Richard Hoshino 星野リチャ‒ドPost-Doctoral Fellow, National Institute of Informatics

Joint work with Ken-ichi Kawarabayashi

Outline of Presentation

Context and Motivation

Intra-league Scheduling for NPB Multi-Round Balanced Traveling Tournament Problem Can cut 60,000 km of travel from NPB intra-league schedule

Inter-league Scheduling for NPB Bipartite Traveling Tournament Problem Can cut 8,000km of travel from NPB inter-league schedule

Implementation Please give us advice and ideas!

Context and Motivation

Life in Makuhari (幕張)

Our Apartment

Kanda University

Chiba Marine Stadium Kaihin-Makuhari Station

Chiba Marines Schedule (2010)

1 2 3 4 5

Saitama Hokkaido Tohoku Orix Fukuoka

6 7 8 9 10

Saitama Hokkaido Orix Tohoku Fukuoka

11 12 13 14 15

Saitama Fukuoka Hokkaido Orix Tohoku

16 17 18 19 20

Orix Hokkaido Fukuoka Saitama Tohoku

23,266 kilometres in total travel.

(HOME sets are marked in red.)

Chiba Marines Schedule (2010)

21 22 23 24 25

Fukuoka Orix Saitama Hokkaido Saitama

26 27 28 29 30

Fukuoka Tohoku Orix Hokkaido Tohoku

31 32 33 34 35

Hokkaido Orix Saitama Fukuoka Tohoku

36 37 38 39 40

Hokkaido Orix Saitama Fukuoka Tohoku

23,266 kilometres in total travel.

(HOME sets are marked in red.)

PL Intra-League TravelTeam Name Distance

(2010)

Chiba 23,266

Tohoku 23,710

Hokkaido 28,599

Orix 24,128

Fukuoka 33,352

Saitama 20,885

TOTAL 153,940

Team Name Trips(2010)

Chiba 36

Tohoku 37

Hokkaido 32

Orix 34

Fukuoka 35

Saitama 34

TOTAL 208

Reducing Travel DistanceTeam Name Distance

(2010)

Chiba 23,266

Tohoku 23,710

Hokkaido 28,599

Orix 24,128

Fukuoka 33,352

Saitama 20,885

TOTAL 153,940

Distance(RH+KK)

Reductionin Travel

16,606 28.6%

17,975 24.2%

20,234 29.2%

18,713 22.4%

21,143 36.6%

19,498 6.6%

114,169 25.8%

Reducing Total TripsTeam Name Trips

(2010)

Chiba 36

Tohoku 37

Hokkaido 32

Orix 34

Fukuoka 35

Saitama 34

TOTAL 208

Trips(RH+KK)

Reductionin Trips

29 19.4%

29 21.6%

27 15.6%

29 14.7%

27 22.9%

28 17.6%

169 18.8%

Nippon Series Champions!

Interview in NII Today

Ken-ichi Kawarabayashi National Institute of Informatics

Mariko TakahashiAsahi Shimbun

Intra-League Scheduling

Intra-League Scheduling

Defining the Traveling Tournament Problem (TTP)

Defining the Multi-Round Balanced TTP (mb-TTP)

Specific to the scheduling requirements of the NPB League.

Reformulation as a Shortest Path Problem

Optimal Solution for Pacific and Central Leagues

~40,000 km reduction for PL (153,940 km to 114,169 km)

~20,000 km reduction for CL (79,067 km to 57,836 km)

Nippon Pro Baseball Schedules

1 2 3 4 5

Saitama Hokkaido Tohoku Orix Fukuoka

6 7 8 9 10

Saitama Hokkaido Orix Tohoku Fukuoka

11 12 13 14 15

Saitama Fukuoka Hokkaido Orix Tohoku

16 17 18 19 20

Orix Hokkaido Fukuoka Saitama Tohoku

At-Most-Three Condition: Home stands and road trips last at most 3 sets.

No-Repeat Condition: No team can play against the same opponent in consecutive sets.

Home-Away Condition: In every ten-set block, each pair of teams plays twice, with one set played at each team’s home stadium.

