A Bigram Based ILP Formulation for Break Minimization in Sports Scheduling Problems
Constructing a suitable schedule for sports competitions is a crucial issue in sports scheduling. The round-robin tournament is a competition adopted in many professional sports. For most round-robin tournaments, it is considered undesirable that a team plays consecutive away or home matches; such a...
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| Published in | IEICE Transactions on Information and Systems Vol. E108.D; no. 3; pp. 192 - 200 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Tokyo
The Institute of Electronics, Information and Communication Engineers
01.03.2025
Japan Science and Technology Agency |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0916-8532 1745-1361 |
| DOI | 10.1587/transinf.2024FCP0003 |
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| Abstract | Constructing a suitable schedule for sports competitions is a crucial issue in sports scheduling. The round-robin tournament is a competition adopted in many professional sports. For most round-robin tournaments, it is considered undesirable that a team plays consecutive away or home matches; such an occurrence is called a break. Accordingly, it is preferable to reduce the number of breaks in a tournament. A common approach is to first construct a schedule and then determine a home-away assignment based on the given schedule to minimize the number of breaks (first-schedule-then-break). In this study, we concentrate on the problem that arises at the second stage of the first-schedule-then-break approach, namely, the break minimization problem (BMP). We propose a novel integer linear programming formulation called the “bigram based formulation.” The computational experiments show its effectiveness over the well-known integer linear programming formulation. We also investigate its valid inequalities, which further enhances the computational performance. |
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| AbstractList | Constructing a suitable schedule for sports competitions is a crucial issue in sports scheduling. The round-robin tournament is a competition adopted in many professional sports. For most round-robin tournaments, it is considered undesirable that a team plays consecutive away or home matches; such an occurrence is called a break. Accordingly, it is preferable to reduce the number of breaks in a tournament. A common approach is to first construct a schedule and then determine a home-away assignment based on the given schedule to minimize the number of breaks (first-schedule-then-break). In this study, we concentrate on the problem that arises at the second stage of the first-schedule-then-break approach, namely, the break minimization problem (BMP). We propose a novel integer linear programming formulation called the “bigram based formulation.” The computational experiments show its effectiveness over the well-known integer linear programming formulation. We also investigate its valid inequalities, which further enhances the computational performance. |
| ArticleNumber | 2024FCP0003 |
| Author | MATSUI, Tomomi FUJII, Koichi |
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| Cites_doi | 10.1287/opre.49.1.163.11193 10.1109/SSCI50451.2021.9660171 10.1016/j.orl.2004.06.004 10.1016/j.ejco.2022.100031 10.1371/journal.pone.0266846 10.1007/s10951-021-00717-3 10.1016/S0304-0208(08)73478-9 10.1007/11757375_15 10.1016/j.ejor.2006.12.050 10.1145/3585516 10.1016/j.orl.2007.09.009 10.1007/978-3-540-45157-0_6 10.1090/dimacs/057/07 10.1287/opre.46.1.1 10.1007/3-540-44629-X_15 10.1016/j.disopt.2005.08.009 10.1080/10556788.2017.1335312 10.1016/j.cor.2011.10.004 10.1007/978-3-642-38189-8_18 10.1016/S0167-6377(03)00025-7 10.1007/978-3-540-45157-0_5 10.1016/j.orl.2020.05.005 10.1016/j.ejor.2019.07.023 10.5540/03.2018.006.01.0311 10.1016/j.cor.2004.09.030 10.1145/227683.227684 10.1007/s12532-023-00236-6 |
