Graph Theory andSports SchedulingRichard Hoshino and Ken-ichi Kawarabayashi

IntroductionThe effects of global warming have been welldocumented, especially in recent years. As a result,the majority of countries have made a commit-ment to reducing their greenhouse gas emissions,including many whose national governments havemade ambitious and unrealistic promises. Meetingthese targets will require a coordinated effort frompolicymakers, businesses, and large industries, andnumerous creative solutions will need to be imple-mented to achieve the desired goal. One potentialsolution is based on discrete mathematics, wherecombinatorial and graph-theoretic techniques areapplied to scheduling optimization, leading toeconomic and environmental benefits.

There are many practical roles for mathemat-ically optimal schedules that reduce total traveldistance, including supply-chain logistics and air-plane flight assignments. In this paper we describehow to optimize the regular-season schedule forNippon Professional Baseball (NPB), Japan’s mostpopular professional sports league, with annualrevenues topping one billion U.S. dollars.

Given the authors’ background as graph theo-rists, this research was motivated by the innocentquestion of whether NPB scheduling could be

Richard Hoshino is a mathematics tutor at Quest UniversityCanada. His email address is richard.hoshino@questu.ca.

Ken-ichi Kawarabayashi is a professor at the National Insti-tute of Informatics, Japan. His email address is k_keniti@nii.ac.jp.

DOI: http://dx.doi.org/10.1090/noti1010

converted into a much simpler shortest-path prob-lem. As we describe in this paper, the answerto this question is affirmative. Consequently, wehave succeeded in generating the distance-optimalNPB regular-season schedule which retains all ofthe league’s constraints that ensure competitivebalance while reducing the total travel distance by24.3%, or nearly 70,000 kilometers, as comparedto the 2010 season schedule.

To solve the NPB scheduling problem, we havegeneralized and extended the Traveling Tourna-ment Problem (TTP), a well-known topic in sportsscheduling [10]. Our research has produced fivepapers, [4], [5], [6], [7], [8], describing the theo-retical aspects of the problem, providing variousheuristics for generating distance-optimal intra-league and inter-league schedules, and applyingthe results to optimize the NPB league schedule.

Shortly after introducing the Traveling Tourna-ment Problem [2], Easton et al. formed a consultingcompany to develop schedules for professionalsports leagues. Their company, the Sports Schedul-ing Group, has received the contract to produce theregular-season schedule for Major League Baseballin six of the past seven years. Having now com-pleted all of our research on NPB scheduling, ourhope is to obtain the contract to produce futureNPB regular-season schedules. We are excited bythe possibility of sharing our expertise and passionwith Nippon Professional Baseball, working in part-nership with the league to produce schedules thatsave money and reduce greenhouse gas emissions,thus making an important contribution to Japan,both economically and environmentally.

726 Notices of the AMS Volume 60, Number 6

Figure 1. Location of the twelve teams in the NPB.

Summary of ResultsNippon Professional Baseball is split into the six-team Pacific League and the six-team Central League.Each team plays 144 games during the regularseason, with 120 intra-league games (against teamsfrom their own league) and 24 inter-league games(against teams from the other league).

The location of each team’s home stadium isgiven in Figure 1. For readability, we label eachteam as follows: the Pacific League teams arep1 (Fukuoka), p2 (Orix), p3 (Saitama), p4 (Chiba),p5 (Tohoku), and p6 (Hokkaido); and the CentralLeague teams are c1 (Hiroshima), c2 (Hanshin), c3

(Chunichi), c4 (Yokohama), c5 (Yomiuri), and c6

(Yakult).Specifically, each NPB team plays twelve home

games and twelve away games against each of theother five teams in its league (24×5), in addition totwo home games and two away games against allsix teams in the other league (4×6). All twenty-fourinter-league games take place during a commonfive-week stretch beginning in mid-May, right nearthe start of the season.

As in Major League Baseball, nearly all NPBgames occur in sets of three games. Thus, wewill adopt the same structure when building ourdistance-optimal schedule. For intra-league play,we have constructed a schedule where each teamplays 40 sets of three games, with each teamhaving an opponent in every time slot. (Similarly,for inter-league play, each team plays 12 sets oftwo games.)

For each team, we define a trip to be any pairof consecutive sets not occurring in the samevenue, i.e., any situation where a team has to travelfrom one venue to another to play its next set ofgames. In Table 1 we list the total distance traveledby all teams under our mathematically optimal

schedule, comparing that to the actual distancetraveled by the teams during the 2010 season (notincluding games rescheduled due to weather). Wealso provide statistics for the number of total tripstaken by the teams. Our optimal schedule reducestotal distance by nearly 70,000 kilometers.

