包装和分区轨道
Packing and Partitioning Orbitopes
摘要：
我们引入轨道位作为0/1-矩阵的凸包，这些矩阵是字典极大的，服从于作用在列上的群。特殊情况是包装和分区轨道位，这是由矩阵的限制引起的，每行最多或正好有一个1-条目。研究这些多面体的目的是通过增加约束来深入了解打破整数规划中某些对称性的方法，例如，图着色问题的一个著名公式。我们提供了一个彻底的多面体调查的情况下，在该集团的作用柱是循环群或对称群的包装和分区轨道位。我们的主要结果是这些多面体的完整的线性不等式描述的刻面定义的不等式。对于循环群的情形，证明是完全幺模的，而对于对称群的情形，证明和描述都是复杂的。然而，在这种情况下，相关的分离问题也可以在线性时间内解决。
英文摘要：
We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal subject to a group acting on the columns. Special cases are packing and partitioning orbitopes, which arise from restrictions to matrices with at most or exactly one 1-entry in each row, respectively. The goal of investigating these polytopes is to gain insight into ways of breaking certain symmetries in integer programs by adding constraints, e.g., for a well-known formulation of the graph coloring problem. We provide a thorough polyhedral investigation of packing and partitioning orbitopes for the cases in which the group acting on the columns is the cyclic group or the symmetric group. Our main results are complete linear inequality descriptions of these polytopes by facet-defining inequalities. For the cyclic group case, the descriptions turn out to be totally unimodular, while for the symmetric group case both the description and the proof are more involved. Nevertheless, the associated separation problem can be solved in linear time also in this case.
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