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| References | [20] M.X. Goemans and D.P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” Journal of the ACM, vol.42, no.6, pp.1115-1145, 1995. 10.1145/227683.227684 [22] M. Kuramata, R. Katsuki, and K. Nakata, “Solving large break minimization problems in a mirrored double round-robin tournament using quantum annealing,” Plos one, vol.17, no.4, p.e0266846, 2022. 10.1371/journal.pone.0266846 [24] D. Van Bulck, D. Goossens, J. Schönberger, and M. Guajardo, “RobinX: A three-field classification and unified data format for round-robin sports timetabling,” European Journal of Operational Research, vol.280, no.2, pp.568-580, 2020. 10.1016/j.ejor.2019.07.023 [9] R. Miyashiro and T. Matsui, “A polynomial-time algorithm to find an equitable home-away assignment,” Operations Research Letters, vol.33, no.3, pp.235-241, 2005. 10.1016/j.orl.2004.06.004 [12] R.V. Rasmussen and M.A. Trick, “The timetable constrained distance minimization problem,” CPAIOR 2006, Lecture Notes in Computer Science, vol.3990, pp.167-181, Springer, 2006. 10.1007/11757375_15 [21] J.C. Peng, A.D. Clark, and A. Dahbura, “Introducing human corrective multi-team SRR sports scheduling via reinforcement learning,” 2021 IEEE Symposium Series on Computational Intelligence (SSCI), pp.1-7, IEEE, 2021. 10.1109/ssci50451.2021.9660171 [27] S. Vigerske and A. Gleixner, “Scip: Global optimization of mixed-integer nonlinear programs in a branch-and-cut framework,” Optimization Methods and Software, vol.33, no.3, pp.563-593, 2018. 10.1080/10556788.2017.1335312 [14] T. Achterberg and R. Wunderling, “Mixed integer programming: Analyzing 12 years of progress,” Facets of combinatorial optimization: Festschrift for Martin Grötschel, pp.449-481, Springer, 2013. 10.1007/978-3-642-38189-8_18 [15] T. Koch, T. Berthold, J. Pedersen, and C. Vanaret, “Progress in mathematical programming solvers from 2001 to 2020,” EURO Journal on Computational Optimization, vol.10, p.100031, 2022. 10.1016/j.ejco.2022.100031 [26] H.L. Urdaneta, J. Yuan, and A.S. Siqueira, “Alternative integer linear and quadratic programming formulations for HA-Assignment problems,” Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, vol.6, no.1, 2018. 10.5540/03.2018.006.01.0311 [18] J.-C. Régin, “Minimization of the number of breaks in sports scheduling problems using constraint programming,” DIMACS workshop on Constraint Programming and Large Scale Discrete Optimization, pp.115-130, 2000. [16] D. de Werra, “Scheduling in sports,” Studies on Graphs and Discrete Programming, vol.59, pp.381-395, 1981. 10.1016/s0304-0208(08)73478-9 [2] D. Briskorn, “Feasibility of home-away-pattern sets for round robin tournaments,” Operations Research Letters, vol.36, no.3, pp.283-284, 2008. 10.1016/j.orl.2007.09.009 [1] G.L. Nemhauser and M.A. Trick, “Scheduling a major college basketball conference,” Operations Research, vol.46, no.1, pp.1-8, 1998. 10.1287/opre.46.1.1 [7] D. Van Bulck and D. Goossens, “On the complexity of pattern feasibility problems in time-relaxed sports timetabling,” Operations Research Letters, vol.48, no.4, pp.452-459, 2020. 10.1016/j.orl.2020.05.005 [17] M. Elf, M. Jünger, and G. Rinaldi, “Minimizing breaks by maximizing cuts,” Operations Research Letters, vol.31, no.5, pp.343-349, 2003. 