Optimizing the Scheduling of Nippon Professional Baseball Using Graph Theory

Richard Hoshino ([email protected]) Joint work with Ken-ichi Kawarabayashi, National Institute of Informatics

1 The Teams of Nippon Professional Baseball (NPB)

The twelve NPB teams are split into the 6-team Pacific League and the 6-team Central League. Each team plays 120 intra-league and 24 inter-league games during the regular season. Teams play intra-league games against teams from their own league (Central vs. Central, or Pacific vs. Pacific), and inter-league games against teams from the other league (Central vs. Pacific).

For readability, we label each team as follows: the Pacific League teams are p1 (Fukuoka), p2 (Orix), p3 (Saitama), p4 (Chiba), p5 (Tohoku), p6 (Hokkaido), and the Central League teams are c1 (Hiroshima), c2 (Hanshin), c3 (Chunichi), c4 (Yokohama), c5 (Yomiuri), and c6 (Yakult).

Team c1 c2 c3 c4 c5 c6 p1 p2 p3 p4 p5 p6 c1 0 323 488 808 827 829 258 341 870 857 895 1288 c2 0 195 515 534 536 577 27 577 564 654 1099 c3 0 334 353 355 742 213 396 383 511 984 c4 0 37 35 916 533 63 58 364 886 c5 0 7 926 552 51 37 331 896 c6 0 923 554 48 39 333 893 p1 0 595 958 934 1100 1466 p2 0 595 582 670 1115 p3 0 86 374 928 p4 0 361 904 p5 0 580 p6 0

Above is the 12 × 12 NPB distance matrix, where all distances are given in kilometres. In this matrix, we only provide Dci,cj and Dpi,pj for i < j, since the case i > j is equivalent by symmetry. 2 Definition of the Traveling Tournament Problem

Let there be n teams in a sports league, and let D be the n × n distance matrix for this league. In the Traveling Tournament Problem (TTP), a double round-robin schedule is sought, i.e., a tournament where every pair of teams plays twice, with one set of games at each home venue. Each team begins the tournament at home, and returns home after having played their last away set. Whenever a team is scheduled for a road trip consisting of multiple away sets, the team doesn’t return to their home city but rather proceeds directly to their next away venue. The solution to the TTP is the tournament schedule that minimizes the total distance traveled by the n teams.

In the TTP, a double round-robin schedule must satisfy the following conditions:

(a) The each-venue condition: Each pair of teams must play two sets, once in each other’s home venue. (b) The at-most-three condition: No team may have a home stand or road trip lasting more than three sets. (c) The no-repeat condition: A team cannot play against the same opponent in two consecutive sets.

For each ordered pair (i, s), define Hi,s and Ri,s to be the number of home and away sets played by team i within the first s sets. By definition, |Hi,s + Ri,s| = s.

In the NPB, we require two additional “balancing” conditions for the 8 rounds (40 sets) of intra-league games:

(d) The each-round condition: Each pair of teams must play exactly once per round, with their matches in rounds 2t − 1 and 2t taking place at different venues. (e) The diff-two condition: |Hi,s − Ri,s| ≤ 2 for all ordered pairs (i, s).

Note that condition (d) automatically implies condition (a).

To illustrate, we provide the first ten sets of the NPB Central League for the 2010 season, where home teams are marked in bold. We see that this is a double round-robin schedule satisfying all of the above conditions.

Team 1 2 3 4 5 6 7 8 9 10 Carp D TG SB SD TG B Tigers B CD GS GB CD S Dragons C S T BG B C S T G Baystars T G S DC D T G S C Giants S BCT D T S BC D Swallows G DBC T C G DB T

We now present the optimal intra-league schedule for both the Pacific and Central Leagues (with 120 games = 40 sets of three games = 8 rounds of five sets), as well as the optimal inter-league schedule (with 24 games = 12 sets of two games). 3 Optimal Intra-League Schedule for the Pacific League

Team R1 R2 R3 R4 R5 R6 R7 R8 Chiba Marines (M) LFEHB EHBLF EHBLF BLFEH FBHLE HLEFB ELFBH FBHEL Tohoku Eagles (E) FHMBL MBLFH MFLHB LHBMF LFBHM BHMLF MFBHL BHLMF Hokkaido Fighters (F) EMBLH BLHEM LEHBM HBMLE MELBH LBHME HEMLB MLBHE Orix Buffaloes (B) HLFEM FEMHL HLMFE MFEHL HMEFL EFLHM LHEMF EMFLH Fukuoka Hawks (H) BELMF LMFBE BMFEL FELBM BLMEF MEFBL FBLEM LEMFB Saitama Lions (L) MBHFE HFEMB FBEMH EMHFB EHFMB FMBEH BMHFE HFEBM

For both the 2010 NPB schedule (first table) and our provably-optimal schedule (second table), we present the 6 by 15 matrix enumerating the number of trips taken by every team between each pair of cities.

