AbstractA real case of sports scheduling problem is presented. A calendar for two leagues has to be constructed; besides the usual restrictions on the alternation of home- and away-games, one has to consider the fact that some pairs of teams in the two leagues share the same facilities and cannot play home-games simultaneously. Furthermore depending on the results in the first games, one of the leagues is divided for the last games into two subleagues. An optimal solution is constructed by using properties of oriented factorizations of complete graphs

de Werra, D.

Jacot-Descombes, L.

Masson, P.

Elsevier - Publisher Connector 

A constrained sports scheduling problem 

Discrete Applied Mathematics 26 (1990) 41-49 
North-Holland 
41 
A CONSTRAINED SPORTS SCHEDULING PROBLEM 
D. de WERRA, L. JACOT-DESCOMBES and P. MASSON 
Dkpartement de Mathkatiques, .&Cole Polytechnique Fpdprale de Lausanne, MA (Emblem) 
Lausanne, CH-IOIS, Switzerland 
Received 14 June 1988 
Revised 24 June 1988 
A real case of sports scheduling problem is presented. A calendar for two leagues has to be 
constructed; besides the usual restrictions on the alternation of home- and away-games, one has 
to consider the fact that some pairs of teams in the two leagues share the same facilities and can- 
not play home-games simultaneously. Furthermore depending on the results in the first games, 
one of the leagues is divided for the last games into two subleagues. An optimal solution is con- 
structed by using properties of oriented factorizations of complete graphs. 
Keywords. Graph factorization, sports scheduling, oriented coloring 
1. Introduction 
The purpose of this paper is to describe a sports scheduling problem with addi- 
tional constraints due to resources. The problem was posed by an Australian basket- 
ball organization. Before formulating the requirements we define the graph- 
theoretical concepts which will be needed. 
A league of 2n teams is represented by a complete graph K2,,: Each game which 
has to be played by teams i and j is associated with an edge [i, j]. A game between 
team i and teamj takes place in the home-city of either team i or team j; in the first 
case, it is a home-game (denoted by H) for i and an away-game (denoted by A) for 
team j. In this case, it will be represented by an oriented arc (j, i). 
A round for a league of 2n teams consists of a collection of n(2n - 1) games: Each 
team meets once every other team. A schedule is thus represented by an oriented 
+ + 
factorization (M,, M2, . . . , $f2n_1): $fk is a collection of n oriented arcs which 
represent the games played on day k. 
With each schedule S we associate a home-away pattern (HAP) denoted by H(S); 
it is a 2n x (2n - 1) array defined by &(S) =A (respectively H) if team i has an 
away-game (respectively a home-game) on day k. Row i of H(S) will be the profile 
Of team i. If hik(S)=hi,&+l(S) f or some k, 1% ks 2n - 2, we say that team i has a 
break on day k+ 1. Figure 1 shows an oriented factorization of K6 and the 
corresponding HAP. 
Generally breaks are undesirable; it is required to construct schedules where for 
each team home-games and away-games are alternating as regularly as possible (see 
[4, 51 for properties of schedules related to the occurrence of breaks). 
0166-218X/90/$3.50 0 1990, Elsevier Science Publishers B.V. (North-Holland) 
42 D. de Werra et al. 
days 
1234 5 
. . . l denotes a home- 
. . game; an empty _ 
entry is an away- 
. . _ game; breaks are 
. . underlined. 
. . . - 
. . L 
Fig. 1. An oriented factorization and its HAP. 
A factorization (F,, . . . , Fznpl) of Kzn is a decomposition of the edge set into 
2n - 1 factors (or matchings of n edges). An example of factorization is given by 
Fj={[2n,i]}U([i+k,i-k]: k= 1,2 ,..., n-l}, (1.1) 
where numbers i + k and i - k are taken modulo 2n - 1 between 1 and 2n - 1. 
An oriented factorization (?,, . . . , szn _ 1) giving exactly 2n - 2 breaks can be ob- 
tained as follows: 
(a) For each i, orient [i+ k, i-k] to obtain arc (i + k, i- k) if k is odd or arc 
(i-k,i+k) if k is even. 
(b) Orient [2n, i] to obtain arc (i, 2n) for i even and arc (2n, i) for i odd. 
Such an oriented factorization will be called canonical. 
