Richard Hoshino 星野リチャ ド Post-Doctoral Fellow, National Institute of Informatics. Joint work with Ken-ichi Kawarabayashi

Transcription

1 Richard Hoshino 星野リチャ ド Post-Doctoral Fellow, National Institute of Informatics Joint work with Ken-ichi Kawarabayashi

2 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!

3 Context and Motivation

4 Life in Makuhari ( 幕張 ) Our Apartment Kanda University Chiba Marine Stadium Kaihin-Makuhari Station

5 Chiba Marines Schedule (2010) Saitama Hokkaido Tohoku Orix Fukuoka Saitama Hokkaido Orix Tohoku Fukuoka Saitama Fukuoka Hokkaido Orix Tohoku Orix Hokkaido Fukuoka Saitama Tohoku 23,266 kilometres in total travel. (HOME sets are marked in red.)

6 Chiba Marines Schedule (2010) Fukuoka Orix Saitama Hokkaido Saitama Fukuoka Tohoku Orix Hokkaido Tohoku Hokkaido Orix Saitama Fukuoka Tohoku Hokkaido Orix Saitama Fukuoka Tohoku 23,266 kilometres in total travel. (HOME sets are marked in red.)

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8 PL Intra-League Travel Team 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

9 Reducing Travel Distance Team 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) Reduction in Travel 16, % 17, % 20, % 18, % 21, % 19, % 114, %

10 Reducing Total Trips Team Name Trips (2010) Chiba 36 Tohoku 37 Hokkaido 32 Orix 34 Fukuoka 35 Saitama 34 TOTAL 208 Trips (RH+KK) Reduction in Trips % % % % % % %

11 Nippon Series Champions!

12 Interview in NII Today Ken-ichi Kawarabayashi National Institute of Informatics Mariko Takahashi Asahi Shimbun

13 Intra-League Scheduling

14 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)

15 Nippon Pro Baseball Schedules Saitama Hokkaido Tohoku Orix Fukuoka Saitama Hokkaido Orix Tohoku Fukuoka Saitama Fukuoka Hokkaido Orix Tohoku 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.

16 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.

17 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.

18 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 2k rounds (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.

19 Concatenating Two Blocks How do we check whether two 10-set blocks can be concatenated to form a 20-set tournament schedule? Team R1 R2 R3 R4 Chiba (C) S H T F O T F O S H S H T F O T F O S H Tohoku (T) H F C O S C O S H F H F C O S C O S H F Hokkaido (H) T C O S F O S F T C T C O S F O S F T C Orix (O) Fukuoka (F) Saitama (S) F S H T C O T S C H C O F H T H T C F S S C H O T F H T C O F S H T C O T S C H C O F H T H T C F S S C H O T F H T C O

20 Concatenating Two Blocks The conditions Home-Away, Each-Round, Diff-Two are automatically satisfied, since each block is feasible. Team R1 R2 R3 R4 Chiba (C) S H T F O T F O S H S H T F O T F O S H Tohoku (T) H F C O S C O S H F H F C O S C O S H F Hokkaido (H) T C O S F O S F T C T C O S F O S F T C Orix (O) Fukuoka (F) Saitama (S) F S H T C O T S C H C O F H T H T C F S S C H O T F H T C O F S H T C O T S C H C O F H T H T C F S S C H O T F H T C O

21 Concatenating Two Blocks To check No-Repeat, just look at the last column in Block #1 and the first column in Block #2. Team R1 R2 R3 R4 Chiba (C) S H T F O T F O S H S H T F O T F O S H Tohoku (T) H F C O S C O S H F H F C O S C O S H F Hokkaido (H) T C O S F O S F T C T C O S F O S F T C Orix (O) Fukuoka (F) Saitama (S) F S H T C O T S C H C O F H T H T C F S S C H O T F H T C O F S H T C O T S C H C O F H T H T C F S S C H O T F H T C O

22 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 R1 R2 R3 R4 Chiba (C) S H T F O T F O S H S H T F O T F O S H Tohoku (T) H F C O S C O S H F H F C O S C O S H F Hokkaido (H) T C O S F O S F T C T C O S F O S F T C Orix (O) Fukuoka (F) Saitama (S) F S H T C O T S C H C O F H T H T C F S S C H O T F H T C O F S H T C O T S C H C O F H T H T C F S S C H O T F H T C O

23 Graph-Theoretic Reformulation Each team starts and ends the season at home (v start, v end ) Each vertex x t,u with 1 u m, represents the first two columns of the t th block (matches = 1 st, home teams = 2 nd ) Each vertex y t,u with 1 u m, represents the last two columns of the t th block (home teams = 9 th, matches = 10 th )

24 Explanation of the variable m Team A B C D E F 1 2 D 1 E 1 F 1 A 0 B 0 C 0 x 1,u There are n n / 2 n 6 20 n / 2 3 ways to select the home teams of any column. So there are 6 3 n / 2! 3! 120 ways to select the three matches of any column. For n = 6, m = = In general, n m n / 2 2 n / 2!

