On weighted graph separation problems and flow-augmentation

TL;DR

The paper investigates the parameterized complexity of weighted graph separation problems under the flow-augmentation paradigm. It extends the approach to obtain randomized FPT algorithms for weighted undirected Multicut, weighted Group Feedback Vertex/Edge Set, and weighted Directed Subset Feedback Edge/Vertex Set, parameterized by the cutset size $k$, while identifying Directed Symmetric Multicut as a major open question. Central to the results is reducing target problems to Generalized Digraph Pair Cut (GDPC) and its tractable variants (e.g., 2K2-free, $b$-bounded, bundled cut with order) via iterative compression, untangling, and labeling-extension gadgets, ultimately leveraging weighted Multiway Cut subroutines. The work broadens flow-augmentation’s applicability from trees and the Directed Feedback Vertex Set problem to a broader class of weighted graph separations, providing a toolkit for future problems and clarifying the boundaries of current techniques. Overall, the paper demonstrates that several weighted separation problems admit randomized FPT algorithms, while highlighting the persisting challenge of Directed Symmetric Multicut and offering a roadmap for tackling it with flow-augmentation and GDPC-based reductions.

Abstract

One of the first application of the recently introduced technique of \emph{flow-augmentation} [Kim et al., STOC 2022] is a fixed-parameter algorithm for the weighted version of \textsc{Directed Feedback Vertex Set}, a landmark problem in parameterized complexity. In this note we explore applicability of flow-augmentation to other weighted graph separation problems parameterized by the size of the cutset. We show the following. -- In weighted undirected graphs \textsc{Multicut} is FPT, both in the edge- and vertex-deletion version. -- The weighted version of \textsc{Group Feedback Vertex Set} is FPT, even with an oracle access to group operations. -- The weighted version of \textsc{Directed Subset Feedback Vertex Set} is FPT. Our study reveals \textsc{Directed Symmetric Multicut} as the next important graph separation problem whose parameterized complexity remains unknown, even in the unweighted setting.

Figures (1)

• Figure 1: Illustration of the reduction of Section \ref{['sec:dsfvs']}. All copies of an edge $e$ of $G$ form a bundle $B_e$: here blue edges form one bundle for an edge $(u,v)$ and magenta edges form another bundle for an edge $(v,w)$. Furthermore, if $(v,w)$ is red, then there is an extra arc $(v^{2a},w^{2a+1})$ for every $1 \leq a \leq |X|$, depicted in red. Intutively, this arc, together with arcs $(s,x_a^{2a})$ and $(x_a^{2a+1},t)$ asks to destroy all closed walks that pass both through $(v,w)$ and $x_a$.

Theorems & Definitions (22)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Corollary 1.6
• proof
• Theorem 2.1: dfl-csp, Theorem 3.3
• Theorem 2.2: dfl-csp, Theorem 3.21
• Lemma 3.1