Journal of Combinatorial Optimization

, Volume 36, Issue 3, pp 916–936 | Cite as

An exact approach for the balanced k-way partitioning problem with weight constraints and its application to sports team realignment

  • Diego RecaldeEmail author
  • Daniel Severín
  • Ramiro Torres
  • Polo Vaca


In this work a balanced k-way partitioning problem with weight constraints is defined to model the sports team realignment. Sports teams must be partitioned into a fixed number of groups according to some regulations, where the total distance of the road trips that all teams must travel to play a double round robin tournament in each group is minimized. Two integer programming formulations for this problem are introduced, and the validity of three families of inequalities associated to the polytope of these formulations is proved. The performance of a tabu search procedure and a branch and cut algorithm, which uses the valid inequalities as cuts, is evaluated over simulated and real-world instances. In particular, an optimal solution for the realignment of the Ecuadorian football league is reported and the methodology can be suitable adapted for the realignment of other sports leagues.


Integer programming models Graph partitioning Tabu search Sports team realignment 



This research was partially supported by the 15-MathAmSud-06 “PACK-COVER: Packing and covering, structural aspects” trilateral cooperation project. We are grateful to the anonymous referees for their useful comments which led to a significantly improved presentation of this work.


  1. Anjos M, Ghaddar B, Hupp L, Liers F, Wiegele A (2013) Solving k-way graph partitioning problems to optimality: the impact of semidefinite relaxations and the bundle method. In: Jünger M, Reinelt G (eds) Facets of combinatorial optimization: Festschrift for Martin Grötschel. Springer, Berlin, pp 355–386CrossRefGoogle Scholar
  2. Buluç A, Meyerhenke H, Safro I, Sanders P, Schulz C (2016) Recent advances in graph partitioning. In: Kliemann L, Sanders P (eds) Algorithm engineering: selected results and surveys. Springer, Cham, pp 117–158CrossRefGoogle Scholar
  3. Catanzaro D, Gourdinb E, Labbé M, Özsoy FA (2011) A branch-and-cut algorithm for the partitioning-hub location-routing problem. Comput Oper Res 38(2):539–549MathSciNetCrossRefzbMATHGoogle Scholar
  4. Fairbrother J, Letchford A, Briggs K (2017) A two-level graph partitioning problem arising in mobile wireless communications. arXiv:1705.08773
  5. Ferreira C, Martin A, de Souza C, Weismantel R, Wolsey L (1998) The node capacitated graph partitioning problem: a computational study. Math Program 81:229–256MathSciNetzbMATHGoogle Scholar
  6. Glover F, McMillan C, Novick B (1985) Interactive decision software and computer graphics for architectural and space planning. Ann Oper Res 5(3):557–573CrossRefGoogle Scholar
  7. Grötschel M, Wakabayashi Y (1989) A cutting plane algorithm for a clustering problem. Math Program 45:59–96MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hendrickson B, Kolda TG (2000) Graph partitioning models for parallel computing. Parallel Comput 26(12):1519–1534MathSciNetCrossRefzbMATHGoogle Scholar
  9. Jaehn F, Pesch E (2013) New bounds and constraint propagation techniques for the clique partitioning problem. Discr Appl Math 161:2025–2037MathSciNetCrossRefzbMATHGoogle Scholar
  10. Ji X, Mitchell JE (2007) Branch-and-price-and-cut on the clique partitioning problem with minimum clique size requirement. Discr Optim 4:87–102MathSciNetCrossRefzbMATHGoogle Scholar
  11. Kahng A, Lienig J, Markov I, Hu J (2011) VLSI physical design: from graph partitioning to timing closure, 1st edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  12. Labbé M, Özsoy FA (2010) Size-constrained graph partitioning polytopes. Discrete Math 310:3473–3493MathSciNetCrossRefzbMATHGoogle Scholar
  13. Lai X, Hao JK, Glover F (2015) Backtracking based iterated tabu search for equitable coloring. Eng Appl Artif Intell 46:269–278CrossRefGoogle Scholar
  14. McDonald B, Pulleyblank W (2014) Realignment in the NHL, MLB, NFL, and NBA. J Quant Anal Sports 10(2):225–240Google Scholar
  15. Méndez Díaz I, Nasini G, Severin D (2014) A tabu search heuristic for the equitable coloring problem. Lect Notes Comput Sci 8596:347–358MathSciNetCrossRefzbMATHGoogle Scholar
  16. Mitchell JE (2001) Branch-and-cut for the k-way equipartition problem. Technical report, Department of Mathematical Sciences, Rensselaer Polytechnic InstituteGoogle Scholar
  17. Mitchell JE (2003) Realignment in the national football league: Did they do it right? Naval Res Logist (NRL) 50(7):683–701MathSciNetCrossRefzbMATHGoogle Scholar
  18. Nguyen DP, Minoux M, Nguyen VH, Nguyen TH, Sirdey R (2017) Improved compact formulations for a wide class of graph partitioning problems in sparse graphs. Discrete Optim 25(C):175–188MathSciNetCrossRefzbMATHGoogle Scholar
  19. Recalde D, Severin D, Torres R, Vaca P (2016) Balanced partition of a graph for football team realignment in ecuador. Lect Notes Comput Sci 9849:357–368MathSciNetCrossRefzbMATHGoogle Scholar
  20. Saltzman R, Bradford RM (1996) Optimal realignments of the teams in the national football league. Eur J Oper Res 93(3):469–475CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Departamento de MatemáticaEscuela Politécnica NacionalQuitoEcuador
  2. 2.Research Center on Mathematical Modelling (MODEMAT)Escuela Politécnica NacionalQuitoEcuador
  3. 3.FCEIAUniversidad Nacional de Rosario and CONICETRosarioArgentina

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