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Annals of Operations Research

, Volume 139, Issue 1, pp 375–388 | Cite as

On Compact Formulations for Integer Programs Solved by Column Generation

  • Daniel Villeneuve
  • Jacques DesrosiersEmail author
  • Marco E. Lübbecke
  • François Soumis
Article

Abstract

Column generation has become a powerful tool in solving large scale integer programs. It is well known that most of the often reported compatibility issues between pricing subproblem and branching rule disappear when branching decisions are based on imposing constraints on the subproblem's variables. This can be generalized to branching on variables of a so-called compact formulation. We constructively show that such a formulation always exists under mild assumptions. It has a block diagonal structure with identical subproblems, each of which contributes only one column in an integer solution. This construction has an interpretation as reversing a Dantzig-Wolfe decomposition. Our proposal opens the way for the development of branching rules adapted to the subproblem's structure and to the linking constraints.

Keywords

integer programming column generation branch-and-bound 

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References

  1. Barnhart, C., C.A. Hane, and P.H. Vance. (1997). “Integer Multicommodity Flow Problems.” Lecture Notes in Economics and Mathematical Systems 450, 17–31.Google Scholar
  2. Barnhart, C., E.L. Johnson, G.L. Nemhauser, M.W.P. Savelsbergh, and P.H. Vance. (1998). “Branch-and-Price: Column Generation for Solving Huge Integer Programs.” Oper. Res. 46(3), 316–329.Google Scholar
  3. Chen, Z.-L. and W.B. Powell. (1999). “Solving Parallel Machine Scheduling Problems by Column Generation.” INFORMS J. Computing 11, 78–94.Google Scholar
  4. Dantzig, G.B. and P. Wolfe. (1960). “Decomposition Principle for Linear Programs.” Oper. Res. 8, 101– 111.Google Scholar
  5. Desaulniers, G., J. Desrosiers, I. Ioachim, M.M. Solomon, F. Soumis, and D. Villeneuve. (1998). “A Unified Framework for Deterministic Time Constrained Vehicle Routing and Crew Scheduling Problems.” In T.G. Crainic and G. Laporte (eds.), Fleet Management and Logistics, Norwell, MA, Kluwer, pp. 57–93.Google Scholar
  6. Desrochers, M., J.K. Lenstra, M.W.P. Savelsbergh, and F. Soumis. (1991). “Vehicle Routing with Time Windows: Optimization and Approximation.” In B.L. Golden and A.A. Assad, (eds.), Vehicle Routing: Methods and Studies, volume 16 of Studies in Management Science and Systems, North-Holland, pp. 65–84.Google Scholar
  7. Desrosiers, J., Y. Dumas, M.M. Solomon, and F. Soumis. (1995). “Time Constrained Routing and Scheduling.” In M.O. Ball, T.L. Magnanti, C.L. Monma, and G.L. Nemhauser (eds.), Network Routing, volume 8 of Handbooks in Operations Research and Management Science, Amsterdam: North-Holland, pp. 35–139.Google Scholar
  8. Desrosiers, J., F. Soumis, and M. Desrochers. (1984). “Routing with Time Windows by Column Generation.” Networks 14, 545–565.Google Scholar
  9. Gamache, M., F. Soumis, D. Villeneuve, J. Desrosiers, and E. Gélinas. (1998). “The Preferential Bidding System at Air Canada.” Transportation Sci. 32(3), 246–255.Google Scholar
  10. Gilmore, P.C. and R.E. Gomory. (1961). “A Linear Programming Approach to the Cutting-Stock Problem.” Oper. Res. 9, 849–859.Google Scholar
  11. Holm, S. and J. Tind. (1988). “A Unified Approach for Price Directive Decomposition Procedures in Integer Programming.” Discrete Appl. Math. 20, 205–219.CrossRefGoogle Scholar
  12. Johnson, E.L. (1989). “Modelling and Strong Linear Programs for Mixed Integer Programming.” In S.W. Wallace (ed.), Algorithms and Model Formulations in Mathematical Programming, Springer: Berlin, pp. 1–43.Google Scholar
