[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content
Springer Nature Link
Log in

Compressing Branch-and-Bound Trees

  • Conference paper
  • First Online:
Integer Programming and Combinatorial Optimization (IPCO 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13904))

Abstract

A branch-and-bound (BB) tree certifies a dual bound on the value of an integer program. In this work, we introduce the tree compression problem (TCP): Given a BB tree T that certifies a dual bound, can we obtain a smaller tree with the same (or stronger) bound by either (1) applying a different disjunction at some node in T or (2) removing leaves from T? We believe such post-hoc analysis of BB trees may assist in identifying helpful general disjunctions in BB algorithms. We initiate our study by considering computational complexity and limitations of TCP. We then conduct experiments to evaluate the compressibility of realistic branch-and-bound trees generated by commonly-used branching strategies, using both an exact and a heuristic compression algorithm.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
GBP 19.95
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
GBP 55.99
Price includes VAT (United Kingdom)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
GBP 69.99
Price includes VAT (United Kingdom)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Aardal, K., Lenstra, A.: Hard equality constrained integer knapsacks. Math. Oper. Res. 29, 724–738 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Achterberg, T., Koch, T., Martin, A.: Branching rules revisited. Oper. Res. Lett. 33(1), 42–54 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Basu, A., Conforti, M., Di Summa, M., Jiang, H.: Complexity of branch-and-bound and cutting planes in mixed-integer optimization - II. In: Singh, M., Williamson, D.P. (eds.) IPCO 2021. LNCS, vol. 12707, pp. 383–398. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-73879-2_27

    Chapter  MATH  Google Scholar 

  4. Beame, P., et al.: Stabbing planes. In: Karlin, A.R. (ed.) 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 94, pp. 10:1–10:20. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl (2018). https://doi.org/10.4230/LIPIcs.ITCS.2018.10, http://drops.dagstuhl.de/opus/volltexte/2018/8341

  5. Bixby, R.: Solving real-world linear programs: a decade and more of progress. Oper. Res. 50(1), 3–15 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bixby, R., Boyd, E., Indovina, R.: MIPLIB: a test set of mixed integer programming problems. SIAM News (1992)

    Google Scholar 

  7. Cheung, K.K.H., Gleixner, A., Steffy, D.E.: Verifying integer programming results. In: Eisenbrand, F., Koenemann, J. (eds.) IPCO 2017. LNCS, vol. 10328, pp. 148–160. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-59250-3_13

    Chapter  MATH  Google Scholar 

  8. Chvátal, V.: Hard knapsack problems. Oper. Res. 28, 1402–1411 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cornuéjols, G., Liberti, L., Nannicini, G.: Improved strategies for branching on general disjunctions. Math. Program. 130, 225–247 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dadush, D., Tiwari, S.: On the complexity of branching proofs. In: Saraf, S. (ed.) 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), vol. 169, pp. 34:1–34:35. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl (2020)

    Google Scholar 

  11. Dey, S., Dubey, Y., Molinaro, M.: Lower bounds on the size of general branch-and-bound trees. Math. Program. (2022)

    Google Scholar 

  12. Dey, S., Dubey, Y., Molinaro, M., Shah, P.: A theoretical and computational analysis of full strong-branching. arXiv:2110.10754 (2021)

  13. Dunning, I., Huchette, J., Lubin, M.: Jump: a modeling language for mathematical optimization. SIAM Rev. 59(2), 295–320 (2017). https://doi.org/10.1137/15M1020575

    Article  MathSciNet  MATH  Google Scholar 

  14. Fischetti, M., Monaci, M.: Backdoor branching. INFORMS J. Comput. 25(4), 693–700 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gamrath, G., Melchiori, A., Berthold, T., Gleixner, A.M., Salvagnin, D.: Branching on multi-aggregated variables. In: Michel, L. (ed.) CPAIOR 2015. LNCS, vol. 9075, pp. 141–156. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-18008-3_10

    Chapter  Google Scholar 

  16. Gläser, M., Pfetsch, M.: On the complexity of finding shortest variable disjunction branch-and-bound proofs. In: Aardal, K., Sanità, L. (eds.) IPCO 2022. LNCS, vol. 13265, pp. 291–304. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-06901-7_22

