Augoust 2004 1/77 Heuristics for the Mirrored Traveling Tournament Problem Celso C. RIBEIRO Sebastián URRUTIA
Jan 02, 2016
Augoust 2004 1/77
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
Example: 1-factorization of K6
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4 3
2
1
5
6
2
1-Factorizations
Example: 1-factorization of K6
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4 3
2
1
5
6
3
1-Factorizations
Example: 1-factorization of K6
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4 3
2
1
5
6
4
1-Factorizations
Example: 1-factorization of K6
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4 3
2
1
5
6
5
1-Factorizations
Example: 1-factorization of K6
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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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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.
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Constructive heuristic
4 3
2
1
5
6
Example: “polygon method” for n=6
1st round
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Constructive heuristic
3 2
1
5
4
6
Example: “polygon method” for n=6
2nd round
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Constructive heuristic
2 1
5
4
3
6
Example: “polygon method” for n=6
3rd round
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Constructive heuristic
1 5
4
3
2
6
Example: “polygon method” for n=6
4th round
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Constructive heuristic
5 4
3
2
1
6
Example: “polygon method” for n=6
5th round
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Constructive heuristic
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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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.
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Constructive 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
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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.
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Constructive heuristic
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)
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)
Round FLU SAN FLA GRE PAL PAY
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)
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 “team swap” (TS):
select two teams and swap their games; also swap the home-away assignment of their own game. 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 “team swap” (TS):
select two teams and swap their games, also swap the home-away assignment of their own game. Real teams (n=6)
Round FLU SAN FLA GRE PAL PAY
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
Max breaks for TTP-constrainedMDRR tournaments
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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
Max breaks for TTP-constrainedMDRR tournaments
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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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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
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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!)
Computational results
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Computational results
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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Computational results
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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Computational results
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
Computational results
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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)