In this paper we present our analysis for theNPB intra-league problem. (We refer the readerto [8] for a detailed analysis of the inter-leagueproblem.) In the following section we presentthe Multi-Round Balanced Traveling TournamentProblem, which generalizes the TTP and preciselymodels the scheduling parameters of the NPB. Wethen present our shortest-path reformulation andapply it to produce distance-optimal intra-leagueschedules for the Pacific and Central Leagues.

Distance Distance Reduction Trips Trips Reduction(2010) (Optimal) in Distance (2010) (Optimal) in Trips

Intra-League (PL) 153,940 114,169 25.8 % 208 169 18.8 %Intra-League (CL) 79,067 57,836 26.8 % 199 170 14.6 %

Inter-League 51,134 42,950 16.0 % 108 101 6.5 %Total 284,141 214,955 24.3 % 515 440 14.6 %

Table 1. The distance-optimal NPB scheduleversus the actual 2010 regular-season schedule.

The Multi-Round Balanced TravelingTournament ProblemConsider the intra-league schedule given in Table 2,which consists of n = 6 teams each playing k = 4blocks of ten sets. Each block consists of tworounds, with each round having n − 1 = 5 sets.Thus, each team plays a total of k(2n−2) = 40 sets.In this schedule,1 as with all subsequent schedulespresented in this paper, home sets are marked inred.

Let n and k be positive integers. Let D bethe n× n distance matrix, where entry Di,j is thedistance between the home stadiums of teams iand j . By definition, Di,j = Dj,i for all 1 ≤ i, j ≤ n,and all diagonal entries Di,i are zero.

For any pair (n, k) and distance matrix D, thesolution to the Multi-Round Balanced TravelingTournament Problem (mb-TTP) is an intra-leaguetournament schedule that minimizes the totaldistance traveled by all n teams, subject to thefollowing conditions:

(a) The compactness condition: The tourna-ment lasts k(2n − 2) sets, i.e., 2k rounds,where each team has one set scheduled ineach time slot. (Thus n must be even.)

(b) The each-round condition: Each pair ofteams must play exactly once per round,with their matches in rounds 2t − 1 and

1For example, team p1 starts by playing a home set againstp2, followed by three consecutive road sets against p5, p3,p4, then returning home to play three consecutive home setsagainst p6, p3, p4, and so on.

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Table 2. A 404040-set (120120120-game) intra-league schedule for the NPB Pacific League.

2t taking place at different venues (for all1 ≤ t ≤ k).

(c) The at-most-three condition: No team mayhave a home stand or road trip lastingmore than three sets.

(d) The no-repeat condition: A team cannotplay against the same opponent in twoconsecutive sets.

(e) The diff-two condition: Let Hi,s and Ri,s bethe number of home and away sets playedby team i within the first s sets. Then|Hi,s − Ri,s| ≤ 2 for all (i, s) with 1 ≤ i ≤ nand 1 ≤ s ≤ k(2n− 2).

For example, one can quickly verify that Table 2satisfies all five conditions. When calculating thetotal distance, we will assume that each team beginsthe tournament at home and returns home afterhaving played their last away set. Furthermore,when a team is scheduled for a road trip consistingof multiple away sets, the team doesn’t return totheir home city but rather proceeds directly totheir next away venue.2 The mb-TTP is an extensionof the well-known NP-hard Traveling SalesmanProblem, asking for an optimal schedule linkingvenues that are close to one another.

We now present an algorithm for solving themb-TTP, for any k ≥ 1, by reformulating it as ashortest-path problem on a directed graph. Wewill create a source node and a sink node andlink them to numerous vertices in a graph whose(weighted) edges represent the possible blocksthat can appear in an optimal schedule. We thenapply Dijkstra’s Algorithm [1] to find the pathof minimum weight between the source and thesink, which is an O(|V | log |V | + |E|) graph searchalgorithm that can be applied to any graph ordigraph with nonnegative edge weights.