  13. Kantorovich, L.V. (1960). “Mathematical Methods of Organising and Planning Production.” Management Sci. 6, 366–422. Translation from the Russian original, dated 1939.Google Scholar
  14. Kohl, N., J. Desrosiers, O.B.G. Madsen, M.M. Solomon, and F. Soumis. (1999). “2-Path Cuts for the Vehicle Routing Problem with Time Windows.” Transportation Sci. 33(1), 101–116.Google Scholar
  15. Mehrotra, A., K.E. Murphy, and M.A. Trick. (2000). “Optimal Shift Scheduling: A Branch-and-Price approach. Naval Res. Logist. 47(3), 185–200.CrossRefGoogle Scholar
  16. Mehrotra, A. and M.A. Trick. (1996). “A Column Generation Approach for Graph Coloring.” INFORMS J. Comput. 8(4), 344–354.CrossRefGoogle Scholar
  17. Nemhauser, G.L. and L.A. Wolsey. (1988). Integer and Combinatorial Optimization. Chichester: John Wiley & Sons.Google Scholar
  18. Ryan, D.M. and J.C. Falkner. (1987). “A Bus Crew Scheduling System Using a Set Partitioning Mode.” Ann. Oper. Res. 4, 39–56.Google Scholar
  19. Ryan, D.M. and B.A. Foster. (1981). “An Integer Programming Approach to Scheduling.” In A. Wren (ed.), Computer Scheduling of Public Transport Urban Passenger Vehicle and Crew Scheduling, Amsterdam: North-Holland, pp. 269–280.Google Scholar
  20. Savelsbergh, M.W.P. (1997). “A Branch-and-Price Algorithm for the Generalized Assignment Problem.” Oper. Res. 45(6), 831–841.Google Scholar
  21. Schrijver, A. (1986). Theory of Linear and Integer Programming. Chichester: John Wiley & Sons.Google Scholar
  22. Sol, M. (1994). Column Generation Techniques for Pickup and Delivery Problems. PhD thesis, Eindhoven University of Technology.Google Scholar
  23. Sweeney, D.J. and R.A. Murphy. (1979). “A Method of Decomposition for Integer Programs.” Oper. Res. 27, 1128–1141.Google Scholar
  24. Valério de Carvalho, J.M. (1999). “Exact Solution of Bin-Packing Problems Using Column Generation and Branch-and-Bound.” Ann. Oper. Res. 86, 629–659.CrossRefGoogle Scholar
  25. Valério de Carvalho, J.M. (2002). “LP Models for Bin-Packing and Cutting Stock Problems.” European J. Oper. Res. 141(2), 253–273.CrossRefGoogle Scholar
  26. van den Akker, J.M., J.A. Hoogeveen, and S.L. van de Velde. (1999). “Parallel Machine Scheduling by Column Generation.” Oper. Res. 47(6), 862–872.Google Scholar
  27. Vance, P.H. (1998). “Branch-and-Price Algorithms for the One-Dimensional Cutting Stock Problem.” Comput. Optim. Appl. 9(3), 211–228.CrossRefGoogle Scholar
  28. Vanderbeck, F. (1994). Decomposition and Column Generation for Integer Programs. PhD thesis, Université catholique de Louvain.Google Scholar
  29. Vanderbeck, F. (2000a). “Exact Algorithm for Minimising the Number of Setups in the One-Dimensional Cutting Stock Problem.” Oper. Res. 48(6), 915–926.CrossRefGoogle Scholar
  30. Vanderbeck, F. (2000b). “On Dantzig-Wolfe Decomposition in Integer Programming and Ways to Perform Branching in a Branch-and-Price Algorithm.” Oper. Res. 48(1), 111–128.CrossRefGoogle Scholar
  31. Vanderbeck, F. and L.A. Wolsey. (1996). “An Exact Algorithm for IP Column Generation. Oper. Res. Lett. 19, 151–159.CrossRefGoogle Scholar
  32. Villeneuve, D. (1999). “Logiciel de Génération de Colonnes.” PhD thesis, École Polytechnique de Montréal.Google Scholar
  33. Villeneuve, D. and G. Desaulniers. (2005). “The Shortest Path Problem with Forbidden Paths.” European J. Oper. Res. 165(1), 97–107.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Daniel Villeneuve
    • 1
  • Jacques Desrosiers
    • 2
    Email author
  • Marco E. Lübbecke
    • 3
  • François Soumis
    • 4
  1. 1.Kronos Inc.MontréalCanada
  2. 2.HEC Montréal and GERAD 3000MontréalCanada
  3. 3.Technische Universität Berlin, Institut für MathematikBerlinGermany
  4. 4.École Polytechnique de Montréal and GERADMontréalCanada

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