Construction of balanced sports schedules with subleagues

Transcription

1 Construction of balanced sports schedules with subleagues 1 A. Geinoz, T. Ekim, D. de Werra Ecole Polytechnique Fédérale de Lausanne, ROSE, CH-1015 Lausanne, Switzerland, s: alexandre.geinoz, tinaz.ekim, dominique.dewerra@epfl.ch May 11, 2007 (Unpublished version) Abstract. In the construction of schedules for sports leagues, one has to balance the games of each team on the different stadiums. We shall exploit here the fact that the league consists of several subleagues. We give a direct construction for tournaments with an odd number of subleagues. Direct constructions known up to now are available to solve problems with 2n teams if 2n mod 3 1 or 2n = 2 p, p 3 or n is odd. Our method can solve more than the half of the missing cases and it provides structured solutions for some known cases. Key words : balanced tournaments, subleagues, Graeco-Latin squares 1 Introduction We consider one of the most extensively studied sports scheduling problems: balanced tournament design (BTD) on multiple venues (also known as Prob026 from CSPLib). A balanced tournament design for 2n teams, denoted by BTD(n), is defined by conditions (1) to (4). 1. There are 2n teams and n different stadiums. 2. Every team plays one game against every other team. 3. Each team plays exactly one game each day of the tournament. 4. The stadiums where a team plays are distributed as equally as possible, i.e., each team plays twice in each stadium except one. It has been shown in [7] that BTD(n) exists for all 2n > 4. Nevertheless, obtaining concrete solutions for real cases is another challenge. Several promising methods have been developed using tabu search [1, 4], constraint programming [10, 11] and integer programming [8]. Although these

2 2 methods achieved great success in recent years, in general, algorithmic approaches (for instance based on integer programming formulation) suffer from the explosion of their time complexity as the size of the problem increases (starting at 2n = 16 for basic methods and 2n = 40 for more involved ones). An alternative to this approach is to provide some direct constructions; [7] suggests solutions if n is odd, [6] if 2n mod 3 = 0 or 2 (this case is also solved in [5] by a linear time algorithm) and [3] if 2n is a power of 2. Note that the latter gives more precisely solutions for a variant of BTD, denoted by BTD*, where each team plays exactly twice on each stadium (each team plays twice against one other team; this is called a duplicated game) with the additional constraint that duplicated games, involving the same teams, are not played on the same stadium. However, the given construction is such that all the duplicated games are played in the same day and therefore it is sufficient to remove that day in order to obtain a solution for BTD. As far as we know, the above cases include all direct constructions for BTD given in the literature so far. To summarize, the only case for which no direct construction is known is 2n mod 12 = 4 and 2n 2 p for some integer p. In this note, we give a direct construction for tournaments with subleagues: a league consists of a certain number of subleagues of the same size. This structure will be explicitly exploited in this construction. Such tournaments with subleagues may be imposed, for instance, by some geographical constraints. To our knowledge, the method described in [3] is so far the only direct construction for schedules with subleagues: the authors give a recursive method for 2n = 2 p with subleagues of size 2 i where i may vary between 3 and p 1. Our method, although not recursive, is another construction with subleagues that may contain several even numbers of teams and not only a number 2 i. Moreover, the solutions obtained have the property that the games played each day are either all internal (i.e., played between teams of the same subleague), or all external (i.e., played between teams of different subleagues). We provide solutions for more than half of the cases not covered by the known constructions. Another advantage of such a direct construction is that solutions can be constructed directly for arbitrarily large values of 2n. Last but not least, it gives very structured solutions using subleagues; even though another solution is already known for some values of 2n, having new types of schedules may be useful in order to satisfy various additional requirements, which may arise in different contexts. It is therefore important to have a wide sample of solutions, among which to choose the best one.

3 3 2 Balanced Tournament Design with subleagues BTD(n) can be alternatively formulated as an arrangement of the n(2n 1) distinct pairs of teams in the set T of 2n teams into a (2n 1) n array in such a way that every element of T occurs exactly once in each row (day) and at most twice in each column (stadium). Let us recall that a Graeco-Latin square of order n is an n n array where the entries contain all distinct ordered pairs of elements from two sets of size n in such a way that each element appears once in each row and once in each column. Lemma 1. Graeco-Latin squares of odd order can be constructed in linear time. Proof. It suffices to number the elements of the two sets from 0 to n 1. In the first row, we group the same numbers together. In the following rows, we increase the numbers by n+1 2 for the first set and by n 1 2 for the second set as can be observed in the example of Table 1. Table 1. Graeco-Latin square of order 5 5 0,0 1,1 2,2 3,3 4,4 3,2 4,3 0,4 1,0 2,1 1,4 2,0 3,1 4,2 0,3 4,1 0,2 1,3 2,4 3,0 2,3 3,4 4,0 0,1 1,2 Note that for the special case of a pair of Graeco-Latin square a construction is given in [9] while for other even values of n no direct construction is known. These Graeco-Latin squares will be useful in our method. We shall provide solutions for tournaments of a league of 2n teams consisting of m subleagues of equal even size (thus 2n = 2mt). Our construction will rely on several known results. Lemma 2 ([7]). For m 3 odd, there is a direct construction to obtain BTD(m). Lemma 3 ([7]). There exists a direct construction of a BTD(m) for m 3 odd such that the two teams playing only once on a stadium play against each other on this same stadium.

