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.

中文翻译：

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混合整数优化中分支定界和切割平面的复杂性--II

我们从理论的角度研究切割平面和分支方案的复杂性。我们为经验观察到的现象提供了一些严格的基础，即将切割平面和分支组合成一个分支和切割框架可以比单独使用这些工具效率高几个数量级。特别是，我们给出了一般条件，在这些条件下，切割平面策略和分支方案在组合成分支和切割时在效率上具有可证明的指数优势。这些算法的效率是使用两个具体措施来评估的：迭代次数和中间线性/凸程序中使用的约束的稀疏性。据我们所知，