It is easy to verify that breaks occur in pairs on days 3,5,7, . . . ,2n - 1. It is known 
that no schedule can have less than 2n - 2 breaks [4]. 
The following constructions will be based on the method described in [4,6]. 
One should observe that in the schedule constructed above the 2n teams can be 
partitioned into n pairs {al, a,+l >, {a2, an+2 },...,{an,a2,,} such that for each i 
teams ai and a,+i have complementary profiles (on any day one is at home and the 
other is away); furthermore aI, a,+l have no break and for i~2, ai, a,,,; have their 
break on day 2i - 1. In the example of Fig. 1, the pairs are { 1,6}, { 2,3}, { 4,5}. More 
properties of oriented factorizations are given in [1,2,3]. 
In the next section we shall formulate the scheduling problem with its constraints 
and a solution will be presented. 
2. Formulation of the scheduling problem 
A sports organization consists of two leagues: 
rounds have to be played (each team meets every 
League A has 2p teams and two 
other team twice); if teams i and 
A constrained sports scheduling problem 43 
j play in the home-city of i in the first round, they should meet in the home-city of 
j in the second round. For such a schedule 4p - 2 days are needed. League B consists 
of 2q teams which play one round; this takes 2q - 1 days. After this the q best teams 
in B form a new subleague C and the q last teams of B form a subleague C’. C and 
C’ play a round simultaneously; this takes q - 1 days. If 4p = 3q the numbers of days 
needed for league A and for league B (including the subleagues C, C’) are equal. 
In our case we have a league A with 2p = 12 teams and a league B with 2q = 16 
teams. Observe that q is necessarily even. 
(a) In the schedules we want to have the breaks spread as evenly as possible 
among the teams; for each league we want the maximum number of breaks that a 
team has in its schedule to be as small as possible. 
(b) For league A which plays two rounds, it is desired that the matches involving 
the same two teams are not scheduled too close to each other; one may for instance 
require that they are separated by at least 2p- 2 days. 
(c) No team in a league should have a break on the last day of the schedule; this 
means that league A cannot have a break on day 4p - 2 and league B cannot have 
a break on day 2q - 1. Also for subleagues C, C’ no break can occur on the last day 
of their round, i.e., on day 3q-2 =4p-2. 
(d) Finally there are q - 2 disjoint pairs { ik, jk} of teams (with ik in league A and 
j, in league B) such that i, and j, have the same home-stadium. So it is required 
that ik and j, never play both at home on the same day (k = 1, . . . , q - 2). A problem 
with constraints related to the use of common facilities is discussed in [3]. The prob- 
lem is now to construct a schedule which satisfies all these requirements. 
3. Construction of a solution 
For league A, we will use a modified oriented factorization obtained as follows: 
Let (@, , . . . , ?2p_ 1) be a canonical oriented factorization of KzP; reverse the orienta- 
tion of arc (2p, 1) to obtain arc (1,2p) and let (fir’, . . . , &_ ,) be the resulting 
oriented factorization. 
For each oriented factorization of KzP we will renumber the teams al, . . . , azp in 
such a way that ai and a,,; have complementary profiles. The days where a; (and 
a,+i) have a break will be denoted by pi, yi, . . . ; pi = 0 means that aj has no break 
in its profile. In all the factorizations used here, there is at most one pair ai, a,+; 
of teams having a break at a given day. 
So we may number the teams in such a way that pr <&< **. <BP; furthermore 
in our numbering ai, . . . , ap start with a home-game. We shall keep this convention 
in the remainder of this paper. In the modified oriented factorization of Fig. 2 we 
have a,=6, a,=2, a,=4, ad-l, a,=3, a,=5 withp,=2, &=3, /3s=5. 
44 D. de Werra et al. 
?;“1glfi??4 t al= 6 
i+‘i =?i (i=2,...,5) e a2 = 2 
a a3 = 4 
m a4 = 1 
s as = 3 
afj = 5 
days 
Fig. 2. A modified oriented factorization with breaks on days 2, 3, 5. 
Proposition 3.1. The modified factorization ($, . . . , &‘,_ ,) has 2p breaks charac- 
terized by 
P, = 2, pi = 2i- 1 (i = 2, . . ..p). 
Proof. For the canonical oriented factorization (s:,, . . . , g2,_ 1) teams ai, aP+ i have 
a break on day 2i- 1, 21iIp; for al = 1, aP+l = 2p we have pi = 0 (no break). 