  17. Gurobi Optimization: Gurobi Optimizer (Version 9.5). https://www.gurobi.com/products/gurobi-optimizer/. Accessed 4 Nov 2022

  18. Jeroslow, R.: Trivial integer programs unsolvble by branch-and-bound. Math. Program. 6, 105–109 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  19. Karamanov, M., Cornuéjols, G.: Branching on general disjunctions. Math. Program. 128, 403–436 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Khalil, E., Vaezipoor, P., Dilkina, B.: Finding backdoors to integer programs: a Monte Carlo tree search framework. In: Proceedings of AAAI (2022)

    Google Scholar 

  21. Legat, B., Dowson, O., Dias Garcia, J., Lubin, M.: MathOptInterface: a data structure for mathematical optimization problems. INFORMS J. Comput. 34(2), 672–689 (2021). https://doi.org/10.1287/ijoc.2021.1067

    Article  MathSciNet  MATH  Google Scholar 

  22. Linderoth, J., Savelsbergh, M.: A computational study of search strategies for mixed integer programming. INFORMS J. Comput. 11(2), 173–187 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mahajan, A., Ralphs, T.: Experiments with branching using general disjunctions. In: Proceedings of Operations Research and Cyber-Infrastructure, pp. 101–118 (2009)

    Google Scholar 

  24. Mahajan, A., Ralphs, T.: On the complexity of selecting disjunctions in integer programming. SIAM J. Optim. 20(5), 2181–2198 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mahmoud, H., Chinneck, J.: Achieving MILP feasibility quickly using general disjunctions. Comput. Oper. Res. 40, 2094–2102 (2013)

    Article  MATH  Google Scholar 

  26. Mehrotra, S., Li, Z.: Branching on hyperplane methods for mixed integer linear and convex programming using adjoint lattices. J. Glob. Optim. (2010)

    Google Scholar 

  27. Owen, J., Mehrotra, S.: Experimental results on using general disjunctions in branch-and-bound for general-integer linear programs. Comput. Optim. Appl. 20, 159–170 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Walter, M.: Sparsity of lift-and-project cutting planes. In: Helber, S., et al. (eds.) Operations Research Proceedings 2012, pp. 9–14. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-00795-3_2

  29. Xavier, A.S., Qiu, F.: MIPLearn: a framework for learning-enhanced mixed-integer optimization (Julia interface) (2022). https://github.com/ANL-CEEESA/MIPLearn.jl

  30. Yang, Y., Boland, N., Savelsbergh, M.: Multivariable branching: a 0–1 knapsack problem case study. INFORMS J. Comput. 33(4), 1354–1367 (2021)

    Google Scholar 

Download references

Acknowledgements

J. Paat was supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant [RGPIN-2021-02475]. Á.S. Xavier was partially supported by the U.S. Department of Energy Office of Electricity. The authors want to thank the referees, whose comments improved the overall presentation of the paper, led to better bounds in Theorem 2, and identified directions of future work.

Author information

Authors and Affiliations

  1. Institute of Engineering Sciences, Universidad de O’Higgins, Rancagua, Chile

    Gonzalo Muñoz

  2. Sauder School of Business, University of British Columbia, Vancouver, BC, Canada

    Joseph Paat

  3. Energy Systems and Infrastructure Analysis Division, Argonne National Laboratory, Lemont, IL, USA

    Álinson S. Xavier

Authors
  1. Gonzalo Muñoz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joseph Paat
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Álinson S. Xavier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joseph Paat .

Editor information

Editors and Affiliations

  1. University of Wisconsin-Madison, Madison, WI, USA

    Alberto Del Pia

  2. Otto-von-Guericke-Universität, Magdeburg, Sachsen-Anhalt, Germany

    Volker Kaibel

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Muñoz, G., Paat, J., Xavier, Á.S. (2023). Compressing Branch-and-Bound Trees. In: Del Pia, A., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2023. Lecture Notes in Computer Science, vol 13904. Springer, Cham. https://doi.org/10.1007/978-3-031-32726-1_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-32726-1_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-32725-4

  • Online ISBN: 978-3-031-32726-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation