Richard Hoshino and Ken-ichi Kawarabayashi N ATIONAL I NSTITUTE OF I - - PowerPoint PPT Presentation

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Richard Hoshino and Ken-ichi Kawarabayashi N ATIONAL I NSTITUTE OF I - - PowerPoint PPT Presentation

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Richard Hoshino and Ken-ichi Kawarabayashi N ATIONAL I NSTITUTE OF I NFORMATICS , T OKYO , J APAN COPLAS C ONFERENCE , J UNE 2011, F REIBURG , G ERMANY Outline of Presentation Context and Motivation Intra-league Scheduling for NPB (ICAPS)

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Richard Hoshino and Ken-ichi Kawarabayashi

NATIONAL INSTITUTE OF INFORMATICS, TOKYO, JAPAN COPLAS CONFERENCE, JUNE 2011, FREIBURG, GERMANY

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Outline of Presentation

 Context and Motivation  Intra-league Scheduling for NPB (ICAPS)

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

 Inter-league Scheduling for NPB (COPLAS)

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

 Implementation

 Please give us advice and ideas!

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SLIDE 3

Context and Motivation

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SLIDE 4

Life in Makuhari, Japan

Our Apartment Kanda University Chiba Marine Stadium Kaihin-Makuhari Station

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SLIDE 5

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.)

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SLIDE 6

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.)

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SLIDE 7
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SLIDE 8

Intra-League Scheduling

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Shameless Plug

 ICAPS Session Va, Applications II

10:30AM to 12:15PM, June 16th Richard Hoshino and Ken-ichi Kawarabayashi The Multi-Round Balanced Traveling Tournament Problem

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Inter-League Scheduling

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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

  • ther 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

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SLIDE 12

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

  • ther 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

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Minimum-Weight Perfect Matching

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Triangle Cover

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Central League Schedule

 For the Central League, the inter-league schedule is

distance-optimal, since the PL minimum-weight perfect matching is {P1 ,P2 }, {P3 ,P4 }, {P5 ,P6 } .

Team R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 Hiroshima (C1) P2 P1 P5 P6 P4 P3 Hanshin (C2) P1 P2 P6 P5 P3 P4 Chunichi (C3) P4 P3 P1 P2 P5 P6 Yokohama (C4) P5 P6 P3 P4 P1 P2 Yomiuri (C5) P6 P5 P4 P3 P2 P1 Yakult (C6) P3 P4 P2 P1 P6 P5

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Minimum-Weight Perfect Matching

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Pacific League Schedule

 For the Pacific League, the inter-league schedule is

almost distance-optimal, since the CL minimum- weight perfect matching is {C1 ,C2 }, {C3 ,C4 }, {C5 ,C6 } .

Team R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 Fukuoka (P1) C3 C6 C4 C5 C1 C2 Orix (P2) C6 C3 C5 C4 C2 C1 Saitama (P3) C4 C5 C1 C2 C6 C3 Chiba (P4) C5 C4 C2 C1 C3 C6 Tohoku (P5) C1 C2 C3 C6 C4 C5 Hokkaido (P6) C2 C1 C6 C3 C5 C4

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Theorem #1

Consider an inter-league tournament between two teams X and Y, each with 2n teams, where each pair of teams xi and yj plays two games, with one game at each team’s home venue. If all teams can play at most two consecutive home/away games, then the distance-optimal schedule must be uniform, and have the HH-RR-…-HH-RR pattern.

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Proof of Theorem #1

 The “perfect matching” construction gives us an inter-

league schedule where the total travel distance is

 Team xi travels a distance of  Team yj travels a distance of



 

 

n i n j y x Y X

j i

D PM PM n

2 1 2 1 ,

2 ) ( 2

n j y x Y

j i

D PM

2 1 ,

n i y x X

j i

D PM

2 1 ,

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Proof of Theorem #1

 Each individual team’s travel distance is minimal, by

the Triangle Inequality

RR – HH – RR – HR – HR – HH – RR – HH RR – HH – RR – HH – RR – HH – RR – HH

 So each team must play their 2n road games in n

blocks of two.

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Proof of Theorem #1

 League X consists of 2n = a+b+c+d teams, where

a teams begin with HH b teams begin with HR c teams begin with RH d teams begin with RR

 League Y consists of 2n = e+f+g+h teams, where

e teams begin with HH f teams begin with HR g teams begin with RH h teams begin with RR

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SLIDE 22

Proof of Theorem #1

 League X consists of 2n = a+b+ d teams, where

a teams begin with HH b teams begin with HR d teams begin with RR

 League Y consists of 2n = e+f+ h teams, where

e teams begin with HH f teams begin with HR h teams begin with RR

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SLIDE 23

Proof of Theorem #1

 League X consists of 2n = a+b+ d teams, where

a teams begin with HH b teams begin with HR d teams begin with RR

 League Y consists of 2n = e+f+ h teams, where

e teams begin with HH f teams begin with HR h teams begin with RR Day 1 implies a+b=h Day 2 implies f+h=a Thus, b+f=0, so b=f=0.