When we reverse the orientation of arc (2p, 1) in @, we have al = 2p, aP+ 1 = 1 and 
pr = 2. The rest of the schedule is unchanged. 0 
If fi denotes an oriented factor in KzP, the factor obtained by reversing the 
orientation of each arc in gi will be denoted by fii. 
Let us now consider the schedule S, for two rounds in league A corresponding 
to the concatenation of the following oriented factorizations: 
S*=& )..., $&0(&E; )..., &_,,$,. (3.1) 
This is to be interpreted as follows: For day 2p + k - 2, 2 I k< 2p - 2 the games are 
the same as on day k except for reversal of arcs; the last day (i.e. day 4p-2) the 
games are the same as on day 1, except for reversal of arcs. Figure 3(a) shows the 
HAP for such a league when p = 6; the profiles of a,, . . . , al2 are complementary to 
those of a,, . . ..a6. 
Proposition 3.2. In the schedule (3.1) for two rounds of a league with 2p teams, 
each team has two breaks. More precisely: 
pr = 2; yi = (2p-l)+l, 
(3.2) 
Pi=2i--1; yi=(2p-1)+2i-2 (25iSp). 
Proof. From Proposition 3.1 we obtain the values of pi, 1 I ilp. The schedule 
S’ = (& Z;, . . . ) $_ 1, &‘) obtained from S = @i’, . . . , &_ 1) by cyclic permutation 
has 2p- 2 breaks occurring in pairs. We have p’=2i-2, 1~ ilp; this does not 
A constrained sports scheduling problem 
days 
45 
al 
a2 
a3 
a4 
a5 
a6 
a7 
1 2 3 4 5 6 7 8 9 1011 
al2 a 
. L . . . . 
. . . . . - 
. . . . . - 
. . . . . - 
. . . . . - 
. . . . . - 
. . . . . - 
. . . . . . - 
. . 2 . . . 
. . . 1_ . . 
. . . . L . 
. . . . . . 
days (w.r. to canonical factorization) 
3 4 5 6 7 8 9 1011 1213 14 15 1 2 
bl . . . . . . . . 
b2. _. . . . . . 
b3. . _. . . . . 
bq . . . _ . . . . q=8 
b5. . . . _. . . 
b6 . . . . . _ . . 
b7. . . . . . . 
bs. . . . . . :._ 
bs 
b 16 b 
Fig. 3. The HAP’s for leagues A and B: (a) league A, (b) league B. 
change when we replace each $ in S’ by 4’. The concatenation (3.1) gives then (by 
replacing the j$ by v,) a break for a, and aP on day (2p - 1) + 1; so we have 
Y1 = (2p-I)+19 y;= (2p-1)+2i-2 (2sisp). q 
Remark 3.3. The schedule (3.1) satisfies furthermore yi - p, = 2p - 2 for each i; the 
two breaks of all teams are equally separated. 
Also there is no break occurring on the last day of the second round (day 4p - 2). 
Remark 3.4. In each one of the rounds there are at least 2p - 2 breaks; so the total 
number of breaks is at least 4p-4. A schedule with 4p-4 breaks would be given 
by (6, . . . , ~~p_,)O$‘~p_~r .. . , Fl). The schedule S, we have constructed in (3.1) 
has 4p breaks; in S, there are at least 2p -2 days separating the two games 
involving the same pair of teams. It is not known in general if a schedule exists with 
less than 4p breaks and with each pair of games in the first and second round 
separated by at least 2p-2 days. 
46 D. de Werra et al 
Let us consider now the schedule for league B which consists of 2q teams which 
have to play one round. From a canonical oriented factorization (F,, . . . , kZq_t) we 
obtain by a cyclic permutation a schedule S, defined by 
(3.3) 
The HAP H(S,) of such a schedule for q = 8 is given in Fig. 3(b). 
Proposition 3.5. The schedule S, defined by (3.3) has 2q-2 breaks occurring in 
pairs with 
PI = 0, /?;=2i-1 (2si5q-l), p, = 2q-2. 
Proof. We start from a canonical oriented factorization k,, . . . , kzq_, having 
j3; = 0, p,!= 2i - 1, 2 5 is q and we call 6: and bi+ i the two teams having a break on 
day &. As before we assume that b;, . . . , bb start with a home-game. 