10.1016/s0167-6377(03)00025-7 [10] R. Miyashiro and T. Matsui, “Semidefinite programming based approaches to the break minimization problem,” Computers & Operations Research, vol.33, no.7, pp.1975-1982, 2006. 10.1016/j.cor.2004.09.030 [25] K. Easton, G. Nemhauser, and M. Trick, “Solving the travelling tournament problem: A combined integer programming and constraint programming approach,” PATAT 2002, Lecture Notes in Computer Science, vol.2740, Springer, pp.100-109, 2002. 10.1007/978-3-540-45157-0_6 [19] P. Van Hentenryck and Y. Vergados, “Minimizing breaks in sport scheduling with local search,” ICAPS, pp.22-29, 2005. [4] M. Henz, “Scheduling a major college basketball conference—revisited,” Operations Research, vol.49, no.1, pp.163-168, 2001. 10.1287/opre.49.1.163.11193 [6] L. Zeng and S. Mizuno, “On the separation in 2-period double round robin tournaments with minimum breaks,” Computers & Operations Research, vol.39, no.7, pp.1692-1700, 2012. 10.1016/j.cor.2011.10.004 [23] K. Bestuzheva, M. Besançon, W.-K. Chen, A. Chmiela, T. Donkiewicz, J. van Doornmalen, L. Eifler, O. Gaul, G. Gamrath, A. Gleixner, L. Gottwald, C. Graczyk, K. Halbig, A. Hoen, C. Hojny, R. van der Hulst, T. Koch, M. Lübbecke, S.J. Maher, F. Matter, E. Mühmer, B. Müller, M.E. Pfetsch, D. Rehfeldt, S. Schlein, F. Schlösser, F. Serrano, Y. Shinano, B. Sofranac, M. Turner, S. Vigerske, F. Wegscheider, P. Wellner, D. Weninger, and J. Witzig, “Enabling research through the SCIP optimization suite 8.0,” ACM Transactions on Mathematical Software, vol.49, no.2, pp.1-21, 2023. 10.1145/3585516 [3] R. Miyashiro, H. Iwasaki, and T. Matsui, “Characterizing feasible pattern sets with a minimum number of breaks,” PATAT 2002, Lecture Notes in Computer Science, vol.2740, pp.78-99, Springer, 2002. 10.1007/978-3-540-45157-0_5 [8] M.A. Trick, “A schedule-then-break approach to sports timetabling,” PATAT 2000, Lecture Notes in Computer Science, vol.2079, pp.242-253, Springer, 2000. 10.1007/3-540-44629-x_15 [11] G. Post and G.J. Woeginger, “Sports tournaments, home-away assignments, and the break minimization problem,” Discrete Optimization, vol.3, no.2, pp.165-173, 2006. 10.1016/j.disopt.2005.08.009 [5] D. Van Bulck and D. Goossens, “Optimizing rest times and differences in games played: an iterative two-phase approach,” Journal of Scheduling, vol.25, no.3, pp.261-271, 2022. 10.1007/s10951-021-00717-3 [13] R.V. Rasmussen, “Scheduling a triple round robin tournament for the best danish soccer league,” European Journal of Operational Research, vol.185, no.2, pp.795-810, 2008. 10.1016/j.ejor.2006.12.050 [28] D. Rehfeldt, T. Koch, and Y. Shinano, “Faster exact solution of sparse MaxCut and QUBO problems,” Mathematical Programming Computation, vol.15, no.3, pp.445-470, 2023. 10.1007/s12532-023-00236-6 22 23 24 25 26 27 28 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 20 21 |
| References_xml | – reference: [12] R.V. Rasmussen and M.A. Trick, “The timetable constrained distance minimization problem,” CPAIOR 2006, Lecture Notes in Computer Science, vol.3990, pp.167-181, Springer, 2006. 10.1007/11757375_15 – reference: [2] D. Briskorn, “Feasibility of home-away-pattern sets for round robin tournaments,” Operations Research Letters, vol.36, no.3, pp.283-284, 2008. 10.1016/j.orl.2007.09.009 – reference: [19] P. Van Hentenryck and Y. Vergados, “Minimizing breaks in sport scheduling with local search,” ICAPS, pp.22-29, 2005. – reference: [3] R. Miyashiro, H. Iwasaki, and T. Matsui, “Characterizing feasible pattern sets with a minimum number of breaks,” PATAT 2002, Lecture Notes in Computer Science, vol.2740, pp.78-99, Springer, 2002. 10.1007/978-3-540-45157-0_5 – reference: [7] D. Van Bulck and D. Goossens, “On the complexity of pattern feasibility problems in time-relaxed sports timetabling,” Operations Research Letters, vol.48, no.4, pp.452-459, 2020. 