Each-Round Condition: In each five-set round, a team plays against all five opponents once.

Diff-Two Condition:The value |H−R| never exceeds 2.

Traveling Tournament Problem

Given an n × n distance matrix, determine the double round-robin tournament schedule that

satisfies At-Most-Three, No-Repeat, and Home-Away.

minimizes the total distance traveled by the n teams.

For example, A-B-C-B-A – C-D-E-D-E is a valid team schedule under the Traveling Tournament Problem (TTP) but would not be for Nippon Pro Baseball as the Each-Round and Diff-Two conditions are not satisfied.

History of the TTP

Proposed by Easton-Nemhauser-Trick (2001) in the context of scheduling for Major League Baseball.

The TTP is NP-complete (2010). To date, only a few benchmark cases have been solved to optimality: NL4(1999), NL6 (1999), NL8 (2008), NL10 (2009).

TTP-solving algorithms are a complex hybrid of integer programming and constraint programming.

Multi-Round Balanced TTP

Motivated by the actual league structure of Nippon Pro Baseball (NPB), we propose the mb-TTP, with multiple rounds and additional balancing constraints.

Given an n × n distance matrix, determine the distance-optimal tournament schedule that lasts 2krounds (k blocks) and satisfies all five conditions: At-Most-Three, No-Repeat, Home-Away, Each-Round, Diff-Two.

We solve this problem by reformulating the mb-TTP as a shortest-path problem on a directed graph.

Concatenating Two Blocks

How do we check whether two 10-set blocks can be concatenated to form a 20-set tournament schedule?

Team

Chiba (C)

Tohoku (T)

Hokkaido (H)

Orix (O)

Fukuoka (F)

Saitama (S)

R1 R2

S H T F O T F O S H

H F C O S C O S H F

T C O S F O S F T C

F S H T C H T C F S

O T S C H S C H O T

C O F H T F H T C O

R3 R4

S H T F O T F O S H

H F C O S C O S H F

T C O S F O S F T C

F S H T C H T C F S

O T S C H S C H O T

C O F H T F H T C O

Concatenating Two Blocks

The conditions Home-Away, Each-Round, Diff-Twoare automatically satisfied, since each block is feasible.

Team

Chiba (C)

Tohoku (T)

Hokkaido (H)

Orix (O)

Fukuoka (F)

Saitama (S)

R1 R2

S H T F O T F O S H

H F C O S C O S H F

T C O S F O S F T C

F S H T C H T C F S

O T S C H S C H O T

C O F H T F H T C O

R3 R4

S H T F O T F O S H

H F C O S C O S H F

T C O S F O S F T C

F S H T C H T C F S

O T S C H S C H O T

C O F H T F H T C O

Concatenating Two Blocks

To check No-Repeat, just look at the last column in Block #1 and the first column in Block #2.

Team

Chiba (C)

Tohoku (T)

Hokkaido (H)

Orix (O)

Fukuoka (F)

Saitama (S)

R1 R2

S H T F O T F O S H

H F C O S C O S H F

T C O S F O S F T C

F S H T C H T C F S

O T S C H S C H O T

C O F H T F H T C O

R3 R4

S H T F O T F O S H

H F C O S C O S H F

T C O S F O S F T C

F S H T C H T C F S

O T S C H S C H O T

C O F H T F H T C O

Concatenating Two Blocks

To check At-Most-Three, look at the last two columns in Block #1 and the first two columns in Block #2.

Team

Chiba (C)

Tohoku (T)

Hokkaido (H)

Orix (O)

Fukuoka (F)

Saitama (S)

R1 R2

S H T F O T F O S H

H F C O S C O S H F

T C O S F O S F T C

F S H T C H T C F S

O T S C H S C H O T

C O F H T F H T C O

R3 R4

S H T F O T F O S H

H F C O S C O S H F

T C O S F O S F T C

F S H T C H T C F S

O T S C H S C H O T

C O F H T F H T C O

Graph-Theoretic Reformulation

Each team starts and ends the season at home (vstart, vend)

Each vertex xt,u with 1 ≤ u ≤ m, represents the first two columns of the tth block (matches = 1st, home teams = 2nd)

Each vertex yt,u with 1 ≤ u ≤ m, represents the last two columns of the tth block (home teams = 9th, matches = 10th)

Explanation of the variable m1 2

D 1

E 1

F 1

A 0

B 0

C 0

Team

A

B

C

D

E

F

x1,u

For n = 6, m = 20 × 120 = 2400. In general,

203

6

2/

n

n

120!33

6!2/

2/

n

n

n

There are

ways to select the home teams of any column.