ME MF MB MH ML EF EB EH EL FB FH FL BH BL HL Total Chiba Marines (M) 576550120001211 36 Tohoku Eagles (E) 511017666002101 37 Hokkaido Fighters (F) 060115102553111 32 Orix Buffaloes (B) 026111601410551 34 Fukuoka Hawks (H) 110600142171605 35 Saitama Lions (L) 011150008106245 34 Total Trips 11 18 14 14 13 13 15 12 19 11 13 13 17 11 14 208

ME MF MB MH ML EF EB EH EL FB FH FL BH BL HL Total Chiba Marines (M) 442353001010420 29 Tohoku Eagles (E) 321116324000312 29 Hokkaido Fighters (F) 410124000333321 27 Orix Buffaloes (B) 213114200201651 29 Fukuoka Hawks (H) 112312023221403 27 Saitama Lions (L) 011242303032043 28 Total Trips 14 10 9 11 14 21 8 4 11 7 9 7 20 14 10 169

We then compare these two schedules with respect to the total distance traveled by all six teams, using our distance matrix.

Distance Distance Reduction Trips Trips Reduction (Existing) (Optimal) in Distance (Existing) (Optimal) in Trips Chiba Marines 23266 16606 28.6 % 36 29 19.4 % Tohoku Eagles 23710 17975 24.2 % 37 29 21.6 % Hokkaido Fighters 28599 20234 29.2 % 32 27 15.6 % Orix Buffaloes 24128 18713 22.4 % 34 29 14.7 % Fukuoka Hawks 33352 21143 36.6 % 35 27 22.9 % Saitama Lions 20885 19498 6.6 % 34 28 17.6 % Total 153940 114169 25.8 % 208 169 18.8 % 4 Optimal Intra-League Schedule for the Central League

Team R1 R2 R3 R4 R5 R6 R7 R8 Hiroshima Carp (C) TGBSD BSDTG DTGSB GSBDT BTDSG DSGBT BTDSG DSGBT Hanshin Tigers (T) CDGBS GBSCD BCDGS DGSBC DCSGB SGBDC DCSGB SGBDC Chunichi Dragons (D) BTSGC SGCBT CSTBG TBGCS TGCBS CBSTG TGCBS CBSTG Yokohama Baystars (B) DSCTG CTGDS TGSDC SDCTG CSGDT GDTCS CSGDT GDTCS Yomiuri Giants (G) SCTDB TDBSC SBCTD CTDSB SDBTC BTCSD SDBTC BTCSD Tokyo Swallows (S) GBDCT DCTGB GDBCT BCTGD GBTCD TCDGB GBTCD TCDGB

For both the 2010 NPB schedule (first table) and our provably-optimal schedule (second table), we present the 6 by 15 matrix enumerating the number of trips taken by every team between each pair of cities.

CT CD CB CG CS TD TB TG TS DB DG DS BG BS GS Total Hiroshima Carp (C) 585530111000022 33 Hanshin Tigers (T) 421015557010200 33 Chunichi Dragons (D) 070015012554210 33 Yokohama Baystars (B) 005031601610740 34 Yomiuri Giants (G) 101602131150507 33 Tokyo Swallows (S) 220040006114166 33 Total Trips 12 19 12 11 12 13 13 10 18 13 13 8 17 13 15 199

CT CD CB CG CS TD TB TG TS DB DG DS BG BS GS Total Hiroshima Carp (C) 443210310013113 27 Hanshin Tigers (T) 611005331110034 29 Chunichi Dragons (D) 041303104441021 28 Yokohama Baystars (B) 412100400520253 29 Yomiuri Giants (G) 402201030151234 28 Tokyo Swallows (S) 430010103005345 29 Total Trips 22 13 9 8 2 9 12 7 8 11 13 10 8 18 20 170

We then compare these two schedules with respect to the total distance traveled by all six teams, using our distance matrix.

Distance Distance % Reduction Trips Trips % Reduction (Existing) (Optimal) in Distance (Existing) (Optimal) in Trips Hiroshima Carp 17850 11741 34.2 % 33 27 18.2 % Hanshin Tigers 14304 8712 39.1 % 33 29 12.1 % Chunichi Dragons 11790 11665 1.1 % 33 28 15.2 % Yokohama Baystars 13104 8929 31.9 % 34 29 14.7 % Yomiuri Giants 11469 9020 21.4 % 33 28 15.2 % Tokyo Swallows 10550 7769 26.4 % 33 29 12.1 % Total 79067 57836 26.8 % 199 170 14.6 % 5 Optimal Inter-League Schedule

Below is the 2010 NPB inter-league schedule, where the Pacific League teams are p1 (Fukuoka), p2 (Orix), p3 (Saitama), p4 (Chiba), p5 (Tohoku), p6 (Hokkaido), and the Central League teams are c1 (Hiroshima), c2 (Hanshin), c3 (Chunichi), c4 (Yokohama), c5 (Yomiuri), and c6 (Yakult).

