In this article, we introduce the parametric matroid $\ell$-interdiction problem, where $\ell\in\mathbb{N}_{>0}$ is a fixed number of elements allowed to be interdicted. Each element of the matroid's ground set is assigned a weight that depends linearly on a real parameter from a given interval. The goal is to compute, for each possible parameter value, a set of $\ell$-most vital elements with corresponding objective value the deletion of which causes a maximum increase of the weight of a minimal basis. We show that such a set, which of course depends on the parameter, can only change polynomially often if the parameter varies. We develop several exact algorithms to solve the problem that have polynomial running times if an independence test can be performed in polynomial time.
Ordinary lines, point degrees, and matroid density
Institute of Discrete Mathematics and Algebra, TU Freiberg（TU Freyberg离散数学与代数研究所）；Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）
Around the Merino--Welsh conjecture: improving Jackson's inequality
HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；ELTE: Eötvös Loránd University Mathematics Institute, Department of Computer Science（罗兰大学数学研究所计算机科学系）
Set of independencies and Tutte polynomial of matroids over a domain
本文研究整环上的拟阵及其相关的几个经典的组合不变量和代数不变式。我们定义了他们的Grothendieck-Tutte多项式$T_{\mathcal{M}}(x,y)$，扩展了Fink和Moci在2016年给出的定义，并且我们证明了这样的多项式具有经典的删除-压缩性质。此外，我们还研究了整环上可实现拟阵的独立集，推广了第二作者在2017年给出的\emph{poset of torsions} $Gr(\mathcal{M})$的定义。这是一个与（拟）算术拟阵相同的单纯偏序集的并集。新的概念通过域上拟阵的面模块$N_\mathcal{M}$自然地协调。当$Gr(\mathcal{M})$是有限偏序集时，它的面模的希尔伯特级数$N_\mathcal{M}(t)$是Tutte多项式$T_{\mathcal{M}}(x,y)$的特殊化。此外，余维1阿贝尔子变量的椭圆曲线承认复乘的安排，我们扩展了毕比的某些结果，我们提供了一个代数解释的椭圆Tutte多项式。
