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An Abstract Model for Branch-and-Cut

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Integer Programming and Combinatorial Optimization (IPCO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13265))

Abstract

Branch-and-cut is the dominant paradigm for solving a wide range of mathematical programming problems—linear or nonlinear—combining intelligent search (via branch-and-bound) and relaxation-tightening procedures (via cutting planes, or cuts). While there is a wealth of computational experience behind existing cutting strategies, there is simultaneously a relative lack of theoretical explanations for these choices, and for the tradeoffs involved therein. Recent papers have explored abstract models for branching and for comparing cuts with branch-and-bound. However, to model practice, it is crucial to understand the impact of jointly considering branching and cutting decisions. In this paper, we provide a framework for analyzing how cuts affect the size of branch-and-cut trees, as well as their impact on solution time. Our abstract model captures some of the key characteristics of real-world phenomena in branch-and-cut experiments, regarding whether to generate cuts only at the root or throughout the tree, how many rounds of cuts to add before starting to branch, and why cuts seem to exhibit nonmonotonic effects on the solution process.

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References

  1. Achterberg, T., Wunderling, R.: Mixed integer programming: analyzing 12 years of progress. In: Jünger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38189-8_18

  2. Al-Khayyal, F.A.: An implicit enumeration procedure for the general linear complementarity problem. In: Hoffman, K.L., Jackson, R.H.F., Telgen, J. (eds.) Computation Mathematical Programming. Mathematical Programming Studies, vol. 31. Springer, Heidelberg (1987). https://doi.org/10.1007/BFb0121176

  3. Anderson, D., Le Bodic, P., Morgan, K.: Further results on an abstract model for branching and its application to mixed integer programming. Math. Program. 190(1), 811–841 (2020). https://doi.org/10.1007/s10107-020-01556-4

    Article  MathSciNet  MATH  Google Scholar 

  4. Basu, A., Conforti, M., Di Summa, M., Jiang, H.: Complexity of cutting planes and branch-and-bound in mixed-integer optimization (2020). https://arxiv.org/abs/2003.05023

  5. 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 

  6. Burer, S., Vandenbussche, D.: A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math. Program. 113(2), 259–282 (2008). https://doi.org/10.1007/s10107-006-0080-6

    Article  MathSciNet  MATH  Google Scholar 

  7. Cornuéjols, G., Liberti, L., Nannicini, G.: Improved strategies for branching on general disjunctions. Math. Program. 130(2, Ser. A), 225–247 (2011). https://doi.org/10.1007/s10107-009-0333-2

  8. Dey, S.S., Dubey, Y., Molinaro, M.: Branch-and-bound solves random binary packing IPs in polytime. In: Marx, D. (ed.) Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference January 10–13, 2021, pp. 579–591. SIAM (2021). https://doi.org/10.1137/1.9781611976465.35

  9. Dey, S.S., Dubey, Y., Molinaro, M.: Lower bounds on the size of general branch-and-bound trees. Math. Program. (2022). https://doi.org/10.1007/s10107-022-01781-z

    Article  Google Scholar 

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

    Google Scholar 

  11. Gasse, M., Chételat, D., Ferroni, N., Charlin, L., Lodi, A.: Exact combinatorial optimization with graph convolutional neural networks. In: Advances in Neural Information Processing Systems, pp. 15580–15592 (2019)

    Google Scholar 

  12. Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bull. Amer. Math. Soc. 64, 275–278 (1958)

    Article  MathSciNet  Google Scholar 

  13. Gomory, R.E.: Solving linear programming problems in integers. Comb. Anal. 10, 211–215 (1960)

    MathSciNet  MATH  Google Scholar 

  14. Gomory, R.E.: An algorithm for integer solutions to linear programs. Recent Adv. Math. Program. 64, 260–302 (1963)

    MATH  Google Scholar 

  15. Huang, Z., et al.: Learning to select cuts for efficient mixed-integer programming. Pattern Recogn. 123, 108353 (2022). https://doi.org/10.1016/j.patcog.2021.108353

    Article  Google Scholar 

  16. Karamanov, M., Cornuéjols, G.: Branching on general disjunctions. Math. Program. 128(1–2), 403–436 (2011). https://doi.org/10.1007/s10107-009-0332-3

    Article  MathSciNet  MATH  Google Scholar 

  17. Kazachkov, A.M., Le Bodic, P., Sankaranarayanan, S.: An abstract model of branch-and-cut (2021). https://arxiv.org/abs/2111.09907

  18. Khalil, E., Dai, H., Zhang, Y., Dilkina, B., Song, L.: Learning combinatorial optimization algorithms over graphs. In: Advances in Neural Information Processing Systems, pp. 6348–6358 (2017)

    Google Scholar 

  19. Khalil, E., Le Bodic, P., Song, L., Nemhauser, G., Dilkina, B.: Learning to branch in mixed integer programming. In: Thirtieth AAAI Conference on Artificial Intelligence (2016)

    Google Scholar 

  20. Land, A.H., Doig, A.G.: An automatic method of solving discrete programming problems. Econometrica 28, 497–520 (1960)

    Article  MathSciNet  Google Scholar 

  21. Le Bodic, P., Nemhauser, G.: An abstract model for branching and its application to mixed integer programming. Math. Prog. 166(1–2), 369–405 (2017). https://doi.org/10.1007/s10107-016-1101-8

    Article  MathSciNet  MATH  Google Scholar 

  22. Mahajan, A.: On selecting disjunctions in mixed integer linear programming. Ph.D. thesis, Lehigh University (2009)

    Google Scholar 

  23. Tang, Y., Agrawal, S., Faenza, Y.: Reinforcement learning for integer programming: learning to cut. In: Proceedings of the 37th International Conference on Machine Learning (ICML 2020), pp. 9367–9376 (2020)

    Google Scholar 

  24. Yang, Y., Boland, N., Savelsbergh, M.: Multivariable branching: a \(0\)-\(1\) knapsack problem case study. INFORMS J. Comput. 33(4), 1354–1367 (2021). https://doi.org/10.1287/ijoc.2020.1052

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank Andrea Lodi, Canada Excellence Research Chair in Data Science for Real-Time Decision Making, for financial support and creating a collaborative environment that facilitated the interactions that led to this paper, as well as Monash University for supporting Pierre’s trip to Montréal.

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Authors and Affiliations

  1. University of Florida, Gainesville, USA

    Aleksandr M. Kazachkov

  2. Monash University, Melbourne, Australia

    Pierre Le Bodic

  3. Indian Institute of Management, Ahmedabad, India

    Sriram Sankaranarayanan

Authors
  1. Aleksandr M. Kazachkov
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  2. Pierre Le Bodic
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  3. Sriram Sankaranarayanan
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Corresponding author

Correspondence to Aleksandr M. Kazachkov .

Editor information

Editors and Affiliations

  1. Delft University of Technology, Delft, The Netherlands

    Karen Aardal

  2. Eindhoven University of Technology, Eindhoven, The Netherlands

    Laura Sanità

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Kazachkov, A.M., Le Bodic, P., Sankaranarayanan, S. (2022). An Abstract Model for Branch-and-Cut. In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_25

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  • DOI: https://doi.org/10.1007/978-3-031-06901-7_25

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-06900-0

  • Online ISBN: 978-3-031-06901-7

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