25 Construction of Edge y t,v x t+1,u Team A B C D E F F 0 E 1 D 1 C 0 B 0 A B 0 A 0 F 0 E 1 D 1 C 1 y 1,v x 2,u y t,v x t+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 y t,v x t+1,u is the distance traveled by the n teams moving from set 2t(n-1) to 2t(n-1)+1.

26 Construction of Edge x t,u y t,v Team A B 0?????? 1 F B A 0?????? 0 E C F 0?????? 1 D D E 1?????? 1 C E D 1?????? 0 B F C 1?????? 0 A x 1,u y 1,v x t,u y t,v is an edge iff there exists a (feasible) block satisfying the five mb-ttp conditions. The weight of edge x t,u y t,v is the minimum possible total distance traveled by the n teams within that block.

27 Dijkstra s Algorithm The directed graph has 2mk+2 vertices and at most 2m+(2k-1)m 2 edges. Each edge has a weight. Now apply Dijkstra s Algorithm to find the shortest path v start x 1,u1 y 1,v1 x k,uk y k,vk v end which produces the optimal solution of the mb-ttp.

28 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

29 Chiba Marines Schedule (2012?) Saitama Hokkaido Tohoku Fukuoka Orix Tohoku Fukuoka Orix Saitama Hokkaido Tohoku Fukuoka Orix Saitama Hokkaido Orix Saitama Hokkaido Tohoku Fukuoka 16,606 kilometres in total travel by this team, a 28.6% reduction. (HOME sets are marked in red.)

30 Chiba Marines Schedule (2012?) Hokkaido Orix Fukuoka Saitama Tohoku Fukuoka Saitama Tohoku Hokkaido Orix Tohoku Saitama Hokkaido Orix Fukuoka Hokkaido Orix Fukuoka Tohoku Saitama 16,606 kilometres in total travel by this team, a 28.6% reduction. (HOME sets are marked in red.)

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32 Results for NPB Central League Team Name Distance (2010) Distance (New) Reduction in Travel Trips (2010) Trips (New) Reduction in Trips Hiroshima 17,850 11, % % Hanshin 14,304 8, % % Chunichi 11,790 11, % % Yokohama 13,104 8, % % Yomiuri 11,469 9, % % Yakult 10,550 7, % % TOTAL 79,067 57, % % For the two leagues, we can reduce total intra-league travel by 60,000 km and eliminate 68 trips!

33 Inter-League Scheduling

34 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

35 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

36 Minimum-Weight Perfect Matching

37 Triangle Cover

38 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)

39 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 Team R1 R2 R3 R4 R5 R6 X1 Y1 Y2 Y3 Y1 Y2 Y3 X1 Y3 Y2 Y1 Y3 Y1 Y2 X2 Y2 Y3 Y1 Y2 Y3 Y1 X2 Y1 Y3 Y2 Y1 Y2 Y3 X3 Y3 Y1 Y2 Y3 Y1 Y2 X3 Y2 Y1 Y3 Y2 Y3 Y1 Y1 X1 X3 X2 X1 X3 X2 Y1 X2 X3 X1 X2 X1 X3 Y2 X2 X1 X3 X2 X1 X3 Y2 X3 X1 X2 X3 X2 X1 Y3 X3 X2 X1 X3 X2 X1 Y3 X1 X2 X3 X1 X3 X2 Trips = = 24 Trips = = 32

40 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 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 X3 X2 X1 X3 X2 X1 Y3 X1 X2 X3 X1 X3 X2 Distance = =16 Distance = =12

41 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.

42 Rooted 4-cycle-covers For each team, find a minimum weight 4-cycle-cover rooted at that vertex.

43 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

44 Best Uniform Tournament The optimal uniform inter-league tournament for NPB has total distance just = 522 km more than the theoretical trivial lower bound. Team ILB Optimal Diff. Fukuoka Orix Saitama Chiba Tohoku Hokkaido TOTAL Team ILB Optimal Diff. Hiroshima Hanshin Chunichi Yokohama Yomiuri Yakult TOTAL

45 Best Non-Uniform Tournament The optimal non-uniform inter-league tournament for NPB has total distance just = 187 km more than the theoretical trivial lower bound. Team ILB Optimal Diff. Fukuoka Orix Saitama Chiba Tohoku Hokkaido TOTAL Team ILB Optimal Diff. Hiroshima Hanshin Chunichi Yokohama Yomiuri Yakult TOTAL

46 Implementation

47 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!