{\sc Vertex $(s, t)$-Cut} and {\sc Vertex Multiway Cut} are two fundamental graph separation problems in algorithmic graph theory. We study matroidal generalizations of these problems, where in addition to the usual input, we are given a representation $R \in \mathbb{F}^{r \times n}$ of a linear matroid $\mathcal{M} = (V(G), \mathcal{I})$ of rank $r$ in the input, and the goal is to determine whether there exists a vertex subset $S \subseteq V(G)$ that has the required cut properties, as well as is independent in the matroid $\mathcal{M}$. We refer to these problems as {\sc Independent Vertex $(s, t)$-cut}, and {\sc Independent Multiway Cut}, respectively. We show that these problems are fixed-parameter tractable ({\sf FPT}) when parameterized by the solution size (which can be assumed to be equal to the rank of the matroid $\mathcal{M}$). These results are obtained by exploiting the recent technique of flow augmentation [Kim et al.~STOC '22], combined with a dynamic programming algorithm on flow-paths \'a la [Feige and Mahdian,~STOC '06] that maintains a representative family of solutions w.r.t.~the given matroid [Marx, TCS '06; Fomin et al., JACM]. As a corollary, we also obtain {\sf FPT} algorithms for the independent version of {\sc Odd Cycle Transversal}. Further, our results can be generalized to other variants of the problems, e.g., weighted versions, or edge-deletion versions.

翻译：顶点$(s, t)$-割与顶点多路割是算法图论中两个基本的图分离问题。本文研究这些问题的拟阵泛化形式：在常规输入之外，我们还给定一个秩为$r$的线性拟阵$\mathcal{M} = (V(G), \mathcal{I})$在域$\mathbb{F}$上的表示矩阵$R \in \mathbb{F}^{r \times n}$，目标是判定是否存在顶点子集$S \subseteq V(G)$，使其既满足所需的割性质，又在拟阵$\mathcal{M}$中独立。我们分别称这些问题为独立顶点$(s, t)$-割与独立多路割。我们证明当以解规模（可假设等于拟阵$\mathcal{M}$的秩）为参数时，这些问题属于固定参数可处理（FPT）类。这些结果通过结合流增强技术[Kim等人，STOC '22]与基于流路径的动态规划算法[Feige和Mahdian，STOC '06]获得，该算法能维持关于给定拟阵的解的代表族[Marx，TCS '06；Fomin等人，JACM]。作为推论，我们还得到了独立版本奇环横截集的FPT算法。此外，我们的结果可推广到其他问题变体，例如加权版本或边删除版本。