3

2

1

5

6

4

1-Factorizations

Example: 1-factorization of K6

Augoust 2004 13/47

4 3

2

1

5

6

5

1-Factorizations

Example: 1-factorization of K6

Augoust 2004 14/47

• 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.

1-Factorizations

Augoust 2004 15/47

Constructive heuristic• Three steps:

1. Schedule games using abstract teams (structure of the draw).

2. Assign real teams to abstract teams.

3. Select stadium for each game (home/away pattern) in the first phase (mirrored tournament).

Augoust 2004 16/47

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.

Augoust 2004 17/47

Constructive heuristic Abstract teams (n=6)

Round

A B C D E F

1/6 F E D C B A2/7 D C B A F E3/8 B A E F C D4/9 E D F B A C5/10 C F A E D B

Augoust 2004 18/47

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.

Augoust 2004 19/47

Constructive heuristicA B C D E F

A 0 1 6 5 2 4B 1 0 2 5 6 4C 6 2 0 2 5 3D 5 5 2 0 2 4E 2 6 5 2 0 3F 4 4 3 4 3 0

Augoust 2004 20/47

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.

Augoust 2004 21/47

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

Augoust 2004 22/47

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.

Augoust 2004 23/47

Constructive heuristic Real teams (n=6)

Round FLU SAN FLA GRE PAL PAY1/6 PAY @PA

L GRE @FLA SAN @FL

U2/7 GRE @FL

A SAN @FLU PAY @PA

L3/8 @SA

N FLU @PAL PAY FLA @GR

E4/9 PAL @GR

E@PA

Y SAN @FLU FLA

5/10 @FLA PAY FLU @PA

L GRE @SAN

Augoust 2004 24/47

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.

Augoust 2004 25/47

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 n8, not always possible).

Rounds ATM SAP CON FLA FLU INT CRU GRE1/82/9 FLA @INT @ATM SAP3/104/11 @SAP ATM @INT FLA5/126/137/14

Augoust 2004 26/47

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 n8, not always possible).

Rounds ATM SAP CON FLA FLU INT CRU GRE1/82/9 @SAP ATM @INT FLA3/104/11 FLA @INT @ATM SAP5/126/137/14

Augoust 2004 27/47

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.

Augoust 2004 28/47

Neighborhoods

4 3

2

1

5

6

2

Enforce game 1vs. 3 at round (factor) 2.

Augoust 2004 29/47

4 3

2

1

5

6

2

Neighborhoods

Teams 1 and 3 are now playing twice in this round.

Augoust 2004 30/47

4 3

2

1

5

6

2

Neighborhoods

Eliminate the other games played by teams 1 and 3 in this round.

Augoust 2004 31/47

4 3

2

1

5

6

2

Neighborhoods

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

Augoust 2004 32/47

4 3

2

1

5

6

4

Neighborhoods

Consider the factor where game 2 vs. 4 was scheduled.

Augoust 2004 33/47

Neighborhoods

4 3

2

1

5

6

4

Enforce game 1 vs. 4 (eliminated from round 2) to be played in this round.

Augoust 2004 34/47

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.

Augoust 2004 35/47

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.

Augoust 2004 36/47

GRASP + ILS heuristicwhile .not.StoppingCriterion

S GenerateRandomizedInitialSolution() S LocalSearch(S)repeatS’ Perturbation(S,history)S’ LocalSearch(S’)S AceptanceCriterion(S,S’,history)S* UpdateBestSolution(S,S*)until ReinitializationCriterion

end

Augoust 2004 37/47

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.

Augoust 2004 38/47

Computational results• All numerical results on a Pentium

IV 2.0 MHz machine.• Comparisons with best known

approximate solutions for the corresponding less constrained not necessarily mirrored instances.

Augoust 2004 39/47

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

Augoust 2004 40/47

• 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!)

Computational results

Augoust 2004 41/47

Computational results

Instance

Best unmirrored

Best mirrored

gap (%)

Time to best (s)

circ12 420 456 8.6 8.5circ14 682 714 4.7 1.1circ16 976 1004 2.9 115.3circ18 1420 1364 -3.9 284.2circ20 1908 1882 -1.4 578.3nl12 112298 120655 7.4 24.0nl14 190056 208086 9.5 69.9nl16 267194 285614 6.9 514.2

Augoust 2004 42/47

Computational results

Instance

Best unmirrored

Best mirrored

gap (%)

Time to best (s)

circ12 420 456 8.6 8.5circ14 682 714 4.7 1.1circ16 976 1004 2.9 115.3circ18 1420 1364 -3.9 284.2circ20 1908 1882 -1.4 578.3nl12 112298 120655 7.4 24.0nl14 190056 208086 9.5 69.9nl16 267194 285614 6.9 514.2

Augoust 2004 43/47

Computational results

Instance

Best unmirrored

Best mirrored

gap (%)

Time to best (s)

circ12 420 456 8.6 8.5circ14 682 714 4.7 1.1circ16 976 1004 2.9 115.3circ18 1364 1364 0.0 284.2circ20 1882 1882 0.0 578.3nl12 112298 120655 7.4 24.0nl14 190056 208086 9.5 69.9nl16 267194 285614 6.9 514.2

Augoust 2004 44/47

• 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

Computational results

Augoust 2004 45/47

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

Augoust 2004 46/47

• 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)

Concluding Remarks

Augoust 2004 47/47

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.