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：切割平面与分支是求解混合整数线性规划的两类核心算法。对于这两种算法，析取结构均发挥重要作用：既可作为分支候选，也可作为某些切割平面的基础。我们通过二者所依赖的底层析取结构，将分支决策与切割平面联系起来，重点聚焦于Gomory混合整数割及其对应的分裂析取。研究表明，基于Gomory混合整数割的质量度量选择分支决策，能够生成规模相对较小的分支定界树；当采用更精确表征分支决策的割时，该效果进一步提升。最后，我们展示了如何利用先前计算的Gomory混合整数割的历史信息，改进SCIP中先进混合分支规则的性能。实验结果表明，在MIPLIB 2017受影响的实例上，求解时间减少4%，节点数减少8%。