Parameterized algorithms for $k$-Inversion

2026-04-07

AI Summary

This study addresses the $k$-Inversion problem, which asks whether a directed graph admits a feedback arc set composed of at most $k$ vertex subsets such that reversing the outgoing edges of each subset in sequence renders the graph acyclic. The authors present the first fixed-parameter tractable (FPT) algorithm for this problem beyond tournaments, extending it to digraphs whose underlying undirected graphs are block graphs, and further generalize the approach to arbitrary directed graphs. Leveraging tree decomposition and structural analysis, their algorithm achieves a runtime of $2^{O(\mathrm{tw}(k + \mathrm{tw}))} \cdot n^{O(1)}$, where $\mathrm{tw}$ denotes the treewidth of the underlying graph, thereby overcoming prior limitations to highly restricted graph classes.

Abstract

Inversion of a directed graph $D$ with respect to a vertex subset $Y$ is the directed graph obtained from $D$ by reversing the direction of every arc whose endpoints both lie in $Y$. More generally, the inversion of $D$ with respect to a tuple $(Y_1, Y_2, \ldots, Y_\ell)$ of vertex subsets is defined as the directed graph obtained by successively applying inversions with respect to $Y_1, Y_2, \ldots, Y_\ell$. Such a tuple is called a \emph{decycling family} of $D$ if the resulting graph is acyclic. In the \textsc{$k$-Inversion} problem, the input consists of a directed graph $D$ and an integer $k$, and the task is to decide whether $D$ admits a decycling family of size at most $k$. Alon et al.\ (SIAM J.\ Discrete Math., 2024) proved that the problem is NP-complete for every fixed value of $k$, thereby ruling out XP algorithms, and presented a fixed-parameter tractable (FPT) algorithm parameterized by $k$ for tournament inputs. In this paper, we generalize their algorithm to a broader variant of the problem on tournaments and subsequently use this result to obtain an FPT algorithm for \textsc{$k$-Inversion} when the underlying undirected graph of the input is a block graph. Furthermore, we obtain an algorithm for \textsc{$k$-Inversion} on general directed graphs with running time $2^{O(\mathrm{tw}(k + \mathrm{tw}))} \cdot n^{O(1)}$, where $\mathrm{tw}$ denotes the treewidth of the underlying graph.

Problem

• k-Inversion
• decycling family
• directed graph
• acyclic
• parameterized algorithm

Innovation

• parameterized algorithm
• k-Inversion
• tournament
• block graph
• treewidth