10.1016/j.orl.2020.05.005 – reference: [18] J.-C. Régin, “Minimization of the number of breaks in sports scheduling problems using constraint programming,” DIMACS workshop on Constraint Programming and Large Scale Discrete Optimization, pp.115-130, 2000. – reference: [10] R. Miyashiro and T. Matsui, “Semidefinite programming based approaches to the break minimization problem,” Computers & Operations Research, vol.33, no.7, pp.1975-1982, 2006. 10.1016/j.cor.2004.09.030 – reference: [25] K. Easton, G. Nemhauser, and M. Trick, “Solving the travelling tournament problem: A combined integer programming and constraint programming approach,” PATAT 2002, Lecture Notes in Computer Science, vol.2740, Springer, pp.100-109, 2002. 10.1007/978-3-540-45157-0_6 – reference: [1] G.L. Nemhauser and M.A. Trick, “Scheduling a major college basketball conference,” Operations Research, vol.46, no.1, pp.1-8, 1998. 10.1287/opre.46.1.1 – reference: [15] T. Koch, T. Berthold, J. Pedersen, and C. Vanaret, “Progress in mathematical programming solvers from 2001 to 2020,” EURO Journal on Computational Optimization, vol.10, p.100031, 2022. 10.1016/j.ejco.2022.100031 – reference: [9] R. Miyashiro and T. Matsui, “A polynomial-time algorithm to find an equitable home-away assignment,” Operations Research Letters, vol.33, no.3, pp.235-241, 2005. 10.1016/j.orl.2004.06.004 – reference: [4] M. Henz, “Scheduling a major college basketball conference—revisited,” Operations Research, vol.49, no.1, pp.163-168, 2001. 10.1287/opre.49.1.163.11193 – reference: [16] D. de Werra, “Scheduling in sports,” Studies on Graphs and Discrete Programming, vol.59, pp.381-395, 1981. 10.1016/s0304-0208(08)73478-9 – reference: [11] G. Post and G.J. Woeginger, “Sports tournaments, home-away assignments, and the break minimization problem,” Discrete Optimization, vol.3, no.2, pp.165-173, 2006. 10.1016/j.disopt.2005.08.009 – reference: [14] T. Achterberg and R. Wunderling, “Mixed integer programming: Analyzing 12 years of progress,” Facets of combinatorial optimization: Festschrift for Martin Grötschel, pp.449-481, Springer, 2013. 10.1007/978-3-642-38189-8_18 – reference: [8] M.A. Trick, “A schedule-then-break approach to sports timetabling,” PATAT 2000, Lecture Notes in Computer Science, vol.2079, pp.242-253, Springer, 2000. 10.1007/3-540-44629-x_15 – reference: [20] M.X. Goemans and D.P. Williamson, “Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,” Journal of the ACM, vol.42, no.6, pp.1115-1145, 1995. 10.1145/227683.227684 – reference: [24] D. Van Bulck, D. Goossens, J. Schönberger, and M. Guajardo, “RobinX: A three-field classification and unified data format for round-robin sports timetabling,” European Journal of Operational Research, vol.280, no.2, pp.568-580, 2020. 10.1016/j.ejor.2019.07.023 – reference: [26] H.L. Urdaneta, J. Yuan, and A.S. Siqueira, “Alternative integer linear and quadratic programming formulations for HA-Assignment problems,” Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, vol.6, no.1, 2018. 10.5540/03.2018.006.01.0311 – reference: [6] L. Zeng and S. Mizuno, “On the separation in 2-period double round robin tournaments with minimum breaks,” Computers & Operations Research, vol.39, no.7, pp.1692-1700, 2012. 10.1016/j.cor.2011.10.004 – reference: [17] M. Elf, M. Jünger, and G. Rinaldi, “Minimizing breaks by maximizing cuts,” Operations Research Letters, vol.31, no.5, pp.343-349, 2003. 10.1016/s0167-6377(03)00025-7 – reference: [28] D. Rehfeldt, T. Koch, and Y. Shinano, “Faster exact solution of sparse MaxCut and QUBO problems,” Mathematical Programming Computation, vol.15, no.3, pp.445-470, 2023. 