So there are

ways to select the three matchesof any column.

!2/2/

2

nn

nm

Construction of Edge yt,v→ xt+1,uTeam

A

B

C

D

E

F

9 10

1 F

0 E

1 D

1 C

0 B

0 A

11 12

B 0

A 0

F 0

E 1

D 1

C 1

y1,v x2,u

yt,v→ xt+1,u is an edge iff the n × 4 concatenation matrix does not violate the at-most-three or no-repeat conditions.

The weight of edge yt,v→ xt+1,u is the distance traveled by the n teams moving from set 2t(n-1) to 2t(n-1)+1.

Construction of Edge xt,u→ yt,v1 2

B 0

A 0

F 0

E 1

D 1

C 1

Team

A

B

C

D

E

F

9 10

1 F

0 E

1 D

1 C

0 B

0 A

x1,u y1,v

xt,u→ yt,v is an edge iff there exists a (feasible) block satisfying the five mb-TTP conditions.

The weight of edge xt,u→ yt,v is the minimum possible total distance traveled by the n teams within that block.

3 4 5 6 7 8

? ? ? ? ? ?

? ? ? ? ? ?

? ? ? ? ? ?

? ? ? ? ? ?

? ? ? ? ? ?

? ? ? ? ? ?

Dijkstra’s Algorithm

The directed graph has 2mk+2 vertices and at most 2m+(2k-1)m2 edges. Each edge has a weight.

Now apply Dijkstra’s Algorithm to find the shortest path vstart → x1,u1→ y1,v1 → … → xk,uk → yk,vk→ vend

which produces the optimal solution of the mb-TTP.

Optimal Intra-League Schedule

In the NPB, each team plays 120 intra-league games (40 sets of 3 games), with eight sets (24 games) against each of the other 5 teams. Thus, there are 8 rounds.

Team R1 R2 R3 R4 R5 R6 R7 R8

Chiba SHTFO TFOSH TFOSH OSHTF HOFST FSTHO TSHOF HOFTS

Tohoku HFCOS COSHF CHSFO SFOCH SHOFC OFCSH CHOFS OFSCH

Hokkaido TCOSF OSFTC STFOC FOCST CTSOF SOFCT FTCSO CSOFT

Orix FSHTC HTCFS FSCHT CHTFS FCTHS THSFC SFTCH TCHSF

Fukuoka OTSCH SCHOT OCHTS HTSOC OSCTH CTHOS HOSTC STCHO

Saitama COFHT FHTCO HOTCF TCFHO TFHCO HCOTF OCFHT FHTOC

Chiba Marines Schedule (2012?)

1 2 3 4 5

Saitama Hokkaido Tohoku Fukuoka Orix

6 7 8 9 10

Tohoku Fukuoka Orix Saitama Hokkaido

11 12 13 14 15

Tohoku Fukuoka Orix Saitama Hokkaido

16 17 18 19 20

Orix Saitama Hokkaido Tohoku Fukuoka

16,606 kilometres in total travel by this team, a 28.6% reduction.

(HOME sets are marked in red.)

Chiba Marines Schedule (2012?)

21 22 23 24 25

Hokkaido Orix Fukuoka Saitama Tohoku

26 27 28 29 30

Fukuoka Saitama Tohoku Hokkaido Orix

31 32 33 34 35

Tohoku Saitama Hokkaido Orix Fukuoka

36 37 38 39 40

Hokkaido Orix Fukuoka Tohoku Saitama

16,606 kilometres in total travel by this team, a 28.6% reduction.

(HOME sets are marked in red.)