Shortest-Path ReformulationBy definition, a block is a two-round tournamentschedule satisfying the conditions of the mb-TTP,

2Thus, after the first block of ten sets, team p1 will have trav-eled a total distance of Dp1,p5 +Dp5,p3 +Dp3,p4 +Dp4,p1 +Dp1,p6 +Dp6,p2 +Dp2,p1 .

with each of the n teams playing 2(n − 1) setsof games. To solve the mb-TTP, we first computethe set of blocks that can appear in a distance-optimal tournament. We then introduce a simple“concatenation matrix” to check whether two pre-computed blocks can be joined together to form amultiblock schedule without violating any of thefive conditions of the mb-TTP. As we will explain,to determine whether two (feasible) blocks B1 andB2 can be concatenated, it suffices to check just thelast two columns of B1 and the first two columnsof B2.

Each column of a block represents a set con-sisting of n

2 different matches, with each matchspecifying the two teams as well as the sta-dium/venue. Thus, a match identifies the hometeam and away team, not just each team’s opponent.

For any column, there are(nn/2

)ways to select the

home teams. Also, there are(nn/2

)·(n2

)! ways to

specify the matches of any column, since there are(n2

)! ways to map any choice of the n

2 home teams

to the unselected n2 away teams to decide the set

of n2 matches. Hence, there are m =

(nn/2

)2·(n2

)!

different ways we can specify the matches of thefirst column and the home teams of the second

column. For n = 6, we have m =(

63

)2× 3! = 2400.

There are m ways that the first two columns ofa block can be chosen as described above, withthe first column listing matches and the secondcolumn listing home teams. Now use any method,such as a lexicographic ordering, to index these moptions with the integers from 1 tom. By symmetry,there are m different ways we can specify the lasttwo columns of a block, with the last columnlisting matches and the second-to-last columnlisting home teams. Thus, we use the same schemeto index these m options. To avoid confusion, wewrite the home teams column in binary form, with1 representing a home game and 0 representingan away game.

728 Notices of the AMS Volume 60, Number 6

Figure 2. Reformulation of the kkk-block mb-TTP as a shortest-path problem.

For example, (p5, p3, p2, p6, p1, p4)T is one ofthe 120 possibilities for the matches column, and(0,1,1,0,1,0)T is one of the 20 possibilities forthe home teams column. We remark that if welisted the column of opponents rather than thecolumn of matches, there would be only 120

23 = 15unique columns, corresponding to the 15 perfectmatchings of the complete graph K6.

For the NPB Pacific League, there exists someinteger q (with 1 ≤ q ≤ 2400) that is the indexof the instance where the home teams column is(0,1,1,0,1,0)T and the matches column is (p5,p3, p2, p6, p1, p4)T . Similarly, there exists somer (with 1 ≤ r ≤ 2400) that is the index of theinstance where the two columns are (p2, p1, p6, p5,p4, p3)T and (1,1,0,0,0,1)T . In Table 2 the lasttwo columns of block 1 have index q and the firsttwo columns of block 2 have index r .

For each pair (u1, u2), with 1 ≤ u1, u2 ≤ m,define Cu2,u1 to be the n× 4 concatenation matrixwhere the first two columns list the home teamsand matches with index u2, and the next twocolumns list the matches and home teams withindex u1. For the indices x and y from the previousparagraph, we have

Cq,r =

0 p5 p2 11 p3 p1 11 p2 p6 00 p6 p5 01 p1 p4 00 p4 p3 1

Note thatCq,r has no row with four home sets, no

row with four away sets, and no row with the sameopponent appearing in columns 2 and 3. As wedescribe in Theorem 1, these three properties are anecessary and sufficient condition for whether twofeasible blocks can be concatenated to produce amultiblock schedule satisfying the conditions ofthe mb-TTP.

Before we proceed with Theorem 1, let us explainthe role of m and Cu2,u1 in the construction of ourdirected graph. Let G consist of a source vertex

vstart , a sink vertex vend , and vertices xt,u and yt,udefined for each 1 ≤ t ≤ k and 1 ≤ u ≤m.

We now describe how these edges are connectedwith a pictorial representation of G in Figure 2.For notational simplicity, denote v1 → v2 as thedirected edge from v1 to v2.

(i) For each 1 ≤ u ≤m, add the edge vstart →x1,u.

(ii) For each 1 ≤ u ≤ m, add the edge yk,u →vend .

(iii) For each 1 ≤ t ≤ k and for each 1 ≤ u1, u2 ≤m, add the edge xt,u1 → yt,u2 iff there existsa (feasible) block for which the first twocolumns have index u1 and the last twocolumns have index u2.

(iv) For each 1 ≤ t ≤ k − 1 and for each1 ≤ u1, u2 ≤m, add the edge yt,u2 → xt+1,u1

iff the concatenation matrix Cu2,u1 has norow with four home sets, no row withfour away sets, and no row with the sameopponent appearing in columns 2 and 3.

