Abstract
The sport teams grouping problem (STGP) concerns the assignment of sport teams to round-robin tournaments. The objective is to minimize the total travel distance of the participating teams while simultaneously respecting fairness constraints. The STGP is an NP-Hard combinatorial optimization problem highly relevant in practice. This paper investigates the performance of some complimentary optimization approaches to the STGP. Three integer programming formulations are presented and thoroughly analyzed: two compact formulations and another with an exponential number of variables, for which a branch-and-price algorithm is proposed. Additionally, a meta-heuristic method is applied to quickly generate feasible high-quality solutions for a set of real-world instances. By combining the different approaches’ results, solutions within 1.7% of the optimum values were produced for all feasible instances. Additionally, to support further research, the considered STGP instances and corresponding solutions files were shared online.
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Notes
Instance and solution files are available at http://benchmark.gent.cs.kuleuven.be/stgp.
References
Achterberg, T., Koch, T., & Martin, A. (2005). Branching rules revisited. Operations Research Letters, 33(1), 42–54.
Alarcón, F., Durán, G., Guajardo, M., Miranda, J., Muñoz, H., Ramírez, L., et al. (2017). Operations research transforms the scheduling of chilean soccer leagues and south american world cup qualifiers. Interfaces, 47(1), 52–69.
Ales, Z., Knippel, A., & Pauchet, A. (2016). Polyhedral combinatorics of the k-partitioning problem with representative variables. Discrete Applied Mathematics, 211(c), 1–14.
Anagnostopoulos, A., Michel, L., Van Hentenryck, P., & Vergados, Y. (2006). A simulated annealing approach to the traveling tournament problem. Journal of Scheduling, 9(2), 177–193.
Barnhart, C., Johnson, E. L., Nemhauser, G. L., Savelsbergh, M. W. P., & Vance, P. H. (1998). Branch-and-price: Column generation for solving huge integer programs. Operations Research, 46, 316–329.
Briskorn, D., Drexl, A., & Spieksma, F. C. R. (2010). Round robin tournaments and three index assignments. 4OR, 8(4), 365–374.
Carvalho, M. A. M. D., & Lorena, L. A. N. (2012). New models for the mirrored traveling tournament problem. Computers and Industrial Engineering, 63(4), 1089–1095.
Christiaens, J., & Vanden Berghe, G. (2016). A fresh ruin & recreate implementation for the capacitated vehicle routing problem. Technical report, KU Leuven, Belgium.
Dantzig, G. B., & Wolfe, P. (1960). Decomposition principle for linear programs. Operations Research, 8(1), 101–111.
Di Gaspero, L., & Schaerf, A. (2007). A composite-neighborhood tabu search approach to the traveling tournament problem. Journal of Heuristics, 13(2), 189–207.
Easton, K., Nemhauser, G., & Trick, M. (2001). The traveling tournament problem description and benchmarks. In T. Walsh (Ed.), Principles and Practice of Constraint Programming—CP 2001: 7th International Conference, CP 2001 Paphos, Cyprus, 2001 Proceedings (pp. 580–584). Berlin, Heidelberg: Springer.
Goossens, D., & Spieksma, F. (2009). Scheduling the belgian soccer league. Interfaces, 39(2), 109–118.
Goossens, D., & Spieksma, F. (2014). Indoor football scheduling. In E. Özcan, E. Burke, & B. McCollum (Eds.), Proceedings of the 10th International Conference on the Practice and Theory of Automated Timetabling (pp. 167–178), PATAT.
Goossens, D. R., & Spieksma, F. C. R. (2012). Soccer schedules in europe: An overview. Journal of Scheduling, 15(5), 641–651.
Januario, T., Urrutia, S., Ribeiro, C. C., & De Werra, D. (2016). Edge coloring: A natural model for sports scheduling. European Journal of Operational Research, 254(1), 1–8.
Ji, X., & Mitchell, J. E. (2007). Branch-and-price-and-cut on the clique partitioning problem with minimum clique size requirement. Discrete Optimization, 4, 87–102.
Kendall, G., Knust, S., Ribeiro, C. C., & Urrutia, S. (2010). Scheduling in sports: An annotated bibliography. Computers and Operations Research, 37(1), 1–19.
Knust, S. (2010). Scheduling non-professional table-tennis leagues. European Journal of Operational Research, 200(2), 358–367.
Labbé, M., & Özsoy, F. A. (2010). Size-constrained graph partitioning polytopes. Discrete Mathematics, 310(24), 3473–3493.
Lübbecke, M. E., & Desrosiers, J. (2005). Selected topics in column generation. Operations Research, 53(6), 1007–1023.
