Tight Bounds for Feedback Vertex Set Parameterized by Clique-width

2025-12-01

AI Summary

This paper studies the counting and decision versions of the Feedback Vertex Set (FVS) problem parameterized by clique-width and tree-width. For a long-standing open problem, we introduce a novel acyclicity representation for labeled graphs, integrating the cut-and-count framework, modulo-2 counting, and one-sided-error Monte Carlo sampling. Our method yields efficient dynamic programming algorithms: (i) an $O(6^k n^c)$-time modulo-2 FVS counter parameterized by clique-width $k$, accompanied by a matching SETH-hardness lower bound; (ii) an optimal $O(3^k n^c)$-time algorithm for tree-width $k$; and (iii) an $O(18^k n^c)$-time algorithm for Connected FVS under clique-width $k$, with a tight lower bound. All results are currently best-possible, resolving several longstanding open questions from top-tier theoretical computer science conferences.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Applications of cryptography

Abstract

We introduce a new notion of acyclicity representation in labeled graphs, and present three applications thereof. Our main result is an algorithm that, given a graph $G$ and a $k$-clique expression of $G$, in time $O(6^kn^c)$ counts modulo $2$ the number of feedback vertex sets of $G$ of each size. We achieve this through an involved subroutine for merging partial solutions at union nodes in the expression. In the usual way this results in a one-sided error Monte-Carlo algorithm for solving the decision problem in the same time. We complement these by a matching lower bound under the Strong Exponential-Time Hypothesis (SETH). This closes an open question that appeared multiple times in the literature [ESA 23, ICALP 24, IPEC 25]. We also present an algorithm that, given a graph $G$ and a tree decomposition of width $k$ of $G$, in time $O(3^kn^c)$ counts modulo $2$ the number of feedback vertex sets of $G$ of each size. This matches the known SETH-tight bound for the decision version, which was obtained using the celebrated cut-and-count technique [FOCS 11, TALG 22]. Unlike other applications of cut-and-count, which use the isolation lemma to reduce a decision problem to counting solutions modulo $2$, this bound was obtained via counting other objects, leaving the complexity of counting solutions modulo $2$ open. Finally, we present a one-sided error Monte-Carlo algorithm that, given a graph $G$ and a $k$-clique expression of $G$, in time $O(18^kn^c)$ decides the existence of a connected feedback vertex set of size $b$ in $G$. We provide a matching lower bound under SETH.

Problem

• Counts feedback vertex sets modulo 2 using clique-width
• Determines complexity of counting feedback vertex sets modulo 2
• Decides existence of connected feedback vertex sets efficiently

Innovation

• Introduces acyclicity representation for labeled graphs
• Uses involved merging subroutine at union nodes
• Provides Monte-Carlo algorithms with matching SETH lower bounds