4 4 Lemma 4 ([3]). If n = 2 p with p 2, Graeco-Latin squares of order n n can be constructed. Lemma 5 ([3], [6] and [7]). For 2t teams, there is a direct construction for BTD(t) if t is odd [7], if 2t is a power of 2 [3] or if 2t mod 3 = 0 or 2 [6]. Theorem 1. For m subleagues of 2t teams each, a BTD(n) such that n = mt can be constructed for all m odd and for all t odd or t = 2 p, p 2 or t = 10. Proof. Let us assume that we have m subleagues of 2t teams each. Consequently, we have 2m semi-subleagues of t teams playing on m complexes of stadiums. Note that each complex of stadiums contains t stadiums. Clearly, every semi-subleague should play against every other, moreover each team in a semi-subleague should play against every other team of the same semi-subleague. We shall describe how our method ensures that each one of these games is present. In our procedure, we need the following hypotheses which are necessary for the constructions that we use. a) m 3 is odd, b) BTD(t) can be constructed (it will correspond to internal games), c) a Graeco-Latin square of order t t can be constructed (it will correspond to external games). For 2n = 18 where m = 3 and t = 3, the structure of the subleagues is represented in Table 2, different stages of the construction are illustrated in Tables 3,4,5 and the complete BTD(n) is given in Table 6. We construct BTD(m) by the construction suggested for m odd in Lemma 2 (see Table 3). In this BTD(m), if we consider each team as one semisubleague, each stadium as one complex of stadiums and each day as one set of days, then it determines the pairs of sets of days (rows) and complexes of stadiums (columns) where different semi-subleagues play against each other. By Lemma 3, for m odd given, there is a row in BTD(m) where all the games are between semi-subleagues playing only once in the corresponding complex of stadiums. We assume without loss of generality that this is the first set of days (up to a renumbering of rows) and it corresponds to the internal games (see the first row of Table 3). Internal games will be played on the corresponding complex of stadiums according to BTD(t) constructed as mentioned in Lemma 5. Thus the first row (set of days 1) contains 2t 1 days and each one of its entries corresponds to t(2t 1) games (see Table 4). Besides this, each entry in

5 all the other sets of days corresponds to external games played between different semi-subleagues. These are played according to a Graeco-Latin square of order t t constructed by Lemma 1 or by Lemma 4 or as the one given for t = 10 in [9]. Table 5 gives the construction outline for the external games for t = 3 teams by semi-subleagues. Such a Graeco- Latin square schedules t 2 games on t days and t stadiums (one complex of stadium). Therefore each one of the remaining 2m 2 rows corresponds to t days. Summing up, the tournament is scheduled on (2m 2)t+(2t 1) = 2n 1 days and the total number of games played is mt(2t 1)+t 2 m(2m 2) = n(2n 1). It follows from the description of the method that each day, we have either only internal games or only external games as in the construction of [2] and [3]. 5 Table 2. Tournament with 3 subleagues of 6 teams each. X i corresponds to the team i in the semi-subleague X Table 3. Construction of BTD(m = 3). Set of days 1 corresponds to internal games constructed by BTD(t) between two semi-subleagues belonging to the same subleague. Sets of days 2 to 5 correspond to external games obtained with a Graeco-Latin square. complex A complex B complex C internal : set of days 1 (5 days) 0,1 2,3 4,5 external : set of days 2 (3 days) 3,4 5,1 0,2 external : set of days 3 (3 days) 3,5 4,0 1,2 external : set of days 4 (3 days) 2,5 4,1 0,3 external : set of days 5 (3 days) 2,4 5,0 1,3

