Heuristics for the Mirrored Traveling Tournament Problem

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Heuristics for the Mirrored Traveling

Tournament Problem

Celso C. RIBEIROSebastián URRUTIA

Augoust 2004 2/77

Summary• Motivation• Formulation• Constructive heuristic• Neighborhoods• Extended GRASP + ILS heuristic• A new class of instances • Computational results• Concluding remarks

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Motivation

• Professional sports leagues are a major economic activity around the world.

• Teams and leagues do not want to waste their investments in players and structure as a consequence of poor schedules of games.

• A tournament schedule determines at which round and in which stadium each game takes place.

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Motivation• Tournament scheduling is a difficult

task, involving different types of constraints, multiple objectives to optimize, and several decision makers (officials, managers, TV, etc…).– Decision makers may have opposite

goals.– Economic issues.– Logistic issues.– Fairness.

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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.

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Formulation• Conditions:

– n (even) teams take part in a tournament.– Each team has its own stadium at its home

city. – Each team is located at its home city in the

beginning, to where it returns at the end.– 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.

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• 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

once in every round)• 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

Open problem:Is the TTP NP-Hard?

– Hard problem: previous largest instance exactly solved to date had only n=6 teams! (n=8 with 20 processors in 4 days CPU time)

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• 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

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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)

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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.

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4 3

2

1

5

6

1-Factorizations

Example: 1-factorization of K6

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4 3

2

1

5

6

1

1-Factorizations


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4 3

2

1

5

6

2

1-Factorizations


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4 3

2

1

5

6

3

1-Factorizations


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4 3

2

1

5

6

4

1-Factorizations


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4 3

2

1

5

6

5

1-Factorizations


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• 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.

– Without considering the stadiums, there are (n-1)! times (number of nonisomorphic 1-factorizations) different “mirrored tournaments”.Dinitz, Garnick, & McKay, “There are 526,915,620 nonisomorphic one-factorizations of K12” (1995)

1-Factorizations

Open problem:How many schedules exist for a single round robin tournament with n teams?

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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).

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• 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.

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4 3

2

1

5

6

Example: “polygon method” for n=6

1st round

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3 2

1

5

4

6


2nd round

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2 1

5

4

3

6


3rd round

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1 5

4

3

2

6


4th round

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5 4

3

2

1

6


5th round

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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

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• 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.

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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

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• 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.

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0 4 4 4 4 4 4 3 4 4 3 4 4 4 4 44 0 2 25 0 0 0 0 0 0 0 0 0 0 25 24 2 0 2 25 0 0 0 0 0 0 0 0 0 0 254 25 2 0 2 25 0 0 0 0 0 0 0 0 0 04 0 25 2 0 2 25 0 0 0 0 0 0 0 0 04 0 0 25 2 0 2 25 0 0 0 0 0 0 0 04 0 0 0 25 2 0 2 25 0 0 0 0 0 0 03 0 0 0 0 25 2 0 2 26 0 0 0 0 0 04 0 0 0 0 0 25 2 0 1 26 0 0 0 0 04 0 0 0 0 0 0 26 1 0 2 25 0 0 0 03 0 0 0 0 0 0 0 26 2 0 2 25 0 0 04 0 0 0 0 0 0 0 0 25 2 0 2 25 0 04 0 0 0 0 0 0 0 0 0 25 2 0 2 25 04 0 0 0 0 0 0 0 0 0 0 25 2 0 2 254 25 0 0 0 0 0 0 0 0 0 0 25 2 0 24 2 25 0 0 0 0 0 0 0 0 0 0 25 2 0

n = 16: note the large number of times inwhich two teams are faced consecutively, which is explored by step 2 of the constructiveheuristic.

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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

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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.

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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

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Neighborhoods

• Neighborhood “home-away swap” (HAS): select a game and exchange the stadium where it takes place.

Real teams (n=6)


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

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Neighborhoods

• Neighborhood “home-away swap” (HAS): select a game and exchange the stadium where it takes place.

