Transcript PVC routing algorithms

Minimizing Travels by
Maximizing Breaks in Round
Robin Tournament
Schedules
Celso RIBEIRO
UFF and PUC-Rio, Brazil
Sebastián URRUTIA
May 2004
PUC-Rio, Brazil
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Minimizing travels by maximizing breaks
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
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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
soccer
championship
(26
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travels by maximizing breaks
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.
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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.
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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
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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
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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)
– 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
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stadium
another)
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Minimizing
travels by to
maximizing
breaks
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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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)
complexity?
Mirrored
TTP: Ribeiro
Urrutiainstance
(2004)
– Hard
problem:
previous&largest
Open!
exactly solved to date had only n=6 teams!
(n=8 with 20 processors in 4 days CPU
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time)
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
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Minimizing travels by maximizing breaks
– Solution of larger
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.
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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
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Connecting breaks with distances
• 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 the first game
travels after playing the last game
travels to play in intermediary rounds if all teams were to travel,
discounted by the number of teams that do not travel (home break
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Connecting breaks with distances
• 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: 15/32
implications
in the solution of the
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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:
2 – 3n + 2
UB
=
2(n-2)
+
(n-2)(n-3)
=
n
SRR
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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 (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
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Polygon method
6
Example:
“polygon method” for n=6
1
5
2
1st round
Phase 1:
game assignment
4
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Polygon method
6
Example:
“polygon method” for n=6
5
4
1
2nd round
Phase 1:
game assignment
3
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Polygon method
6
Example:
“polygon method” for n=6
4
3
5
3rd round
Phase 1:
game assignment
2
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Polygon method
6
Example:
“polygon method” for n=6
3
2
4
4th round
Phase 1:
game assignment
1
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Polygon method
6
Example:
“polygon method” for n=6
2
1
3
5th round
Phase 1:
game assignment
5
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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
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• Final extremity of each arc is the home team.
Polygon method
Phase 2:
stadium assignment
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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. 25/32
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Max breaks for TTP-constrained
MDRR tournaments
• Upper bounds to the number of breaks can
be derived using similar (although much
more elaborated) counting arguments:
UBTTP
14,
4(n 2  n) / 3  4n  20,

 2
4(n  2n) / 3,
4n 2 / 3  4n,
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if n  4
if n  1mod 3  0 and n  4
if n  1mod 3  1
if n  1mod 3  2
Minimizing travels by maximizing breaks
Max breaks for TTP-constrained
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.
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Max breaks for TTP-constrained
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 nMinimizing
= 4, travels
6, 8,
..., 18, 20
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by maximizing breaks
Max breaks for TTP-constrained
MDRR tournaments
n
4
6
8
D(S)
17
48
80
LB
17
48
80
gap
-
B(S)
14
24
64
10
12
14
130
192
256
130
192
252
4
100
144
216
16
18
342
434
342
432
2
276
356
20
526
520
6
468
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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
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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.
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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):
Our
solution: 506,433 kms (52% reduction)
(official
draw): 1,048,134
kms
• Realized
Approximate
corresponding
potential
savings
in airfares:
US$ 1,700,000
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