10.1007/s12532-023-00236-6 – reference: [5] D. Van Bulck and D. Goossens, “Optimizing rest times and differences in games played: an iterative two-phase approach,” Journal of Scheduling, vol.25, no.3, pp.261-271, 2022. 10.1007/s10951-021-00717-3 – reference: [13] R.V. Rasmussen, “Scheduling a triple round robin tournament for the best danish soccer league,” European Journal of Operational Research, vol.185, no.2, pp.795-810, 2008. 10.1016/j.ejor.2006.12.050 – reference: [22] M. Kuramata, R. Katsuki, and K. Nakata, “Solving large break minimization problems in a mirrored double round-robin tournament using quantum annealing,” Plos one, vol.17, no.4, p.e0266846, 2022. 10.1371/journal.pone.0266846 – reference: [27] S. Vigerske and A. Gleixner, “Scip: Global optimization of mixed-integer nonlinear programs in a branch-and-cut framework,” Optimization Methods and Software, vol.33, no.3, pp.563-593, 2018. 10.1080/10556788.2017.1335312 – reference: [21] J.C. Peng, A.D. Clark, and A. Dahbura, “Introducing human corrective multi-team SRR sports scheduling via reinforcement learning,” 2021 IEEE Symposium Series on Computational Intelligence (SSCI), pp.1-7, IEEE, 2021. 10.1109/ssci50451.2021.9660171 – reference: [23] K. Bestuzheva, M. Besançon, W.-K. Chen, A. Chmiela, T. Donkiewicz, J. van Doornmalen, L. Eifler, O. Gaul, G. Gamrath, A. Gleixner, L. Gottwald, C. Graczyk, K. Halbig, A. Hoen, C. Hojny, R. van der Hulst, T. Koch, M. Lübbecke, S.J. Maher, F. Matter, E. Mühmer, B. Müller, M.E. Pfetsch, D. Rehfeldt, S. Schlein, F. Schlösser, F. Serrano, Y. Shinano, B. Sofranac, M. Turner, S. Vigerske, F. Wegscheider, P. Wellner, D. Weninger, and J. Witzig, “Enabling research through the SCIP optimization suite 8.0,” ACM Transactions on Mathematical Software, vol.49, no.2, pp.1-21, 2023. 10.1145/3585516 – ident: 4 doi: 10.1287/opre.49.1.163.11193 – ident: 21 doi: 10.1109/SSCI50451.2021.9660171 – ident: 9 doi: 10.1016/j.orl.2004.06.004 – ident: 15 doi: 10.1016/j.ejco.2022.100031 – ident: 22 doi: 10.1371/journal.pone.0266846 – ident: 5 doi: 10.1007/s10951-021-00717-3 – ident: 16 doi: 10.1016/S0304-0208(08)73478-9 – ident: 12 doi: 10.1007/11757375_15 – ident: 13 doi: 10.1016/j.ejor.2006.12.050 – ident: 23 doi: 10.1145/3585516 – ident: 2 doi: 10.1016/j.orl.2007.09.009 – ident: 25 doi: 10.1007/978-3-540-45157-0_6 – ident: 18 doi: 10.1090/dimacs/057/07 – ident: 1 doi: 10.1287/opre.46.1.1 – ident: 8 doi: 10.1007/3-540-44629-X_15 – ident: 11 doi: 10.1016/j.disopt.2005.08.009 – ident: 19 – ident: 27 doi: 10.1080/10556788.2017.1335312 – ident: 6 doi: 10.1016/j.cor.2011.10.004 – ident: 14 doi: 10.1007/978-3-642-38189-8_18 – ident: 17 doi: 10.1016/S0167-6377(03)00025-7 – ident: 3 doi: 10.1007/978-3-540-45157-0_5 – ident: 7 doi: 10.1016/j.orl.2020.05.005 – ident: 24 doi: 10.1016/j.ejor.2019.07.023 – ident: 26 doi: 10.5540/03.2018.006.01.0311 – ident: 10 doi: 10.1016/j.cor.2004.09.030 – ident: 20 doi: 10.1145/227683.227684 – ident: 28 doi: 10.1007/s12532-023-00236-6 |
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| Snippet | Constructing a suitable schedule for sports competitions is a crucial issue in sports scheduling. The round-robin tournament is a competition adopted in many... |
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| SubjectTerms | Design of experiments graph theory integer linear programming Integer programming Linear programming Optimization round robin Schedules Scheduling sports scheduling tournament scheduling Tournaments & championships |
| Title | A Bigram Based ILP Formulation for Break Minimization in Sports Scheduling Problems |
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