Rank axiom of modular supermatroids: A connection with directional DR submodular functions
Rank axiom of modular supermatroids: A connection with directional DR submodular functions
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模超拟阵的秩公理：与定向 DR 子模函数的联系
DOI：
10.1016/j.aam.2021.102304
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发表时间：
2022
影响因子：
1.1
通讯作者：
So Nakashima
中科院分区：
文献类型：
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作者：
Takanori Maehara;So Nakashima
自从惠特尼将拟阵作为线性独立性的抽象引入以来，拟阵一直是最重要的组合结构之一。作为拟阵的一个重要性质，它可以用几个不同（但等价）的公理来表征，例如增广、基交换或秩公理。超拟阵是在格上定义的拟阵的推广。这里，核心问题是超拟阵是否可以用几个类似于拟阵的等价公理来表征。 Barnabei、Nicoletti 和 Pezzoli 描述了分布格子上的超拟阵，Fujishige、Koshevoy 和 Sano 推广了 cg-拟阵（较低局部分布格子上的超拟阵）的结果。在这项研究中，我们重点关注模格子，它是分配格子的重要超类，并提供了模格子上超拟阵的等效表征。我们使用秩公理来描述模格上的超拟阵，其中秩函数是方向 DR 子模函数，这是作者引入的子模函数的推广。使用基于秩函数的表征，我们进一步证明了超拟阵的强交换性质，这在优化中具有应用。我们还揭示了下半模格上的超拟阵公理之间的关系，下半模格是下局部分布格和模格的共同超类。
A matroid has been one of the most important combinatorial structures since it was introduced by Whitney as an abstraction of linear independence. As an important property of a matroid, it can be characterized by several different (but equivalent) axioms, such as the augmentation, the base exchange, or the rank axiom.A supermatroid is a generalization of a matroid defined on lattices. Here, the central question is whether a supermatroid can be characterized by several equivalent axioms similar to a matroid. Barnabei, Nicoletti, and Pezzoli characterized supermatroids on distributive lattices, and Fujishige, Koshevoy, and Sano generalized the results for cg-matroids (supermatroids on lower locally distributive lattices).In this study, we focus on modular lattices, which are an important superclass of distributive lattices, and provide equivalent characterizations of supermatroids on modular lattices. We characterize supermatroids on modular lattices using the rank axiom in which the rank function is a directional DR-submodular function, which is a generalization of a submodular function introduced by the authors. Using a characterization based on rank functions, we further prove the strong exchange property of a supermatroid, which has application in optimization.We also reveal the relation between the axioms of a supermatroid on lower semimodular lattices, which is a common superclass of a lower locally distributive lattice and a modular lattice.