Cuts in Graphs with Matroid Constraints
<|reference_start|>Cuts in Graphs with Matroid Constraints: {\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.<|reference_end|>
@article{banik2024cuts,
    title={Cuts in Graphs with Matroid Constraints},
    author={Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar,
  Satyabrata Jana, Saket Saurabh},
    journal={arXiv preprint arXiv:2406.19134},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.19134},
    primaryClass={cs.DM cs.DS math.CO}
}