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Complexity of branch-and-bound and cutting planes in mixed-integer optimization

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Abstract

We investigate the theoretical complexity of branch-and-bound (BB) and cutting plane (CP) algorithms for mixed-integer optimization. In particular, we study the relative efficiency of BB and CP, when both are based on the same family of disjunctions. We extend a result of Dash (International Conference on Integer Programming and Combinatorial Optimization (IPCO), pp. 145–160, 2002) to the nonlinear setting which shows that for convex 0/1 problems, CP does at least as well as BB, with variable disjunctions. We sharpen this by giving instances of the stable set problem where we can provably establish that CP does exponentially better than BB. We further show that if one moves away from 0/1 sets, this advantage of CP over BB disappears; there are examples where BB finishes in O(1) time, but CP takes infinitely long to prove optimality, and exponentially long to get to arbitrarily close to the optimal value (for variable disjunctions). We next show that if the dimension is considered a fixed constant, then the situation reverses and BB does at least as well as CP (up to a polynomial blow up factor), for quite general families of disjunctions. This is also complemented by examples where this gap is exponential (in the size of the input data).

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Notes

  1. We make the standard notational convention that the trivial intersection \(\bigcap _{i=1}^0 X_i = \mathbb {R}^n\).

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Acknowledgements

Daniel Dadush pointed out to us the example by Dash et al. from [35] that establishes an exponential lower bound on any BB proof based on general split disjunctions. The manuscript also benefited greatly from discussions with Aleksandr Kazachkov, Andrea Lodi and Sriram Sankaranarayanan. Part of this work was done when the first author was visiting the Centre de Recherches Mathématiques (CRM) at Université de Montréal, Canada as part of the Simons-CRM scholar-in-residence program. The visit and the research done was supported in part by funding from the Simons Foundation and the Centre de Recherches Mathématiques.

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  1. Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, USA

    Amitabh Basu & Hongyi Jiang

  2. Dipartimento di Matematica “Tullio Levi-Civita”, Università degli Studi Padova, Padova, Italy

    Michele Conforti & Marco Di Summa

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  1. Amitabh Basu
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  2. Michele Conforti
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Correspondence to Amitabh Basu.

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Michele Conforti and Marco Di Summa were supported by the Italian Ministry of Education, University and Research (Ministero dell’Istruzione, dell’Università e della Ricerca MIUR) grant 2015B5F27W. Amitabh Basu and Hongyi Jiang gratefully acknowledge support from the National Science Foundation (NSF) grants CMMI1452820, CCF2006587, the Office of Naval Research (ONR) grant N000141812096 and the Air Force Office of Scientific Research (AFOSR) grant FA95502010341.

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Basu, A., Conforti, M., Di Summa, M. et al. Complexity of branch-and-bound and cutting planes in mixed-integer optimization. Math. Program. 198, 787–810 (2023). https://doi.org/10.1007/s10107-022-01789-5

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