Split cuts are prominent general-purpose cutting planes in integer programming. The split closure of a rational polyhedron is what is obtained after intersecting the half-spaces defined by all the split cuts for the polyhedron. In this paper, we prove that deciding whether the split closure of a rational polytope is empty is NP-hard, even when the polytope is contained in the unit hypercube. As a direct corollary, we prove that optimization and separation over the split closure of a rational polytope in the unit hypercube are NP-hard, extending an earlier result of Caprara and Letchford.

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关于在0,1超立方体中确定有理多义位的拆分封闭是否为空的NP难度

分割切割是整数编程中突出的通用切割平面。有理多面体的分割闭合是在将由所有多面体的所有分割切口定义的半空间相交后获得的。在本文中，我们证明即使有单位超立方体中包含有多态性，判定有理多态性的拆分封闭是否为NP也是困难的。作为直接的推论，我们证明了在单元超立方体中有理多义位点的拆分闭合上的优化和分离是NP难的，扩展了Caprara和Letchford的早期结果。