When we perform the cyclic permutation on the days which gives S, we observe 
the following: 
(a) A new break is created for 6; and bi,, in F, (which is day 2q- 2 of the 
schedule S,). 
(b) b; and bi+2 no longer have a break in 2s (which is the first day of S,). 
Renaming the teams bl, . . . , bzq we get the required property. 0 
For league C (and C’) we construct a schedule 
by taking the factors of a canonical oriented factorization in the reverse order. If 
we call c t, . . . , cq (respectively c;, . . . , cb) the teams in league C (respectively C’), we 
have 
a, = 0, pi=2i-2 (24iI+q). 
Here as before ci, . . . , cq12 and c;, . . . , cb12 start with a home-game. 
In our problem, it is not known before the games of day 2q- 1 are played to 
which one of each subleague (C or C’) a team bi in league B will be assigned. 
Notice however that teams Ci and ci have the same profile. 
In Section 2 constraints (d) required that some pairs of teams never played a 
home-game on the same day. Let us say that teams ak in league A and bj in league 
B form a compatible pair if they never have a home-game on the same day. It will 
be convenient to identify a team ci or bj with its profile. 
For the last q - 1 days of the schedule of league B we will have to assign a profile 
ci or cl (1 I isq) to each one of the 2q teams. Equivalently we have to give each 
one of the profiles Ci (15 is q) to exactly two teams in league B. This assignment 
should be done in such a way that as few new breaks as possible are created on day 
2q. We cannot do this for all teams in B before day 2q- 1; however we can assign 
profiles of some ci to some teams bi by the following concatenation operation: 
A constrained sports scheduling problem 41 
6: = b@c;_q,‘$, q+i = k/+iOCi+q/4 (i = $q+2, . ..) +q>. (3.4) 
An example of this construction is given in Fig. 5. The teams in B are now called 
b;, . . . . b&. Notice that we do not create any break on day 2q for the teams b:, b;+; 
in (3.4). Some of the teams bT, b;+i have now two breaks; this cannot be avoided 
since in the first 2q - 1 days there must be at least 2q - 2 teams which have one break 
and in the last q- 1 days, when both leagues C, C’ play at the same time there are 
at least 2q-4 teams which have one break. 
In order to solve our problem, we have to show that there are q - 2 teams among 
b:, . . . . bz*, and q - 2 teams among a,, . . . , a,, which can be matched (i.e., which can 
use the same home-stadium). 
Proposition 3.6. For i=+q+2, . . ..+q {ai,b;+;} and {a,+i,bT) are compatible 
pairs. 
Proof. For a team aj (2s+q+2<i<+q=p) we have pi=2i-1 and ri=(2p-l)+ 
2i-2 from Proposition 3.2. For team bT (25i1 $qSq- 1) when q24, we also 
have pi = 2i - 1. From the construction (3.4) for +q + 2 5 i 5 +q the second break of 
bf occurs on day ~i=(2q-1)+2(i-+q-l)=+g-l+2i-2=(2p-1)+2i-2. SO ai 
and bjr have identical profiles. Similarly a,,+i and bG+i have identical profiles; since 
ai and aP+i on one hand as well as b? and b;+i on the other hand have complemen- 
tary profiles, the result follows. 0 
Proposition 3.7. For teams b:, b4*+i ($4 + 2 5 is $q), the profiles of the last q - 1 
days are all different. 
Proof. It follows from the construction (3.4) that the profiles used for the teams 
b;, b:+i (+q+2<i< +q) are c2, c3, . . . . cqj2, c~/~+~, . . . . cq. They are all different; in 
fact there is a schedule based on a canonical oriented factorization for a league of 
q teams whose HAP contains any specified subset of these profiles (in addition to 
the missing profiles). 0 
4. The schedule 
In Section 3 we have constructed a schedule in two rounds for a league A con- 
sisting of 2p teams. These teams have been called a,, . . . , a2,. The HAP associated 
to this schedule was also given. For league B (with 2q teams) a schedule in one round 
with the corresponding HAP was first obtained. Then we constructed a schedule 
with a HAP for a subleague of B consisting of q (unknown) teams. Since B is divid- 
ed for the last q - 1 days into two subleagues C, C’ we introduced the profiles of 
C and C’ at the end of the profiles of some bi’s. All teams in B were called b:; for 
keeping the notations simple, we identified teams with their profiles when no confu- 
sion was possible. 