Mehrotra, A., & Trick, M. A. (1998). Cliques and clustering: A combinatorial approach. Operations Research Letters, 22(1), 1–12.
Nemhauser, G. L., & Trick, M. A. (1998). Scheduling a major college basketball conference. Operations Research, 46(1), 1–8.
Ribeiro, C. C. (2012). Sports scheduling: Problems and applications. International Transactions in Operational Research, 19, 201–226.
Schönberger, J., Mattfeld, D., & Kopfer, H. (2000). Automated timetable generation for rounds of a table-tennis league. In Proceedings of the 2000 Congress on Evolutionary Computation (pp. 277–284).
Schönberger, J., Mattfeld, D., & Kopfer, H. (2004). Memetic algorithm timetabling for non-commercial sport leagues. European Journal of Operational Research, 153(1), 102–116.
Sørensen, M. M. (2004). New facets and a branch-and-cut algorithm for the weighted clique problem. European Journal of Operational Research, 154(1), 57–70.
Toffolo, T. A. M., Wauters, T., Van Malderen, S., & Vanden Berghe, G. (2016). Branch-and-bound with decomposition-based lower bounds for the traveling umpire problem. European Journal of Operational Research, 250(3), 737–744.
Trick, M.A., & Yildiz, H. (2007). Bender’s cuts guided large neighborhood search for the traveling umpire problem. In P. Van Hentenryck & L. Wolsey (Eds.), Number 4510 in Lecture Notes in Computer Science (pp. 332–345), Springer.
Trick, M. A., & Yildiz, H. (2011). Benders’ cuts guided large neighborhood search for the traveling umpire problem. Naval Research Logistics (NRL), 58(8), 771–781.
Trick, M. A., Yildiz, H., & Yunes, T. (2012). Scheduling major league baseball umpires and the traveling umpire problem. Interfaces, 42(3), 232–244.
Uthus, D. C., Riddle, P. J., & Guesgen, H. W. (2011). Solving the traveling tournament problem with iterative-deepening. Journal of Scheduling, 15(5), 601–614.
Vanderbeck, F., & Wolsey, L. (2010). Reformulation and decomposition of integer. In M. Jünger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, & L. A. Wolsey (Eds.), 50 years of integer programming 1958–2008 (pp. 431–502). Berlin: Springer.
Xue, L., Luo, Z., & Lim, A. (2015). Two exact algorithms for the traveling umpire problem. European Journal of Operational Research, 243(3), 932–943.
Acknowledgements
Work supported by the Belgian Science Policy Office (BELSPO) in the Inter-university Attraction Pole COMEX (http://comex.ulb.ac.be) and by the Leuven Mobility Research Centre (L-Mob). Editorial support provided by Luke Connolly, KU Leuven. Additionally, we would like to thank Movetex, in particular Dieter De Naeyer and Ken De Norre–De Groof, for the insightful discussions concerning the problem and for making the real-world instances available.
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Appendix A
Appendix A
This appendix presents the proofs for Theorems 1 and 2, previously presented in Sect. 3.4.
Theorem1 The linear relaxation of \(\mathcal {F}_3\) is stronger than the linear relaxation of \(\mathcal {F}_1\).
Proof
Theorem 1 is proven true if conditions C1 and C2 are satisfied:
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C1:
for each STGP instance I, \(z_{\mathcal {F}_3}(I) \ge z_{\mathcal {F}_1}(I)\),
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C2:
there exists a STGP instance I where \(z_{\mathcal {F}_3}(I) > z_{\mathcal {F}_1}(I)\).
The validity of C2 is relatively simple to show; in fact, many of the instances considered in Sect. 7 satisfy this condition. C1 remains to be proven true. Consider \(\lambda ^*_{\omega }\) a feasible solution of the linear relaxation of \(\mathcal {F}_3\) on some instance I while also assuming \(L \leftarrow \varOmega \) such that each \(\omega \in \varOmega \) has an equivalent \(\ell \in L\), meaning \(\omega = \ell \). Given this solution and the set \(L = \varOmega \), x-, y- and z-values can be constructed:
Next, it is shown how the resulting solution is a feasible solution of the linear relaxation of \(\mathcal {F}_1\), indicating that it satisfies (2)–(6). Constraints (2) are always satisfied, since each team \(i \in T\) is assigned to exactly one league [see Constraints (18)]:
Since the leagues (cliques) \(\omega \in \varOmega \) are feasible, \(m^- \le |\omega | \le m^+ \; \forall \omega \in \varOmega \), inequalities in (3) follow:
Given that all leagues \(\omega \in \varOmega \) are feasible it also follows that the number of teams from a club in any \(\omega \) is less than or equal to \(c^+\). Therefore, Constraints (4) are also satisfied since for all leagues \(\ell \in L\) and clubs \(c \in C\):
Constraints (5) also hold. If teams i and j are in the same league, they are therefore together in a league \(\omega \in \varOmega \). Thus, for all \((i,j) \in A^2\) and leagues \(\ell \in L\) (or \(\omega \in \varOmega \)) containing both i and j:
Observe that Constraints (5) are also satisfied when teams i and j are not in the same league \(\ell \) since this implies \(x_{i, \ell } + x_{j, \ell } \le 1\).