Real teams (n=6)


1/6 PAY PAL GRE@FL

A@SA

N@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

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Neighborhoods• Neighborhood “team swap” (TS):

select two teams and swap their games, also swap the home-away assignment of their own game. Real teams (n=6)


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

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select two teams and swap their games; also swap the home-away assignment of their own game. Real teams (n=6)


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

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select two teams and swap their games, also swap the home-away assignment of their own game. Real teams (n=6)


1/6 PAY@PA

LSAN

@FLA

GRE@FL

U

2/7 GRE@FL

APAY

@FLU

SAN@PA

L

3/8@SA

NFLU PAL PAY

@FLA

@GRE

4/9 PAL@GR

E@FL

USAN

@PAY

FLA

5/10@FL

APAY GRE

@PAL

FLU@SA

N

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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

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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

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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.

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Neighborhoods

4 3

2

1

5

6

2

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

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4 3

2

1

5

6

2

Neighborhoods

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

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4 3

2

1

5

6

2

Neighborhoods

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

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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.

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4 3

2

1

5

6

4

Neighborhoods

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

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Neighborhoods

4 3

2

1

5

6

4

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

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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.

Open problem:Is the GR neighborhood complete?

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Iterated Local Search

S GenerateInitialSolution() S LocalSearch(S)repeat

S’ Perturbation(S,history)S’ LocalSearch(S’)S AceptanceCriterion(S,S’,history)S* UpdateBestSolution(S,S*)

until StoppingCriterion

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GRASP

while .not.StoppingCriterionS GenerateRandomizedInitialSolution() S LocalSearch(S)S* UpdateBestSolution(S,S*)

end

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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 + fast tabu search to restore feasibility.

– Algorithm fully described in the paper.

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Extended GRASP + ILS heuristic

while .not.StoppingCriterionS GenerateRandomizedInitialSolution() S LocalSearch(S)repeat

S’ Perturbation(S,history)S’ LocalSearch(S’)S AceptanceCriterion(S,S’,history)S* UpdateBestSolution(S,S*)

until ReinitializationCriterionend

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Test Instances

• 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.

• New uniform instances.

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Home-away pattern (HAP)

• Matrix with as many rows as teams (n) and as many columns as rounds in the tournament.

• Each row of a HAP is a sequence of H’s and A’s.

• An H (resp. A) in position r of row t means that team t has a home (resp. an away) game in round r.

• A team has a break in round r if it has two consecutive home (or away) games in rounds r-1 and r.

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HAP & Breaks

• Single round-robin tournament (SRR):– Each team plays every other team

exactly once in n-1 prescheduled rounds.

– There are no two equal rows in a HAP of an SRR tournament (every two teams have to play against each other at some round)

Team/Round

1 2 3

1 H H H

2 A H A

3 A A A

4 H A H

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Tournament schedules• Schedule S:

– B(S) = total number of breaks (sum of the number of breaks over all teams in the tournament)• Number of home breaks = number of away

breaks = B(S)/2

– D(S) = total distance traveled (sum of the distances traveled by all teams in the tournament)

– T(S) = total number of travels (number of times any team must travel from one stadium to another)

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Tournament schedules

• Breaks minimization problems:– Schedules with a minimum number of

breaks De Werra (1981,1988): constraints on geographical locations (complementary HAPs for teams in the same location, e.g. Mets and Yankees in NY), teams organized in divisions (weekday vs. weekend games), minimize the number of rounds with breaks

– Minimize breaks when the order of games is fixed Elf, Junger & Rinaldi (2003)

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discounted by the number of teams that do not travel (home breaks)

Connecting breaks with travels

• R = number of rounds• T(S) = n/2 + n(R-1) – B(S)/2 + n/2 =

nR – B(S)/2

travels to play in intermediary rounds if all teams were to travel,

travels after playing the last gametravels to play the first game

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Connecting breaks with distances

• New uniform instances: all distances equal to one.

• In the particular case of a uniform instance:D(S) = T(S)Then, D(S) = nR – B(S)/2

• maximize breaks => minimize travels => => minimize distance traveled for uniform instances

• Motivation: UB to breaks gives LB to distance• Consequence: implications in the solution of

the TTP

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Max breaks for SRR tournaments

• SRR tournaments: maximum number of breaks for any team is (n-2): all home games or all away games

• Only two teams may have (n-2) breaks: all games away and all games at home

• Remaining (n-2) teams: at most (n-3) breaks each

• Upper bound to the number of breaks:UBSRR = 2(n-2) + (n-2)(n-3) = n2 – 3n + 2

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Polygon method

• Upper bound to the number of breaks:UBSRR = 2(n-2) + (n-2)(n-3) = n2 – 3n + 2

• UBSRR bound is tight.

