Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cuts in Graphs with Matroid Constraints

Aritra Banik, Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Satyabrata Jana, Saket Saurabh

Abstract

{\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.

Cuts in Graphs with Matroid Constraints

Abstract

{\sc Vertex -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 of a linear matroid of rank in the input, and the goal is to determine whether there exists a vertex subset that has the required cut properties, as well as is independent in the matroid . We refer to these problems as {\sc Independent Vertex -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 ). 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.
Paper Structure (19 sections, 12 theorems, 2 equations, 5 figures, 1 table)

This paper contains 19 sections, 12 theorems, 2 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Results and Methods.
  3. Independent Multiway Cut.
  4. Motivation for independence constraints.
  5. Preliminaries
  6. FPT Algorithm for Independent Vertex $(s, t)$-Cut
  7. Transforming the solution into a (directed) minimum cut via Flow Augmentation
  8. Phase I. Undirected Vertex $(s, t)$-cut $\Rightarrow$ Directed Vertex $(s, t)$-cut.
  9. Matroid Transformation.
  10. Phase II. Directed Vertex $(s, t)$-cut $\Rightarrow$ Directed Edge $(s, t)$-cut.
  11. Matroid Transformation.
  12. Algorithm.
  13. Phase III. Flow augmentation
  14. Matroid Transformation.
  15. Algorithm.
  16. ...and 4 more sections

Key Result

Proposition 2

There is an algorithm that, given a matrix $R$ over a field $\mathbb{F}$, representing a matroid $\mathcal{M} = (U, \mathcal{I})$ of rank $k$, a $p$-family ${\cal A}$ of independent sets in $\mathcal{M}$, and an integer $q$ such that $p + q = k$, computes a $q$-representative family ${\cal A}'\subse

Figures (5)

  • Figure 1: Schematic depiction of sequence of reductions from undirected vertex $(s, t)$-cut to directed vertex $(s, t)$-cut via flow augmentation. These details are omitted from the short version.
  • Figure 2: Illustration of Phase I.
  • Figure 3: Illustration of Phase II.
  • Figure 4: Illustration of Phase IV.
  • Figure 5: An illustration of a valid coloring on critical vertices.

Theorems & Definitions (24)

  • Definition 1: fomin2014efficientDBLP:books/sp/CyganFKLMPPS15
  • Proposition 2: fomin2014efficientDBLP:books/sp/CyganFKLMPPS15
  • Proposition 3: CyganFKLMPPS15
  • Theorem 4
  • Proposition 5: Flow-augmentation (randomized) for Directed Graphs, kim2021flowDBLP:conf/stoc/0002KPW22
  • Proposition 6: Flow-augmentation (deterministic) for Directed Graphs kim2021flowDBLP:conf/stoc/0002KPW22
  • Claim 7
  • Claim 8
  • Lemma 9
  • Definition 10: Compatible augomenting set
  • ...and 14 more