Document Open Access Logo

A Computational Investigation on the Strength of Dantzig-Wolfe Reformulations

Authors Michael Bastubbe, Marco E. Lübbecke, Jonas T. Witt

Thumbnail PDF


  • Filesize: 0.86 MB
  • 12 pages

Document Identifiers

Author Details

Michael Bastubbe
  • Lehrstuhl für Operations Research, RWTH Aachen University, Kackertstr. 7, D-52072 Aachen, Germany
Marco E. Lübbecke
  • Lehrstuhl für Operations Research, RWTH Aachen University, Kackertstr. 7, D-52072 Aachen, Germany
Jonas T. Witt
  • Lehrstuhl für Operations Research, RWTH Aachen University, Kackertstr. 7, D-52072 Aachen, Germany

Cite AsGet BibTex

Michael Bastubbe, Marco E. Lübbecke, and Jonas T. Witt. A Computational Investigation on the Strength of Dantzig-Wolfe Reformulations. In 17th International Symposium on Experimental Algorithms (SEA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 103, pp. 11:1-11:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


In Dantzig-Wolfe reformulation of an integer program one convexifies a subset of the constraints, leading to potentially stronger dual bounds from the respective linear programming relaxation. As the subset can be chosen arbitrarily, this includes the trivial cases of convexifying no and all constraints, resulting in a weakest and strongest reformulation, respectively. Our computational study aims at better understanding of what happens in between these extremes. For a collection of integer programs with few constraints we compute, optimally solve, and evaluate the relaxations of all possible (exponentially many) Dantzig-Wolfe reformulations (with mild extensions to larger models from the MIPLIBs). We observe that only a tiny number of different dual bounds actually occur and that only a few inclusion-wise minimal representatives exist for each. This aligns with considerably different impacts of individual constraints on the strengthening the relaxation, some of which have almost no influence. In contrast, types of constraints that are convexified in textbook reformulations have a larger effect. We relate our experiments to what could be called a hierarchy of Dantzig-Wolfe reformulations.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Integer programming
  • Dantzig-Wolfe reformulation
  • strength of reformulations
  • Lagrangean relaxation
  • partial convexification
  • column generation
  • hierarchy of relaxations


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. T. Achterberg, T. Koch, and A. Martin. MIPLIB 2003. Oper. Res. Lett., 34(4):361-372, 2006. Google Scholar
  2. M. Bergner, A. Caprara, A. Ceselli, F. Furini, M.E. Lübbecke, E. Malaguti, and E. Traversi. Automatic Dantzig-Wolfe reformulation of mixed integer programs. Math. Programming, 149(1-2):391-424, 2015. Google Scholar
  3. R Bixby, Sebastian Ceria, C McZeal, and M Savelsbergh. An updated mixed integer programming library: Miplib 3.0, 1996. Google Scholar
  4. D.G. Cattrysse, M. Salomon, and L.N. Van Wassenhove. A set partitioning heuristic for the generalized assignment problem. European J. Oper. Res., 72(1):167-174, 1994. Google Scholar
  5. P.C. Chu and JE Beasley. A genetic algorithm for the generalised assignment problem. Comput. Oper. Res., 24(1):17-23, 1997. Google Scholar
  6. G. Gamrath and M.E. Lübbecke. Experiments with a generic Dantzig-Wolfe decomposition for integer programs. In P. Festa, editor, Proceedings of the 9th International Symposium on Experimental Algorithms (SEA), volume 6049 of Lect. Notes Comput. Sci., pages 239-252, Berlin, 2010. Springer-Verlag. Google Scholar
  7. A.M. Geoffrion. Lagrangean relaxation for integer programming. Math. Programming Stud., 2:82-114, 1974. Google Scholar
  8. Kaj Holmberg, Mikael Rönnqvist, and Di Yuan. An exact algorithm for the capacitated facility location problems with single sourcing. European J. Oper. Res., 113(3):544-559, 1999. Google Scholar
  9. T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R. E. Bixby, E. Danna, G. Gamrath, A. M. Gleixner, S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs, D. Salvagnin, D. E. Steffy, and K. Wolter. MIPLIB 2010. Math. Program. Comput., 3(2):103-163, 2011. Google Scholar
  10. Anuj Mehrotra and Michael A Trick. A column generation approach for graph coloring. INFORMS J. Comput., 8(4):344-354, 1996. Google Scholar
  11. Rashed Sahraeian and Payman Kaveh. Solving capacitated p-median problem by hybrid k-means clustering and fixed neighborhood search algorithm. In Proceedings of the 2010 International Conference on Industrial Engineering and Operation Management, pages 1-6, 2010. Google Scholar
  12. P. Schwerin and G. Wäscher. The bin-packing problem: A problem generator and some numerical experiments with ffd packing and mtp. Internat. Trans. Oper. Res., 4(5-6):377-389, 1997. Google Scholar
  13. F. Vanderbeck and M.W.P. Savelsbergh. A generic view of Dantzig-Wolfe decomposition in mixed integer programming. Oper. Res. Lett., 34(3):296-306, 2006. Google Scholar
  14. J.T. Witt and M.E. Lübbecke. Dantzig-Wolfe reformulations for the stable set problem. repORt 2015-029, Lehrstuhl für Operations Research, RWTH Aachen University, Nov 2015. URL:
  15. Y. Wu and P. Yang. Sample complexity of the distinct elements problem. ArXiv e-prints, Dec 2016. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail