Skip to main content
Springer Nature Link
Log in

The Mcf-separator: detecting and exploiting multi-commodity flow structures in MIPs

  • Full Length Paper
  • Published:
Mathematical Programming Computation Aims and scope Submit manuscript

Abstract

Given a general mixed integer program, we automatically detect block structures in the constraint matrix together with the coupling by capacity constraints arising from multi-commodity flow formulations. We identify the underlying graph and generate cutting planes based on cuts in the detected network. Our implementation adds a separator to the branch-and-cut libraries of Scip and Cplex. We make use of the complemented mixed integer rounding framework but provide a special purpose aggregation heuristic that exploits the network structure. Our separation scheme speeds-up the computation for a large set of mixed integer programs coming from network design problems by a factor two on average. We show that almost 10% of the instances in general testsets contain consistent embedded networks. For these instances the computation time is decreased by 18% on average.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achterberg, T.: Constraint Integer Programming. PhD thesis, Technische Universität Berlin. http://opus.kobv.de/tuberlin/volltexte/2007/1611/ (2007)

  2. Achterberg, T., Raack, C.: The Mcf-separator—Detecting and Exploiting Multi-Commodity Flow Structures in MIPs. ZIB-Report 07-38, Konrad-Zuse-Zentrum für Informationstechnik Berlin. http://www.zib.de/ (2009)

  3. Achterberg, T., Koch, T., Martin, A.: MIPLIB Oper. Res. Lett. 34(4):361–372, (2003). http://miplib.zib.de/ (2006)

    Google Scholar 

  4. Ahuja R., Magnanti T., Orlin J.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)

    Google Scholar 

  5. Atamtürk A.: Flow pack facets of the single node fixed-charge flow polytope. Oper. Res. Lett. 29, 107–114 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Atamtürk A.: On capacitated network design cut-set polyhedra. Math. Program. 92, 425–437 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Atamtürk A.: On the facets of the mixed-integer knapsack polyhedron. Math. Program. 98, 145–175 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Atamtürk A.: Cover and pack inequalities for mixed integer programming. Ann. Oper. Res. 139(1), 21–38 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Atamtürk, A.: MIP instances. University of California, Berkeley. http://www.ieor.berkeley.edu/~atamturk/data/ (2009)

  10. Atamtürk A., Rajan D.: On splittable and unsplittable capacitated network design arc-set polyhedra. Math. Program. 92, 315–333 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Atamtürk A., Nemhauser G.L., Savelsbergh M.W.P.: Valid inequalities for problems with additive variable upper bounds. Math. Program. 91, 145–162 (2001)

    MATH  MathSciNet  Google Scholar 

  12. Balas E.: Facets of the knapsack polytope. Math. Program. 8, 146–164 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bienstock D., Günlük O.: Capacitated network design—polyhedral structure and computation. INFORMS J. Comput. 8, 243–259 (1996)

    Article  MATH  Google Scholar 

  14. Bienstock D., Chopra S., Günlük O., Tsai C.Y.: Minimum cost capacity installation for multicommodity network flows. Math. Program. 81, 177–199 (1998)

    Google Scholar 

  15. Bixby R., Fourer R.: Finding embedded network rows in linear programs I. Extraction heuristics. Manage. Sci. 34(3), 342–376 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bixby R., Rothberg E.: Progress in computational mixed integer programming—a look back from the other side of the tipping point. Ann. Oper. Res. 149(1), 37–41 (2007)

    Article  MathSciNet  Google Scholar 

  17. Bixby, R.E., Ceria, S., McZeal, C.M., Savelsbergh, M.W.P.: An updated mixed integer programming library: MIPLIB 3.0. Optima 58:12–15. http://www.caam.rice.edu/~bixby/miplib/miplib.html (1998)

    Google Scholar 

  18. Brockmüller B., Günlük O., Wolsey L.A.: Designing private line networks: polyhedral analysis and computation. Trans. Oper. Res. 16, 7–24 (2004)