Results for NPB Central LeagueTeam Name Distance

(2010)Distance

(New)Reduction in Travel

Trips(2010)

Trips (New)

Reduction in Trips

Hiroshima 17,850 11,741 34.2% 33 27 18.2%

Hanshin 14,304 8,712 39.1% 33 29 12.1%

Chunichi 11,790 11,665 1.1% 33 28 15.2%

Yokohama 13,104 8,929 31.9% 34 29 14.7%

Yomiuri 11,469 9,020 21.4% 33 28 15.2%

Yakult 10,550 7,769 26.4% 33 29 12.1%

TOTAL 79,067 57,836 26.8% 199 170 14.6%

For the two leagues, we can reduce total intra-league travel by 60,000 km and eliminate 68 trips!

Inter-League Scheduling

2010 Inter-League Schedule

In the NPB, each team plays 24 inter-league games (12 sets of 2 games), against each of the 6 teams from the other league. The home game slots are uniform.

Team R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12

Fukuoka (P1) C3 C6 C2 C1 C4 C5 C3 C6 C1 C2 C4 C5

Orix (P2) C6 C3 C1 C2 C5 C4 C6 C3 C2 C1 C5 C4

Saitama (P3) C4 C5 C6 C3 C1 C2 C4 C5 C6 C3 C2 C1

Chiba (P4) C5 C4 C3 C6 C2 C1 C5 C4 C3 C6 C1 C2

Tohoku (P5) C1 C2 C4 C5 C3 C6 C1 C2 C4 C5 C3 C6

Hokkaido (P6) C2 C1 C5 C4 C6 C3 C2 C1 C5 C4 C6 C3

2010 Inter-League Schedule

In the NPB, each team plays 24 inter-league games (12 sets of 2 games), against each of the 6 teams from the other league. The home game slots are uniform.

Team R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12

Hiroshima (C1) P5 P6 P2 P1 P3 P4 P5 P6 P1 P2 P4 P3

Hanshin (C2) P6 P5 P1 P2 P4 P3 P6 P5 P2 P1 P3 P4

Chunichi (C3) P1 P2 P4 P3 P5 P6 P1 P2 P4 P3 P5 P6

Yokohama (C4) P3 P4 P5 P6 P1 P2 P3 P4 P5 P6 P1 P2

Yomiuri (C5) P4 P3 P6 P5 P2 P1 P4 P3 P6 P5 P2 P1

Yakult (C6) P2 P1 P3 P4 P6 P5 P2 P1 P3 P4 P6 P5

Minimum-Weight Perfect Matching

Triangle Cover

Inter-League Scheduling

Defining the 2n-team Bipartite TTP (BTTP)

Replacing At-Most-Two with At-Most-Three.

Retaining the Home-Away and No-Repeat conditions.

Considering both uniform and non-uniform schedules .

NP-completeness of both BTTP variants.

Optimal Solution for NPB Inter-League Play (n=6)

Uniform: ~7,700 km reduction (51,134 km to 43,285 km)

Non-Uniform: ~8,000 km reduction (51,134 km to 42,950 km)

Simple Illustration for n=3

Here are two feasible bipartite tournament schedules with teams X = (X1, X2, X3) and Y = (Y1, Y2, Y3). Which schedule has lower total travel distance?

Team R1 R2 R3 R4 R5 R6

X1 Y1 Y2 Y3 Y1 Y2 Y3

X2 Y2 Y3 Y1 Y2 Y3 Y1

X3 Y3 Y1 Y2 Y3 Y1 Y2

Y1 X1 X3 X2 X1 X3 X2

Y2 X2 X1 X3 X2 X1 X3

Y3 X3 X2 X1 X3 X2 X1

Team R1 R2 R3 R4 R5 R6

X1 Y3 Y2 Y1 Y3 Y1 Y2

X2 Y1 Y3 Y2 Y1 Y2 Y3

X3 Y2 Y1 Y3 Y2 Y3 Y1

Y1 X2 X3 X1 X2 X1 X3

Y2 X3 X1 X2 X3 X2 X1

Y3 X1 X2 X3 X1 X3 X2

Trips = 4+4+4+4+4+4 = 24 Trips = 5+5+5+6+6+5 = 32

Simple Illustration for n=3

It all depends on where the teams are located!

