Cuts in Graphs with Matroid Constraints
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arXiv
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| Natura: | Preprint |
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2024
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| author | Banik, Aritra Fomin, Fedor V. Golovach, Petr A. Inamdar, Tanmay Jana, Satyabrata Saurabh, Saket |
| author_facet | Banik, Aritra Fomin, Fedor V. Golovach, Petr A. Inamdar, Tanmay Jana, Satyabrata Saurabh, Saket |
| contents | {\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 á 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19134 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cuts in Graphs with Matroid Constraints Banik, Aritra Fomin, Fedor V. Golovach, Petr A. Inamdar, Tanmay Jana, Satyabrata Saurabh, Saket Discrete Mathematics Data Structures and Algorithms Combinatorics {\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 á 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. |
| title | Cuts in Graphs with Matroid Constraints |
| topic | Discrete Mathematics Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2406.19134 |