Team 1 2 3 4 5 6 7 8 9 10 11 12 p1 c3 c6 c2 c1 c4 c5 c3 c6 c1 c2 c4 c5 p2 c6 c3 c1 c2 c5 c4 c6 c3 c2 c1 c5 c4 p3 c4 c5 c6 c3 c1 c2 c4 c5 c6 c3 c2 c1 p4 c5 c4 c3 c6 c2 c1 c5 c4 c3 c6 c1 c2 p5 c1 c2 c4 c5 c3 c6 c1 c2 c4 c5 c3 c6 p6 c2 c1 c5 c4 c6 c3 c2 c1 c5 c4 c6 c3 c1 p5 p6 p2 p1 p3 p4 p5 p6 p1 p2 p4 p3 c2 p6 p5 p1 p2 p4 p3 p6 p5 p2 p1 p3 p4 c3 p1 p2 p4 p3 p5 p6 p1 p2 p4 p3 p5 p6 c4 p3 p4 p5 p6 p1 p2 p3 p4 p5 p6 p1 p2 c5 p4 p3 p6 p5 p2 p1 p4 p3 p6 p5 p2 p1 c6 p2 p1 p3 p4 p6 p5 p2 p1 p3 p4 p6 p5

We improve this schedule in two ways. First, we determine the distance-optimal schedule where the home slots are uniform, i.e., all Pacific League teams play their home games on the same day.

Team 1 2 3 4 5 6 7 8 9 10 11 12 p1 c2 c3 c1 c2 c3 c6 c4 c5 c6 c1 c4 c5 p2 c4 c6 c5 c3 c6 c1 c2 c3 c1 c2 c5 c4 p3 c3 c1 c2 c6 c4 c2 c5 c6 c4 c5 c3 c1 p4 c5 c4 c6 c5 c2 c4 c3 c1 c2 c6 c1 c3 p5 c1 c2 c3 c4 c1 c5 c6 c4 c5 c3 c2 c6 p6 c6 c5 c4 c1 c5 c3 c1 c2 c3 c4 c6 c2 c1 p5 p3 p1 p6 p5 p2 p6 p4 p2 p1 p4 p3 c2 p1 p5 p3 p1 p4 p3 p2 p6 p4 p2 p5 p6 c3 p3 p1 p5 p2 p1 p6 p4 p2 p6 p5 p3 p4 c4 p2 p4 p6 p5 p3 p4 p1 p5 p3 p6 p1 p2 c5 p4 p6 p2 p4 p6 p5 p3 p1 p5 p3 p2 p1 c6 p6 p2 p4 p3 p2 p1 p5 p3 p1 p4 p6 p5 We now remove the uniformity restriction and consider non-uniform schedules. This is a much harder problem.

Team 1 2 3 4 5 6 7 8 9 10 11 12 p1 c3 c5 c1 c3 c2 c1 c6 c2 c4 c5 c6 c4 p2 c5 c3 c2 c1 c3 c6 c1 c4 c5 c6 c4 c2 p3 c4 c2 c6 c5 c4 c3 c5 c1 c3 c2 c1 c6 p4 c2 c4 c5 c4 c6 c5 c3 c6 c1 c3 c2 c1 p5 c1 c6 c4 c6 c5 c2 c4 c3 c2 c1 c5 c3 p6 c6 c1 c3 c2 c1 c4 c2 c5 c6 c4 c3 c5 c1 p5 p6 p1 p2 p6 p1 p2 p3 p4 p5 p3 p4 c2 p4 p3 p2 p6 p1 p5 p6 p1 p5 p3 p4 p2 c3 p1 p2 p6 p1 p2 p3 p4 p5 p3 p4 p6 p5 c4 p3 p4 p5 p4 p3 p6 p5 p2 p1 p6 p2 p1 c5 p2 p1 p4 p3 p5 p4 p3 p6 p2 p1 p5 p6 c6 p6 p5 p3 p5 p4 p2 p1 p4 p6 p2 p1 p3

We note that both the optimal uniform and non-uniform schedules are extremely close to the theoretical lower bound, found by adding up the minimum possible travel distance for each of the twelve teams. Both schedules achieve a > 15% reduction compared to the 2010 NPB schedule.

Distance Optimal Optimal Theoretical (Existing) Uniform Non-Uniform Lower Bound Central League 27205 26587 26258 26077 Pacific League 23929 16698 16692 16686 Total Distance 51134 43285 42950 42763 Improvement 15.3% 16.0% 16.4%