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Proof of Theorem #1

 League X consists of 2n = a+ d teams, where

a teams begin with HH d teams begin with RR

 League Y consists of 2n = e+ h teams, where

e teams begin with HH h teams begin with RR Day 1 implies a+b=h Day 2 implies f+h=a Thus, b+f=0, so b=f=0.

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Proof of Theorem #1

 League X consists of 2n = a+ d teams, where

a teams begin with HH d teams begin with RR

 League Y consists of 2n = e+ h teams, where

e teams begin with HH h teams begin with RR Day 1 implies a+b=h Day 2 implies f+h=a Thus, b+f=0, so b=f=0. Since b=f=0, we have: a=h and d=e.

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Proof of Theorem #1

 League X consists of 2n = a+ d teams, where

a teams begin with HH d teams begin with RR

 League Y consists of 2n = a+ d teams, where

d teams begin with HH a teams begin with RR Day 1 implies a+b=h Day 2 implies f+h=a Thus, b+f=0, so b=f=0. Since b=f=0, we have: a=h and d=e.

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SLIDE 27

Proof of Theorem #1

 League X consists of 2n = a+d teams, where

a teams begin with HH d teams begin with RR

 League Y consists of 2n = a+d teams, where

d teams begin with HH a teams begin with RR

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SLIDE 28

Proof of Theorem #1

 League X consists of 2n = a+d teams, where

a teams begin with HHRR d teams begin with RRH

 League Y consists of 2n = a+d teams, where

d teams begin with HHRR a teams begin with RRH

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SLIDE 29

Proof of Theorem #1

 League X consists of 2n = a+d teams, where

a teams begin with HHRR d teams begin with RRHH

 League Y consists of 2n = a+d teams, where

d teams begin with HHRR a teams begin with RRHH

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Proof of Theorem #1

 League X consists of 2n = a+d teams, where

a teams begin with HHRRH d teams begin with RRHHRR

 League Y consists of 2n = a+d teams, where

d teams begin with HHRRH a teams begin with RRHHRR

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Proof of Theorem #1

 League X consists of 2n = a+d teams, where

a teams begin with HHRRHHRR d teams begin with RRHHRR

 League Y consists of 2n = a+d teams, where

d teams begin with HHRRHHRR a teams begin with RRHHRR

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Proof of Theorem #1

 League X consists of 2n = a+d teams, where

a teams begin with HHRRHHRR….HHRR d teams begin with RRHHRRHH….RRHH

 League Y consists of 2n = a+d teams, where

d teams begin with HHRRHHRR….HHRR a teams begin with RRHHRRHH….RRHH

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SLIDE 33

Proof of Theorem #1

 League X consists of 2n = a+d teams, where

a teams begin with HHRRHHRR….HHRR d teams begin with RRHHRRHH….RRHH

 League Y consists of 2n = a+d teams, where

d teams begin with HHRRHHRR….HHRR a teams begin with RRHHRRHH….RRHH

If a>0 and d>0, then we have a contradiction.

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Proof of Theorem #1

 League X consists of 2n = a teams, where

a teams begin with HHRRHHRR….HHRR

 League Y consists of 2n = a teams, where

a teams begin with RRHHRRHH….RRHH

If a>0 and d>0, then we have a contradiction. And so the inter-league schedule must be uniform.

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At Most Two to At Most Three

 If all teams can play at most two consecutive

home/away games, then the distance-optimal schedule must be uniform… and there is a simple O(n3) algorithm to construct an optimal schedule.

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At Most Two to At Most Three

 If all teams can play at most two consecutive

home/away games, then the distance-optimal schedule must be uniform… and there is a simple O(n3) algorithm to construct an optimal schedule.

 If all teams can play at most three consecutive

home/away games, then constructing a distance-

  • ptimal schedule is NP-complete, even when

restricted to the set of uniform schedules!

(Hoshino + Kawarabayshi, 2011 AAAI Conference)

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Theorem #2

 We prove that the Bipartite Traveling Tournament

Problem (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.