The following theorem [4] shows that the k-blockmb-TTP can be reformulated in a graph-theoreticcontext for any k ≥ 1.

Theorem 1. Every feasible solution of the mb-TTPcan be described by a path from vstart to vend ingraphG. Conversely, any path from vstart to vend inG corresponds to a feasible solution of the mb-TTP.

Having constructed our digraph, we now assigna weight to each edge using the distance matrix Dso that the shortest path (i.e., path of minimumtotal weight) from vstart to vend corresponds to theoptimal solution of the mb-TTP that minimizes thetotal distance traveled by the n teams.

For any block, we define its in-distance to be thetotal distance traveled by the n teams within thatblock, i.e., starting from set 1 and ending at set2(n−1). Note that the in-distance does not includethe distance traveled by the teams heading to thevenue of set 1 or from the venue of set 2(n − 1).We will use this definition in part (c) below:

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Table 3. The distance-optimal intra-league schedule for the NPB Central League.

Table 4. Comparison of intra-league schedulesfor the Pacific League.

(a) For each 1 ≤ u ≤ m, the weight of edgevstart → x1,u is the total distance traveledby the n

2 teams making the trip from theirhome city to the venue of their opponentin set 1.

(b) For each 1 ≤ u ≤ m, the weight of edgeyk,u → vend is the total distance traveled

by the n2 teams making the trip from the

venue of their opponent in set 2k(n − 1)back to their home city.

(c) For each 1 ≤ t ≤ k, and for each 1 ≤u1, u2 ≤m, the weight of edge xt,u1 → yt,u2

is the minimum in-distance of a block,selected among all blocks for which thefirst two columns have index u1 and thelast two columns have index u2.

(d) For each 1 ≤ t ≤ k − 1 and for each1 ≤ u1, u2 ≤m, the weight of edge yt,u2 →xt+1,u1 is the total distance traveled by theteams that travel from their match in set2t(n−1) to their match in set 2t(n−1)+1,where the last two columns of the tth blockhave index u2 and the first two columns ofthe (t + 1)th block have index u1.

To illustrate (d), consider the first two blocks inTable 2, where the last two columns of block 1 haveindex q and the first two columns of block 2 haveindex r . When we concatenate these two blocks,the weight of edge y1,q → x2,r is the total distancetraveled by the teams from their matches in set10 to their matches in set 11. This sum equals

Dp3,p1 +Dp1,p4 +Dp4,p3 , the distances traveled byteams p2, p5, and p6, respectively.

By this construction, we have produced aweighted digraph. In part (c), suppose there existtwo blocks B and B′ for which the first twocolumns have index u1 and the last two columnshave index u2. If the in-distance of B is less thanthe in-distance of B′, then block B′ cannot be ablock in an optimal solution, since we can justreplace B′ by B to create a feasible solution witha lower objective value. This trivial observation,based on Bellman’s Principle of Optimality, allowsus to assign the minimum in-distance as the weightof edge xt,u1 → yt,u2 , for all 1 ≤ u1, u2 ≤ m. As aresult, we have a digraph G on 2mk+ 2 verticesand at most 2m+ (2k− 1)m2 edges, with a uniqueweight for each edge. Combined with the previoustheorem, we have established the following.

Theorem 2. Let P = vstart → x1,p1 → y1,q1 →x2,p2 → y2,q2 → ·· · → xk,pk → yk,qk → vend be ashortest path in G from vstart to vend , i.e., a paththat minimizes the total weight. For each 1 ≤ t ≤ k,let Bt be the block of minimum in-distance selectedamong all blocks for which the first two columnshave index pt and the last two columns have indexqt . Then the multiblock schedule S = B1, B2, . . . , Bk,created by concatenating the k blocks consecutively,is an optimal solution of the mb-TTP.

Application to the NPBTherefore, we have shown that the mb-TTP isisomorphic to finding the shortest weighted pathin the directed graph G. For the case n = 6, wecan show that G consists of 4800k + 2 verticesand 2400+ 2400+ 2618520k+ 1486320(k− 1) =4104840k − 1481520 edges. Given the distancematrices for the NPB Pacific and Central Leagues,we can determine the appropriate weights for eachedge in G to determine the shortest path fromvstart to vend using Dijkstra’s Algorithm, which inturn generates the distance-optimal schedule. Wewrote our code using Maplesoft; for each league,Maplesoft produced the distance-optimal intra-league schedule after five hours of computation

730 Notices of the AMS Volume 60, Number 6

time. The large majority of the running time wasspent determining the correct edge weight for part(c) of our construction. For more information, werefer the reader to [7].

