离散递减最小化，第II部分：离散凸分析的观点
Discrete Decreasing Minimization, Part II: Views from Discrete Convex Analysis
摘要：
我们继续考虑在第一部分中讨论的整基多面体上的离散递减极小化问题。问题是在整基多面体中寻找一个字典序最小整向量，其中向量的分量按降序排列。本文的研究可以看作是Fujishige（1980）关于连续变量的基多面体的字典序最优基和主剖分的工作的离散对应部分。第二部分有两个目标。第一个是提供结构性意见，从离散凸分析（DCA）的结果，第一部分得到的建设性和算法的方法。第二个目标是为DCA方法在其他离散结构如M-凸集的交集、流和子模流上的离散减极小化铺平道路。我们从DCA中M-凸集和M-凸函数的基本事实推导出第一部分的结构结果。本文给出了在第一部分中用迭代方法构造的典型划分的一个直接特征。这揭示了离散情况下的典型划分和连续情况下的主划分之间的精确关系。此外，这个结果需要一个邻近定理，说明每个递减极小元包含在包含（唯一）分数递减极小元（最小范数点）的小盒子中，进一步导致一个连续松弛算法，用于寻找M-凸集的递减极小元。这样就完全阐明了连续情况和离散情况之间的关系。此外，我们提出了网络流，两个M-凸集的交集，和子模流的DCA最小-最大公式。
英文摘要：
We continue to consider the discrete decreasing minimization problem on an integral base-polyhedron treated in Part I. The problem is to find a lexicographically minimal integral vector in an integral base-polyhedron, where the components of a vector are arranged in a decreasing order. This study can be regarded as a discrete counter-part of the work by Fujishige (1980) on the lexicographically optimal base and the principal partition of a base-polyhedron in continuous variables. The objective of Part II is two-fold. The first is to offer structural views from discrete convex analysis (DCA) on the results of Part I obtained by the constructive and algorithmic approach. The second objective is to pave the way of DCA approach to discrete decreasing minimization on other discrete structures such as the intersection of M-convex sets, flows, and submodular flows. We derive the structural results in Part I from fundamental facts on M-convex sets and M-convex functions in DCA. A direct characterization is given to the canonical partition, which was constructed by an iterative procedure in Part I. This reveals the precise relationship between the canonical partition for the discrete case and the principal partition for the continuous case. Moreover, this result entails a proximity theorem, stating that every decreasingly minimal element is contained in the small box containing the (unique) fractional decreasingly minimal element (the minimum-norm point), leading further to a continuous relaxation algorithm for finding a decreasingly minimal element of an M-convex set. Thus the relationship between the continuous and discrete cases is completely clarified. Furthermore, we present DCA min-max formulas for network flows, the intersection of two M-convex sets, and submodular flows.
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