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Application of the BTTP

 While the 2n-team BTTP is NP-complete, we can still

solve it to optimality for n=6, for both

 Uniform schedules  Non-uniform schedules

 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)

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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

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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

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Rooted 4-cycle-covers

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

rooted at that vertex.

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Individual Team Lower Bound

 The minimum-weight rooted 4-cycle-cover determines

each team’s individual lower bound. We use this to first build a trivially-feasible uniform tournament.

Team R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 Fukuoka (P1) C1 C2 C3 C1 C2 C3 C4 C5 C6 C4 C5 C6 Orix (P2) C2 C3 C1 C2 C3 C1 C5 C6 C4 C5 C6 C4 Saitama (P3) C3 C1 C2 C3 C1 C2 C6 C4 C5 C6 C4 C5 Chiba (P4) C4 C5 C6 C4 C5 C6 C1 C2 C3 C1 C2 C3 Tohoku (P5) C5 C6 C4 C5 C6 C4 C2 C3 C1 C2 C3 C1 Hokkaido (P6) C6 C4 C5 C6 C4 C5 C3 C1 C2 C3 C1 C2

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Trivial Uniform Tournament

 We then compare each team’s individual lower bound

(ILB) with the distance traveled via this trivial uniform schedule.

Team ILB Trivial Diff. Fukuoka 3401 3401 Orix 2178 2182 4 Saitama 1939 1940 1 Chiba 1895 1901 6 Tohoku 2660 2660 Hokkaido 4613 4614 1 TOTAL 16686 16698 12 Team ILB Trivial Diff. Hiroshima 5027 5077 50 Hanshin 4549 4550 1 Chunichi 4459 4652 193 Yokohama 4023 4349 326 Yomiuri 4008 4019 11 Yakult 4011 4017 6 TOTAL 26077 26664 587

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SLIDE 44

Heuristic #1

 This trivial uniform inter-league tournament has total

distance just 12 + 587 = 599 km more than the theoretical trivial lower bound.

 For each team t, we can just restrict the set of

schedules St to just those satisfying the inequality:

 “Reduction prior to propagation” is a simple yet

powerful heuristic!

} 599 ) ( : {   

t t t t

ILB d S  

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SLIDE 45

Best Uniform Tournament

 From Heuristic #1, we can construct the distance-

  • ptimal uniform tournament, which has total distance

just 12 + 510 = 522 km more than the theoretical trivial lower bound.

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

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Heuristic #2

 Take into account the

locations of the teams.

 Notice that P5 and P6

are located quite far from the other 10 teams.

 Key insight: each Cj

must play back-to-back road games against teams P5 and P6.

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SLIDE 47

Best Non-Uniform Tournament

 From Heuristics #1 and #2, we can construct the

distance-optimal non-uniform tournament. Notice that each Cj plays against P5 and P6 back-to-back.

Team R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 Hiroshima (C1) P5 P6 P1 P2 P6 P1 P2 P3 P4 P5 P3 P4 Hanshin (C2) P4 P3 P2 P6 P1 P5 P6 P1 P5 P3 P4 P2 Chunichi (C3) P1 P2 P6 P1 P2 P3 P4 P5 P3 P4 P6 P5 Yokohama (C4) P3 P4 P5 P4 P3 P6 P5 P2 P1 P6 P2 P1 Yomiuri (C5) P2 P1 P4 P3 P5 P4 P3 P6 P2 P1 P5 P6 Yakult (C6) P6 P5 P3 P5 P4 P2 P1 P4 P6 P2 P1 P3

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SLIDE 48

Best Non-Uniform Tournament

 From Heuristics #1 and #2, we can construct the

distance-optimal non-uniform tournament. Notice that each Cj plays against P5 and P6 back-to-back.

Team R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 Hiroshima (C1) P5 P6 P1 P2 P6 P1 P2 P3 P4 P5 P3 P4 Hanshin (C2) P4 P3 P2 P6 P1 P5 P6 P1 P5 P3 P4 P2 Chunichi (C3) P1 P2 P6 P1 P2 P3 P4 P5 P3 P4 P6 P5 Yokohama (C4) P3 P4 P5 P4 P3 P6 P5 P2 P1 P6 P2 P1 Yomiuri (C5) P2 P1 P4 P3 P5 P4 P3 P6 P2 P1 P5 P6 Yakult (C6) P6 P5 P3 P5 P4 P2 P1 P4 P6 P2 P1 P3

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SLIDE 49

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 Orix 2178 2182 4 Saitama 1939 1939 Chiba 1895 1895 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

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SLIDE 50

Implementation

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SLIDE 51

Implementation

 Additional scheduling constraints (e.g. certain

stadiums are unavailable on various days).

 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!