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

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)  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!

Context and Motivation

Life in Makuhari, Japan 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 23,266 kilometres in total travel. 16 17 18 19 20 Orix Hokkaido Fukuoka Saitama Tohoku (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 23,266 kilometres in total travel. 36 37 38 39 40 Hokkaido Orix Saitama Fukuoka Tohoku (HOME sets are marked in red.)

Intra-League Scheduling

Shameless Plug  ICAPS Session Va, Applications II 10:30AM to 12:15PM, June 16 th Richard Hoshino and Ken-ichi Kawarabayashi The Multi-Round Balanced Traveling Tournament Problem

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

Central League Schedule  For the Central League, the inter-league schedule is distance-optimal, since the PL minimum-weight perfect matching is {P 1 ,P 2 }, {P 3 ,P 4 }, {P 5 ,P 6 } . 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

Minimum-Weight Perfect Matching

Pacific League Schedule  For the Pacific League, the inter-league schedule is almost distance-optimal, since the CL minimum- weight perfect matching is {C 1 ,C 2 }, {C 3 ,C 4 }, {C 5 ,C 6 } . 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

Theorem #1 Consider an inter-league tournament between two teams X and Y , each with 2n teams, where each pair of teams x i and y j 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.

Proof of Theorem #1  The “perfect matching” construction gives us an inter - league schedule where the total travel distance is 2 n 2 n    2 n ( PM PM ) 2 D X Y x , y i j   i 1 j 1 2 n    Team x i travels a distance of PM D Y x , y i j  j 1 2 n    Team y j travels a distance of PM D X x , y i j  i 1

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.

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

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

Proof of Theorem #1  League X consists of 2n = a+b+ d teams, where a teams begin with HH Day 1 implies a+b=h b teams begin with HR Day 2 implies f+h=a d teams begin with RR Thus, b+f=0 , so b=f=0 .  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

Proof of Theorem #1  League X consists of 2n = a+ d teams, where a teams begin with HH Day 1 implies a+b=h Day 2 implies f+h=a d teams begin with RR Thus, b+f=0 , so b=f=0 .  League Y consists of 2n = e+ h teams, where e teams begin with HH h teams begin with RR

Proof of Theorem #1  League X consists of 2n = a+ d teams, where a teams begin with HH Day 1 implies a+b=h Day 2 implies f+h=a d teams begin with RR Thus, b+f=0 , so b=f=0 .  League Y consists of 2n = e+ h teams, where e teams begin with HH Since b=f=0 , we have: a=h and d=e . h teams begin with RR

Proof of Theorem #1  League X consists of 2n = a+ d teams, where a teams begin with HH Day 1 implies a+b=h Day 2 implies f+h=a d teams begin with RR Thus, b+f=0 , so b=f=0 .  League Y consists of 2n = a+ d teams, where d teams begin with HH Since b=f=0 , we have: a=h and d=e . a teams begin with RR

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

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

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

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

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

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

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.

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.

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(n 3 ) algorithm to construct an optimal schedule.

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(n 3 ) algorithm to construct an optimal schedule.  If all teams can play at most three consecutive home/away games, then constructing a distance- optimal schedule is NP -complete, even when restricted to the set of uniform schedules! (Hoshino + Kawarabayshi, 2011 AAAI Conference)