6 6 Table 4. Internal games constructed by BTD(t = 3) [3] for the set of days 1 for teams 0 0, 0 1, 0 2 and 1 0, 1 1, 1 2 of semi-subleagues 0 and 1 of the same subleague. Stadium Y j corresponds to the stadium j in the complex Y. stadium A 0 stadium A 1 stadium A 2 set of days 1, day 1 0 0, , , 1 2 set of days 1, day 2 1 0, , , 0 2 set of days 1, day 3 1 0, , , 0 2 set of days 1, day 4 0 2, , , 1 0 set of days 1, day 5 0 2, , , 1 0 Table 5. Example of a Graeco-Latin square of order 3 3 constructed for the external games between semi-subleagues 3 and 4 played during the sets of days 2 on complex A. Day Z k corresponds to the day k in the set of days Z. stadium A 0 stadium A 1 stadium A 2 external games, day , , , 4 2 external games, day , , , 4 0 external games, day , , , 4 1 Table 6. BTD(n = 9) given by the construction : stadiums A 0 A 1 A 2 B 0 B 1 B 2 C 0 C 1 C 2 day 1 0 0, , , , , , , , , 5 2 day 2 1 0, , , , , , , , , 4 2 day 3 1 0, , , , , , , , , 4 2 day 4 0 2, , , , , , , , , 5 0 day 5 0 2, , , , , , , , , 5 0 day 6 3 0, , , , , , , , , 2 2 day 7 3 2, , , , , , , , , 2 0 day 8 3 1, , , , , , , , , 2 1 day 9 3 0, , , , , , , , , 2 2 day , , , , , , , , , 2 0 day , , , , , , , , , 2 1 day , , , , , , , , , 3 2 day , , , , , , , , , 3 0 day , , , , , , , , , 3 1 day , , , , , , , , , 3 2 day , , , , , , , , , 3 0 day , , , , , , , , , 3 1

7 7 Table 7. Summary of 2n = 2mt teams for which BTD(n) is provided. t m Results and discussion Clearly, Table 7 can be extended to infinity while leaving some holes, i.e. values of 2n for which no direct construction is known so far. Programming approaches (integer programming or constraint programming) leave no such holes but they are practically limited to relatively small values of n (see for instance [4,10]). As can be observed in Table 7, our construction gives BTD(mt) for all m odd and for all values of t such that a Graeco-Latin square of order t is known and BTD(t) can be constructed. Note that no Graeco-Latin square of order 6 exists. The bold numbers in Table 7 indicate the sizes of BTDs which are such that 2n mod 12 = 4 and 2n 2 p, hence not constructible by already known direct constructions. More precisely, we provide solutions for more than half of these unknown cases. In fact, if 2n mod 24 = 16 and 2n 2 p then 2n can be written as m2 k with m odd, k 3. It follows that BTD(n) with m subleagues can be constructed since BTD(2 k 1 ) and a Graeco-Latin square of order 2 k 1 can be obtained by [3]. Moreover, if 2n mod 24 = 4 and 2n mod 10 = 0, one can obtain a solution by setting t = 10 (therefore m odd since m mod 6 = 5). Consequently, the only remaining unsolved case is if 2n mod 24 = 4, 2n mod 5 0 and 2n 2 p, the most significant values from a practical point of view being 28, 52 and 76. Another advantage of our construction with subleagues is the regular structure which allows spreading for each team its games against seeded teams (these teams correspond to the favourite teams during a tournament). If suffices to include one seeded team in each semi-subleague. Thus, during sets of days 2 to 2m 1 (in our example 2 to 5 in Table 3), each

8 8 team will play against a seeded team once in every set of t days. We conclude with another property of our construction. Under hypotheses a) and c), whenever 2n = 2mt and BTD*(t) is known, for instance, if 2t = 2 p, one can easily derive solutions for the slightly harder problem of BTD*(n) using the solutions we provide: the duplicated games are played during internal games according to BTD*(n) and the rest of the schedule is the same as BTD(n). To solve the balancing problem of games in the stadiums, we have at this stage a number of different direct constructions according to the value of 2n. It would be interesting to explore other types of constructions with the hope of finding a much more limited number of constructions or even a unique type of construction to cover all cases. This has been possible for the problem of minimizing the breaks of alternances between home games and away games with the so called canonical factorizations (see [3]). There are good reasons to explore whether it is also the case for our problem (although we know that it could not simply be the canonical factorizations). References 1. D. Costa. An evolutinary tabu search algorithm and the NHL scheduling problem. INFOR, 33(3): , D. de Werra. Geography, games and graphs. Disc. App. Math., 2: , D. de Werra, T. Ekim, and C. Raess. Construction of sports schedules with multiple venues. Disc. App. Math., 154:47 58, J.-P. Hamiez and J.-K. Hao. Solving the sports league scheduling problem with tabu search. Lecture Notes in Artificial Intelligence, 2148:24 36, J.-P. Hamiez and J.-K. Hao. A linear-time algorithm to solve the sports league scheduling problem (prob026 of CSPLib). Disc. App. Math., 143(1-3): , J. Haselgrove and J. Leech. A tournament design problem. American Mathematical Monthly, 84(3): , E.R. Lamken and S.A. Vanstone. The existence of factored balanced tournament designs. Ars Combinatoria, 19: , G. Nemhauser and M. Trick. Scheduling a major college basketball conference. Operations Research, 46:1 8, E.T. Parker. Orthogonal latin squares. Proc. Nat. Acad. Sci. U.S.A., 45: , J.-C. Régin. Constraint programming and sports scheduling problems. In Proc. INFORMS 99, T.L. Urban and R.A. Russell. A constraint-programming approach to the multiplevenue, sport-scheduling problem. Computers & Operations Research, special issue on OR in Sport, 33(7): , 2006.