    Google Scholar 

  19. Brown G., Wright W.: Automatic identification of embedded network rows in large-scale optimization models. Math. Program. 29, 41–56 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  20. Bussieck M.R., Kreuzer P., Zimmermann U.T.: Discrete optimization in public rail transport. Math. Program. 79(1–3), 415–444 (1997)

    Google Scholar 

  21. Chopra S., Gilboa I., Sastry S.T.: Source sink flows with capacity installation in batches. Discret. Appl. Math. 86, 165–192 (1998)

    Article  MathSciNet  Google Scholar 

  22. COmputational INfrastructure for Operations Research (COIN-OR) Cut Generation Library (CGL). https://projects.coin-or.org/Cgl (2009)

  23. Dahl G., Stoer M.: A polyhedral approach to multicommodity survivable network design. Numer. Math. 68, 149–167 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Dahl G., Stoer M.: A cutting plane algorithm for multicommodity survivable network design problems. INFORMS J. Comput. 10, 1–11 (1998)

    Article  MathSciNet  Google Scholar 

  25. Garey M., Johnson D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman and Company, New York (1979)

    MATH  Google Scholar 

  26. Gonçalves, J.P.M., Ladanyi, L.: An implementation of a separation procedure for mixed integer rounding inequalities. IBM Res. Report RC23686 (W0508-022), IBM (2005)

  27. Gottlieb, J., Mittelmann, H.: FCTP instances. Arizona State University. http://plato.la.asu.edu/ftp/fctp/ (2009)

  28. Gu Z., Nemhauser G.L., Savelsbergh M.W.P.: Lifted flow cover inequalities for mixed 0-1 integer programs. Math. Program. 85, 436–467 (1999)

    Article  MathSciNet  Google Scholar 

  29. Gu, Z., Nemhauser, G.L., Savelsbergh, M.W.P.: Sequence independent lifting in mixed integer programming. INFORMS J. Comput. pp. 109–129 (2000)

  30. Günlük O.: A branch and cut algorithm for capacitated network design problems. Math. Program. 86, 17–39 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  31. Hoesel S.P.M., Koster A.M.C.A., van de Leensel R.L.M.J., Savelsbergh M.W.P.: Polyhedral results for the edge capacity polytope. Math. Program. 92(2), 335–358 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  32. Hoesel S.P.M., Koster A.M.C.A., van de Leensel R.L.M.J., Savelsbergh M.W.P.: Bidirected and unidirected capacity installation in telecommunication networks. Discret. Appl. Math. 133, 103–121 (2004)

    Google Scholar 

  33. IBM-ILOG: Cplex. http://www.ilog.com/products/cplex/ (2009)

  34. Louveaux Q., Wolsey L.A.: Lifting, superadditivity, mixed integer rounding and single node flow sets revisited. 4OR 1(3), 173–207 (2003)

    MATH  MathSciNet  Google Scholar 

  35. Magnanti T.L., Mirchandani P.: Shortest paths, single origin-destination network design and associated polyhedra. Networks 33, 103–121 (1993)

    Article  MathSciNet  Google Scholar 

  36. Magnanti T.L., Wong R.T.: Network design and transportation planning: models and algorithms. Trans. Sci. 18(1), 1–55 (1984)

    Article  Google Scholar 

  37. Magnanti T.L., Mirchandani P., Vachani R.: The convex hull of two core capacitated network design problems. Math. Program. 60, 233–250 (1993)

    Article  MathSciNet  Google Scholar 

  38. Magnanti T.L., Mirchandani P., Vachani R.: Modelling and solving the two-facility capacitated network loading problem. Oper. Res. 43, 142–157 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  39. Marchand, H.: A polyhedral study of the mixed knapsack set and its use to solve mixed integer programs. PhD thesis, Université Catholique de Louvain, Louvain-la-Neuve, Belgium (1997)

