We study the complexity of cutting planes and branching schemes from a theoretical point of view. We give some rigorous underpinnings to the empirically observed phenomenon that combining cutting planes and branching into a branch-and-cut framework can be orders of magnitude more efficient than employing these tools on their own. In particular, we give general conditions under which a cutting plane strategy and a branching scheme give a provably exponential advantage in efficiency when combined into branch-and-cut. The efficiency of these algorithms is evaluated using two concrete measures: number of iterations and sparsity of constraints used in the intermediate linear/convex programs. To the best of our knowledge, our results are the first mathematically rigorous demonstration of the superiority of branch-and-cut over pure cutting planes and pure branch-and-bound.
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We are very grateful to an anonymous referee who raised some subtle, but very important, points that were unclear in a previous version of the paper. Addressing those concerns has helped to clarify these delicate points. Amitabh Basu and Hongyi Jiang gratefully acknowledge support from ONR Grant N000141812096, NSF Grant CCF2006587, and AFOSR Grant FA95502010341. Michele Conforti and Marco Di Summa were supported by a SID grant of the University of Padova.
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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 — II. Combinatorica 42 (Suppl 1), 971–996 (2022). https://doi.org/10.1007/s00493-022-4884-7
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