• We use the polygon method to build a schedule with exactly UBSRR breaks.

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Polygon method

• Extend the polygon method giving orientation to each edge

• Edge connecting nodes 1 and n is always oriented from 1 to n (in every round)

• k=2,...,n/2: the edge connecting nodes k and n+1-k is oriented from the even (resp. odd) numbered node to the odd (resp. even) numbered node in even (resp. odd) rounds

• Final extremity of each arc is the home team.

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Polygon method

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Max breaks for TTP-constrainedMDRR tournaments

• Similar tight bounds can also be obtained for equilibrated SRR, DRR, and MDRR tournaments.

• Mirrored DRR tournaments in which each schedule must follow the same constraints of the traveling tournament problem:– No team can play more than three

consecutive home games or more than three consecutive away games.

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• Upper bounds to the number of breaks can be derived using similar (although much more elaborated) counting arguments:

2

2

2

14, if 4

4( ) / 3 4 20, if 1mod3 0 and 4

4( 2 ) / 3, if 1mod3 1

4 / 3 4 , if 1mod3 2

TTP

n

n n n n nUB

n n n

n n n


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• Since T(S) = 2n(n-1) – B(S)/2, the upper bound UBTTP can be used in the computation of lower bounds to T(S) and, for the uniform instances, also to D(S) = T(S).

• Contrarily to the previous problems, a construction method to build schedules for TTP-constrained MDRR tournaments with exactly UBTTP breaks does not seem to exist to date.

• Use an effective TTP heuristic to find good approximate solutions


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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.

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• 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

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• 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!)


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Instance

Best unmirrored

Best mirrored

gap (%)

Time to best (s)

circ12 420 456 8.6 8.5

circ14 682 714 4.7 1.1

circ16 976 1004 2.9 115.3

circ18 1420 1364 -3.9 284.2

circ20 1908 1882 -1.4 578.3

nl12 112298 120655 7.4 24.0

nl14 190056 208086 9.5 69.9

nl16 267194 285614 6.9 514.2

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Instance

Best unmirrored

Best mirrored

gap (%)

Time to best (s)

circ12 420 456 8.6 8.5

circ14 682 714 4.7 1.1

circ16 976 1004 2.9 115.3

circ18 1420 1364 -3.9 284.2

circ20 1908 1882 -1.4 578.3

nl12 112298 120655 7.4 24.0

nl14 190056 208086 9.5 69.9

nl16 267194 285614 6.9 514.2

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Instance

Best unmirrored

Best mirrored

gap (%)

Time to best (s)

circ12 420 456 8.6 8.5

circ14 682 714 4.7 1.1

circ16 976 1004 2.9 115.3

circ18 1364 1364 0.0 284.2

circ20 1882 1882 0.0 578.3

nl12 112298 120655 7.4 24.0

nl14 190056 208086 9.5 69.9

nl16 267194 285614 6.9 514.2

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• 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


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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

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Computational results for Uniform Instances

n D(S) LB gap B(S)

4 17 17 - 14

6 48 48 - 24

8 80 80 - 64

10 130 130 - 100

12 192 192 - 144

14 256 252 4 216

16 342 342 - 276

18 434 432 2 356

20 526 520 6 468

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• 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

• Effectiveness of the ejection chain neighborhood.

• Mirrored schedules are good schedules.• Significant savings in airfare costs and

traveled distance in the real instance.

Concluding Remarks

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Concluding Remarks• Connection between breaks maximization

and distance minimization problems• This connection is used to prove the

optimality of approximate solutions found by an effective heuristic for the TTP.

• New largest TTP instance exactly solved to date: n=16

• In spite of being easier than other classes of TTP instances, uniform instances could not be exactly solved for n > 16.

Open problem:Is the TTP NP-Hard in the special case of uniform instances?

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• Ribeiro & Urrutia, “Heuristics for the Mirrored Traveling Tournament Problem”, PATAT 2004 - Practice and Theory of Automated Timetabling (2004)

• Urrutia & Ribeiro, “Minimizing travels by maximizing breaks in round robin tournament schedules”, Electronic Notes in Discrete Mathematics, (2004)

Documents

Heuristics for the Mirrored Traveling Tournament Problem