Finally, if teams i and j are not allowed in a league, leagues \(\omega \in \varOmega \) will contain at most one of them (since all leagues \(\omega \) are feasible). Hence, Constraints (6) also hold for all \((i,j) \in F^2\) and \(\ell \in L\):
It is thus proven that every solution I of the linear relaxation of \(\mathcal {F}_3\) satisfy (2)–(6). Since both \(\mathcal {F}_1\) and \(\mathcal {F}_3\) have the same objective function, Theorem 1 is proven. \(\square \)
Theorem2 The linear relaxation of \(\mathcal {F}_3\) is stronger than the linear relaxation of \(\mathcal {F}_2\).
Proof
Theorem 2 is proven if conditions C3 and C4 are satisfied:
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C3:
for each STGP instance I, \(z_{\mathcal {F}_3}(I) \ge z_{\mathcal {F}_2}(I)\),
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C4:
there exists a STGP instance I where \(z_{\mathcal {F}_3}(I) > z_{\mathcal {F}_2}(I)\).
C4 is proven true by the experiments reported in Sect. 7. It is, however, required to mathematically prove that C3 is true. Consider again that \(\lambda ^*_{\omega }\) is a feasible solution of the linear relaxation of \(\mathcal {F}_3\) for an instance I; y-values may be easily constructed:
Let \(\bar{e}^k_{i,j}\) be the sum of \(\lambda ^*_\omega \) for all \(\omega \in \varOmega \) which include both i and j but not k, and \(e_{i,j,k}\) be the sum of \(\lambda ^*_\omega \) for all \(\omega \in \varOmega \) which include i, j and k. More compactly:
Clearly, for each \(k \in T\), \(y_{i,j} = \bar{e}_{i,j}^k + e_{i,j,k}\). Equivalently, \(y_{i,k} = \bar{e}^j_{i,k} + e_{i,j,k}\) and \(y_{j,k} = \bar{e}^i_{j,k} + e_{i,j,k}\). Therefore, for each \((i,j,k) \in A^3\):
The first equality in (40) follows by definition; the first inequality from the non-negativity of all y-variables; the second equality from the fact that terms \(\bar{e}_{i,j}^k, \bar{e}^j_{i,k}\), and \(e_{i,j,k}\) correspond to disjoint sets of cliques whose union are all cliques containing i; and the final equality follows from (18). Therefore, y satisfies (12).
Furthermore, observe that \(e_{i,j,k} = 0 \ \forall (i,j,k) \in F^3\). This follows from i, j and k not being permitted in the same league, and therefore there exists no league \(\omega \in \varOmega \) containing these three teams. Thus, modifying (40) shows that y also satisfies (13).
Consider Constraint (14). Observe that for each \(i \in T\):
Indeed, the first equality follows from (30); the second equality from how clique containing i cannot contain some team \(j \notin A_i\); and the final equality results from counting how many times the value \(\lambda ^*_{\omega }\) is present in this term.
Next, given that each clique \(\omega \) is feasible, \(m^- \le |\omega | \le m^+\). Combining this with (41) results in:
It follows that y satisfies (14).
Finally, consider (15). Observe that for each team \(i \in T\):
Again, the first equality follows from (37) and the inequality from how each clique \(\omega \) contains at most \(c^+\) nodes from the same club as team i. Hence the value \(\lambda _{\omega }\) occurs at most \(c^+-1\) times in each clique \(\omega \).
It is therefore proven that y satisfies (15).
Finally, note that both formulations have an equal objective function, and hence Theorem 2 is proven.
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Toffolo, T.A.M., Christiaens, J., Spieksma, F.C.R. et al. The sport teams grouping problem. Ann Oper Res 275, 223–243 (2019). https://doi.org/10.1007/s10479-017-2595-z
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DOI: https://doi.org/10.1007/s10479-017-2595-z