Cutting planes and branching are two of the most important algorithms for solving mixed-integer linear programs. For both algorithms, disjunctions play an important role, being used both as branching candidates and as the foundation for some cutting planes. We relate branching decisions and cutting planes to each other through the underlying disjunctions that they are based on, with a focus on Gomory mixed-integer cuts and their corresponding split disjunctions. We show that selecting branching decisions based on quality measures of Gomory mixed-integer cuts leads to relatively small branch-and-bound trees, and that the result improves when using cuts that more accurately represent the branching decisions. Finally, we show how the history of previously computed Gomory mixed-integer cuts can be used to improve the performance of the state-of-the-art hybrid branching rule of SCIP. Our results show a 4% decrease in solve time, and an 8% decrease in number of nodes over affected instances of MIPLIB 2017.

翻译：割平面与分支是求解混合整数线性规划的两类核心算法。在这两种算法中，析取结构均发挥着重要作用，既作为分支候选，又为部分割平面提供理论基础。本文通过底层析取结构关联分支决策与割平面，重点研究Gomory混合整数割及其对应的分割析取。研究表明，基于Gomory混合整数割质量指标选择分支决策，可生成规模较小的分支定界树；当采用更准确反映分支决策的割平面时，该效果进一步优化。最后，我们展示了如何利用历史Gomory混合整数割计算结果提升SCIP最新混合分支规则的性能。实验结果表明，在MIPLIB 2017受影响实例上，求解时间降低4%，节点数量减少8%。