  40. Marchand H., Wolsey L.A.: The 0–1 knapsack problem with a single continuous variable. Math. Program. 85, 15–33 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  41. Marchand H., Wolsey L.A.: Aggregation and mixed integer rounding to solve MIPs. Oper. Res. 49(3), 363–371 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  42. Mittelmann, H.: Benchmarks for optimization software. http://plato.asu.edu/bench.html (2009)

  43. Nemhauser G.L., Wolsey L.A.: Integer and Combinatorial Optimization. Wiley, London (1988)

    MATH  Google Scholar 

  44. Orlowski , S. , Pióro M., Tomaszewski A., Wessäly R.: SNDlib 1.0–Survivable Network Design Library. Networks 55(3), 276–286 (2010)

    Google Scholar 

  45. Ortega F., Wolsey L.A.: A branch-and-cut algorithm for the single-commodity, uncapacitated, fixed-charge network flow problem. Networks 41, 143–158 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  46. Padberg M.W., Roy T.J.V., Wolsey L.A.: Valid linear inequalities for fixed charge problems. Oper. Res. 33, 842–861 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  47. Pióro M., Medhi D.: Routing, Flow, and Capacity Design in Communication and Computer Networks. Morgan Kaufmann Publishers, Menlo Park (2004)

    MATH  Google Scholar 

  48. Pochet Y., Wolsey L.A.: Integer knapsack and flow covers with divisible coefficients. Discret. Appl. Math. 59, 57–74 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  49. Raack, C., Koster, A.M.C.A., Orlowski, S., Wessäly, R.: Capacitated network design using general flow-cutset inequalities. In: Proceedings of the Third International Network Optimization Conference (INOC 2007), Spa, Belgium (2007)

  50. Raack, C., Koster, A.M.C.A., Orlowski, S., Wessäly, R.: On cut-based inequalities for capacitated network design polyhedra. Networks (2010, to appear)

  51. Resende, M., Pardalos, P. (eds): Handbook of Optimization in Telecommunications. Springer, Berlin (2006)

    MATH  Google Scholar 

  52. Van Roy T.J., Wolsey L.A.: Valid inequalities for mixed 0-1 programs. Discret. Appl. Math. 14, 199–213 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  53. Weismantel R.: On the 0/1 knapsack polytope. Math. Program. 77, 49–68 (1997)

    MathSciNet  Google Scholar 

  54. Wessäly, R.: Dimensioning Survivable Capacitated NETworks. PhD thesis, Technische Universitä Berlin (2000)

  55. Wolsey, L.: UFCN instances. CORE, Université catholique de Louvain. http://www.core.ucl.ac.be/wolsey/ufcn.htm (2009)

  56. Wolsey L.A.: Faces for a linear inequality in 0–1 variables. Math. Program. 8, 165–178 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  57. Wolsey L.A.: Valid inequalities and superadditivity for 0–1 integer programs. Math. Oper. Res. 2, 66–77 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  58. Wolsey L.A.: Integer Programming. Wiley, London (1998)

    MATH  Google Scholar 

  59. Wolter, K.: Implementation of Cutting Plane Separators for Mixed Integer Programs. Master’s thesis, Technische Universität Berlin (2006)

  60. Zuse Institut Berlin: —Solving Constraint Integer Programs. http://scip.zib.de/ (2009)

  61. Zuse Institut Berlin: SNDlib—Survivable Network Design Library. http://sndlib.zib.de/ (2009)

Download references

Author information

Authors and Affiliations

  1. IBM Schoenaicher, Str. 220, 71003, Boeblingen, Germany

    Tobias Achterberg

  2. Zuse Institute Berlin (ZIB), Takustr. 7, 14195, Berlin, Germany

    Christian Raack

Authors
  1. Tobias Achterberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Raack
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Raack.

Additional information

This research has been supported by the DFG research Center Matheon.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Achterberg, T., Raack, C. The Mcf-separator: detecting and exploiting multi-commodity flow structures in MIPs. Math. Prog. Comp. 2, 125–165 (2010). https://doi.org/10.1007/s12532-010-0015-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12532-010-0015-3

Keywords

Mathematics Subject Classification (2000)