48 D. de Werra et al. 
cl 
c2 
c3 
c4 
. . 
C8 
7 - 
. 
. 
654321 
. . . 
L . . 
. . . - 
! 
. . L 
q=8 
Fig. 4. The HAP of each subleague C, C’. 
According to Propositions 3.6 and 3.7, we now let ai and u,,+~ (aq+2si<p= 
44) be the teams which share their home-stadium with a team in league B; we let 
bG+i (respectively bjr) be the team of league B which shares its home-stadium with 
aj (respectively ap+i). Constraints (d) will then be satisfied according to Proposi- 
tion 3.6. After 2q - 1 days, the division of league B into two leagues C, C’of q teams 
each is known. Let B*={bT,b,*+i: $q+2sis+q}; for each team bz in B* we 
know in which league it must be included. Since from Proposition 3.7 all profiles 
of teams bt in B are different, we can for each subleague C (respectively C’) assign 
a different profile ci (not used by any team in Bfl C (respectively Bn C’)). A 
schedule associated with the HAP H(S,) constructed for league C (see example in 
Fig, 4) can then be constructed. 
This assignment can be made by constructing a square weight matrix as follows: 
For subleague C, introduce a column ci for each profile in {c,, . . . , cq} not used by 
any team bz in B * fl C and a row b;” for each team in C-B *. Each entry in column 
ci and row b: contains a weight equal to 1 if the concatenation b,?@c; creates a new 
break (on day 2q) or 0 otherwise. We repeat the same for league C’. Then we com- 
pute a minimum weight assignment; this gives a schedule with a minimum number 
League B 
b’lo . 
b’z . a 
b.3. -i 
b-4. 0 
b-6. 0 
b.6. 0 
b’7. l 
b’6. l 
“‘12 
b +I 3 
b i 4 
i 
Fig. 5. The HAP for bl: . . . . b?G 
A constrained sports scheduling problem 49 
b*l 
b*z 
b*3 
b*7 
b*l6 
League C = (b*l,b*2,b*3,b*5,b*6.b*7,b*l3,b*l6] 
League c’ = (b*4,b*~,b*9.b*1~.b*ll.b*~2,b*~~,b*15) 
Cl C2 C5 C6 C8 Cl c3 c4 c5 c7 
1 1uo 0 b*8 0 1oJ 0 1 1 
oa1 11 b*9 TrJo 0 11 
rjlo1 11 b*lo 1 1 1 
OOlDl 
OiEl 
b*ll 1 1 liEI 
1 1 0 0 l-d b*15 11 flo 0 
Fig. 6. An assignment of profiles to the unconstrained teams in leagues C and C’. 
of new breaks on day 2q. The assignment matrices for the example of Fig. 5 are 
given in Fig. 6 with optimal solutions. Here two new breaks are created on day 2q. 
Notice that the simple structure of the weight matrix suggests to use a simplified 
assignment technique. Remark 4.1 below will generate a more general weight 
matrix. 
The assignment of profiles to teams in league C in the above example would be 
the following: 
profiles: cl? c2, c3, c4, c5, c6, c7, c8, 
teams: b:, b,*, b3, bg, b:, bh bh, b;. 
Remark 4.1. Instead of simply penalizing new breaks created on day 2q, we can 
penalize a break only if the corresponding team would have more than two breaks 
in its global profile. For instance b;E, = bll@c3 should be more penalized than 
b,*, = bl, @cl in the example of Fig. 6, because there would be three breaks in the 
profile of b:, in the first case and only two in the second. 
References 
[l] E. Mendelsohn and A. Rosa, One-factorization of the complete graph-a survey, J. Graph Theory 
9 (1985) 43-65. 
[2] T.H. Straley, Scheduling designs for a league tournament, Ars Combin. 15 (1983) 193-200. 
[3] W.D. Wallis, A tournament problem, J. Austral. Math. Sot. Ser. B 24 (1983) 289-291. 
[4] D. de Werra, Scheduling in sports in: P. Hansen, ed., Studies on Graphs and Discrete Programming 
(North-Holland, Amsterdam, 1981) 381-395. 
[5] D. de Werra, Geography, games and graphs, Discrete Appl. Math. 2 (1980) 327-337. 
[6] D. de Werra, On the multiplication of divisions: The use of graphs for sports scheduling, Networks 
15 (1985) 125-136. 


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