The optimal Pacific League intra-league scheduleappears in Table 2; the optimal Central Leagueintra-league schedule appears in Table 3.

In Table 1 we produced an overall summaryof the results. In Table 4 we provide a team-by-team breakdown for the NPB Pacific League,comparing our distance-optimal schedule to theactual schedule played by the teams during the2010 regular season.

In addition to the significant 25.8% reduction intotal distance traveled, we also remark that this isa more equitable schedule. In the 2010 NPB intra-league schedule, team p1 traveled nearly 12,500kilometers more than team p3. Under our schedule,the difference between the most traveled and leasttraveled would reduce to just 4,500 kilometers.For the Central League, the difference betweenthe most traveled and least traveled would reducefrom 7,500 kilometers to just 4,000 kilometers.

ImplementationNaturally, there are additional factors involvedwith the actual scheduling of NPB games at thesehome stadiums. For example, one of the ballparkshosts a three-day concert each August, and anotherballpark is used as the locale of the national highschool baseball tournament. Hence those teamsmust play away games on those particular days.In many sports leagues, rival teams play “derbymatches” that need to be scheduled on particulardays to optimize revenue and that are oftendictated by the wishes of television broadcasters toadd drama and boost TV ratings. These constraintsmust be taken into account when producing anoptimal schedule that can be implemented by NPBto ensure no conflicts occur and that the scheduleis the best possible for all parties involved.

EpilogueAfter the publication of our initial results, theNippon Professional Baseball (NPB) League invitedus (the authors) to visit their head office. Over thecourse of three meetings in September 2012, weconsulted the NPB and helped them design theCentral League’s 2013 intra-league schedule [3].

When we met with the NPB head scheduler, welearned that the league has additional schedulingconstraints, such as the “revenue-balancing” re-quirement that each team play the same number ofweekend home games, weekday home games, week-end road games, and weekday road games. In [9],we describe how we fully solved this TTP-variantfor the case n = 6, and helped the NPB designthe intra-league schedule for the 2013 Central

League, reducing the number of total trips by 12and cutting over six thousand kilometers of totaltravel distance, as compared to the schedule fromthe previous season.

We look forward to partnering with the NPB onceagain, and hope to have the opportunity to help thisleague produce future regular-season schedulesthat will result in annual win-wins for the peopleof Japan, both economically and environmentally.

References[1] E. W. Dijkstra (1959), A note on two problems in

connexion with graphs, Numerische Mathematik, 1,269–271.

[2] K. Easton, G. Nemhauser, and M. Trick (2001), Thetraveling tournament problem: description and bench-marks, Proceedings of the 7th International Conferenceon Principles and Practice of Constraint Programming,pp. 580–584.

[3] S. Hesse, Canadian scientist uses math to greenJapanese baseball, http://www.japantimes.co.jp/life/2012/11/25/environment/canadian-scientist-uses-math-to-green-japanese-baseball/, Japan Times, 2012.

[4] R. Hoshino and K. Kawarabayashi (2011), Themulti-round balanced traveling tournament prob-lem, Proceedings of the 21st International Conferenceon Automated Planning and Scheduling (ICAPS), pp.106–113.

[5] (2011), The inter-league extension of the travel-ing tournament problem and its application to sportsscheduling, Proceedings of the 25th AAAI Conferenceon Artificial Intelligence, pp. 977–984.

[6] (2011), The distance-optimal inter-league sched-ule for Japanese pro baseball, Proceedings of the ICAPS2011 Workshop on Constraint Satisfaction Techniquesfor Planning and Scheduling Problems (COPLAS),71–78.

[7] (2011), A multi-round generalization of thetraveling tournament problem and its application toJapanese baseball, European Journal of OperationalResearch, 215, 481–497.

[8] (2011), Scheduling bipartite tournaments tominimize total travel distance, Journal of ArtificialIntelligence Research, 42, 91–124.

[9] , Balancing the Traveling Tournament Problemfor Weekday and Weekend Games, Proceedings of the2013 AAAI Conference, to appear.

[10] G. Kendall, S. Knust, C. C. Ribeiro, and S. Ur-rutia (2010), Scheduling in sports: an annotatedbibliography, Computers and Operations Research, 37,1–19.

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