On weighted graph separation problems and flow-augmentation

TL;DR

This paper explores the application of flow-augmentation techniques to various weighted graph separation problems, establishing fixed-parameter tractability for several problems and identifying open challenges in others.

Contribution

It demonstrates that flow-augmentation can be used to show FPT results for multiple weighted graph separation problems, expanding its applicability beyond the directed feedback vertex set.

Findings

Weighted undirected Multicut is FPT in both edge- and vertex-deletion variants.

Weighted Group Feedback Vertex Set is FPT with group oracle access.

Weighted Directed Subset Feedback Vertex Set is FPT.

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.