may 2004 minimizing travels by maximizing breaks1/32 minimizing travels by maximizing breaks in...

May 2004 Minimizing travels by maximizing breaks1/32

Minimizing Travels by Maximizing Breaks in Round Robin Tournament Schedules

Celso RIBEIROUFF and PUC-Rio, Brazil

Sebastián URRUTIAPUC-Rio, Brazil


Summary• Motivation• Tournament schedules and

the traveling tournament problem• Connecting breaks with distances• Maximum number of breaks for SRR

tournaments• Polygon method• Maximum number of breaks for TTP-

constrained MDRR tournaments• Numerical results• Concluding remarks


Motivation• Motivation for this work:

– Context: research group on applications of OR techniques to problems in sports management and scheduling

– Effective algorithms for the Traveling Tournament Problem: the total distance traveled is an important variable to be minimized in tournament scheduling, to reduce traveling costs and to give more time to the players for resting and training.

– Real life application: finding a good schedule to the Brazilian national soccer championship (26 teams)


Tournament schedules• 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.


Tournament schedules• Conditions (cont’d):

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

exactly once in n-1 prescheduled rounds.

– Double round-robin tournament (DRR):• Each team plays every other team

exactly twice in 2(n-1) prescheduled rounds (each of them with exactly n/2 games), once at home and once away.


Tournament schedules• Conditions (cont’d):

– Mirrored double round-robin tournament (MDRR):• Each team plays every other team exactly twice in

2(n-1) prescheduled rounds (each of them with exactly n/2 games), once at home and once away.

• MDRR is a SRR tournament in the first (n-1) rounds, followed by the same SRR tournament with reversed venues in the last (n-1) rounds.

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


Tournament schedules

• 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.


Tournament schedules• Schedule S:

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

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

– 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)


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)


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

Tournament schedules• Distance minimization problems:

– NHL schedule: minimize the total distance traveled (evolutionary tabu search) - Costa (1995)

– Traveling tournament problem: minimize the total distance traveled, such that no team plays more than three consecutive away games or three consecutive home games - Easton, Nemhauser & Trick (2001,2004)

– Mirrored TTP: Ribeiro & Urrutia (2004)

complexity?Open!


Tournament schedules• In this work:

– Connection between breaks and distance problems

– New class of instances for which distance minimization is equivalent to breaks maximization

– Construction of schedules with maximum number of breaks and minimum distance traveled

– Mirrored DRR schedules satisfying TTP contraints– Solution of larger TTP instances


Tournament schedules• Variants:

– no-repeaters– no synchronized rounds– multiple games (more than two, variable)– teams with complementary patterns in

the same city– pre-scheduled games and TV constraints– stadium availability– minimize airfare and hotel costs, etc.


Connecting breaks with distances

• Benchmark instances for distance minimization problems:– Structured circular instances with n =

4 to 20 teams– MLB instances with n = 4 to 16 teams– All available from Michael Trick’s web

page– 2003 edition of the Brazilian national

soccer championship with 24 teams


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


• New uniform instances: all distances equal to one D(S) = T(S)

• 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



• 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


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


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 (or circle) method to build a schedule with exactly UBSRR breaks.

• Phase 1: assign games to rounds– Graph with one edge for each game at each

round


Polygon method

4 3

2

1

5

6

Example: “polygon method” for n=6

1st roundPhase 1: game assignment


Polygon method

3 2

1

5

4

6


2nd roundPhase 1: game assignment


Polygon method

2 1

5

4

3

6


3rd roundPhase 1: game assignment


Polygon method

1 5

4

3

2

6


4th roundPhase 1: game assignment


Polygon method

5 4

3

2

1

6


5th roundPhase 1: game assignment


Polygon method• Phase 2: extension of the polygon method

an orientation to each edge (oriented edge coloring)

• 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 odd (resp. even) rounds

• Final extremity of each arc is the home team.


Polygon method

Phase 2: stadium assignment


Max breaks for TTP-constrained

MDRR 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.


• 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


MDRR tournaments


• 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.


MDRR tournaments


• Use an effective TTP heuristic to find good approximate solutions (10 minutes):– Ribeiro & Urrutia (2004): better solutions in 10

minutes of CPU for benchmark instances than Anagnostopoulos, Michel, Van Hentenryck & Vergados (2003) in 5 days (similar machine); also best known solutions to circ18 and circ20

• 2.0 GHz Pentium IV with 512 Mb RAM memory

• Uniform instances with n = 4, 6, 8, ..., 18, 20


MDRR tournaments



MDRR tournamentsn 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


Concluding remarks

• New class of uniform instances• 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


Concluding remarks

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

• Complexity results for this new class will possibly shed some light on the complexity of the traveling tournament problem.


Concluding remarks

• Total distance traveled for the 2003 edition of the Brazilian soccer championship with 24 teams (instance br24) in 12 hours (Pentium IV 2.0 MHz):Realized (official draw): 1,048,134 kms

Our solution: 506,433 kms (52% reduction)• Approximate corresponding potential savings in airfares:US$ 1,700,000

may 2004 minimizing travels by maximizing breaks1/32 minimizing travels by maximizing breaks in...

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