Locate X1, X3, Y1, Y2 at (0,0) and X2, Y3 at (1,0). Then we get a counter-intuitive result!

Team R1 R2 R3 R4 R5 R6

X1 Y1 Y2 Y3 Y1 Y2 Y3

X2 Y2 Y3 Y1 Y2 Y3 Y1

X3 Y3 Y1 Y2 Y3 Y1 Y2

Y1 X1 X3 X2 X1 X3 X2

Y2 X2 X1 X3 X2 X1 X3

Y3 X3 X2 X1 X3 X2 X1

Team R1 R2 R3 R4 R5 R6

X1 Y3 Y2 Y1 Y3 Y1 Y2

X2 Y1 Y3 Y2 Y1 Y2 Y3

X3 Y2 Y1 Y3 Y2 Y3 Y1

Y1 X2 X3 X1 X2 X1 X3

Y2 X3 X1 X2 X3 X2 X1

Y3 X1 X2 X3 X1 X3 X2

Distance = 2+4+2+2+2+4=16 Distance = 2+2+2+2+2+2=12

NP-completeness of BTTP

We prove that BTTP is NP-complete, even when the restricted to the set of uniform schedules!

We accomplish this by reducing from 3-SAT, using a special “gadget” and carefully defining edge weights.

Rooted 4-cycle-covers For each team, find a minimum weight 4-cycle-cover

rooted at that vertex.

Individual Team Lower Bound

The minimum-weight rooted 4-cycle-cover determines each team’s individual lower bound. We use this to build a feasible uniform bipartite tournament.

Team R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12

Fukuoka (P1) C2 C3 C1 C2 C3 C6 C4 C5 C6 C1 C4 C5

Orix (P2) C4 C6 C5 C3 C6 C1 C2 C3 C1 C2 C5 C4

Saitama (P3) C3 C1 C2 C6 C4 C2 C5 C6 C4 C5 C3 C1

Chiba (P4) C5 C4 C6 C5 C2 C4 C3 C1 C2 C6 C1 C3

Tohoku (P5) C1 C2 C3 C4 C1 C5 C6 C4 C5 C3 C2 C6

Hokkaido (P6) C6 C5 C4 C1 C5 C3 C1 C2 C3 C4 C6 C2

Best Uniform Tournament

The optimal uniform inter-league tournament for NPB has total distance just 12 + 510 = 522 km more than the

theoretical trivial lower bound.

Team ILB Optimal Diff.

Fukuoka 3401 3401 0

Orix 2178 2178 0

Saitama 1939 1940 1

Chiba 1895 1906 11

Tohoku 2660 2660 0

Hokkaido 4613 4613 0

TOTAL 16686 16698 12

Team ILB Optimal Diff.

Hiroshima 5027 5027 0

Hanshin 4549 4550 1

Chunichi 4459 4612 153

Yokohama 4023 4362 339

Yomiuri 4008 4019 11

Yakult 4011 4017 6

TOTAL 26077 26587 510

Best Non-Uniform Tournament

The optimal non-uniform inter-league tournament for NPB has total distance just 6 + 181 = 187 km more than

the theoretical trivial lower bound.

Team ILB Optimal Diff.

Fukuoka 3401 3401 0

Orix 2178 2182 4

Saitama 1939 1939 0

Chiba 1895 1895 0

Tohoku 2660 2661 1

Hokkaido 4613 4614 1

TOTAL 16686 16692 6

Team ILB Optimal Diff.

Hiroshima 5027 5078 51

Hanshin 4549 4558 9

Chunichi 4459 4490 31

Yokohama 4023 4081 58

Yomiuri 4008 4027 19

Yakult 4011 4024 13

TOTAL 26077 26258 181

Implementation

Implementation

Additional scheduling constraints (e.g. certain stadiums are unavailable on various days).

Minimizing carry-over effects for competitive balance.

Rivalry matches that must be scheduled on specific dates (e.g. holidays, end-of-season).

How do we make this happen? Your ideas and advice would be very much appreciated!