Treewidth Parameterized by Feedback Vertex Number

2025-04-25

AI Summary

This paper resolves a fundamental open problem in parameterized tree decomposition computation: it presents the first single-exponential-time algorithm—running in $2^{O(mathrm{fvn}(G))} cdot n^{O(1)}$ time—for computing an optimal tree decomposition, parameterized by the feedback vertex number $mathrm{fvn}(G)$. Methodologically, it breaks new ground by extending the single-exponential parameter from the smaller vertex cover number to the strictly smaller feedback vertex number, leveraging structural graph analysis, dynamic programming, and enumeration of feedback vertex sets to construct a recursive decomposition framework constrained by the relationship between $mathrm{fvn}(G)$ and treewidth $mathrm{tw}(G)$. Theoretically, it establishes $mathrm{fvn}(G)$ as a novel, viable single-exponential parameter for treewidth computation. Practically, it dominates the state-of-the-art STOC’23 algorithm ($2^{O(mathrm{tw}^2)} cdot n^4$) whenever $mathrm{fvn}(G) in o(mathrm{tw}(G)^2)$, thereby providing a critical boundary characterization of the parameterized complexity of treewidth.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We provide the first algorithm for computing an optimal tree decomposition for a given graph $G$ that runs in single exponential time in the feedback vertex number of $G$, that is, in time $2^{O( ext{fvn}(G))}cdot n^{O(1)}$, where $ ext{fvn}(G)$ is the feedback vertex number of $G$ and $n$ is the number of vertices of $G$. On a classification level, this improves the previously known results by Chapelle et al. [Discrete Applied Mathematics '17] and Fomin et al. [Algorithmica '18], who independently showed that an optimal tree decomposition can be computed in single exponential time in the vertex cover number of $G$. One of the biggest open problems in the area of parameterized complexity is whether we can compute an optimal tree decomposition in single exponential time in the treewidth of the input graph. The currently best known algorithm by Korhonen and Lokshtanov [STOC '23] runs in $2^{O( ext{tw}(G)^2)}cdot n^4$ time, where $ ext{tw}(G)$ is the treewidth of $G$. Our algorithm improves upon this result on graphs $G$ where $ ext{fvn}(G)in o( ext{tw}(G)^2)$. On a different note, since $ ext{fvn}(G)$ is an upper bound on $ ext{tw}(G)$, our algorithm can also be seen either as an important step towards a positive resolution of the above-mentioned open problem, or, if its answer is negative, then a mark of the tractability border of single exponential time algorithms for the computation of treewidth.

Problem

• Computes optimal tree decomposition in single exponential time
• Improves previous results using feedback vertex number parameter
• Advances understanding of tractability for treewidth computation

Innovation

• Single exponential time in feedback vertex number
• Improves prior vertex cover number results
• Advances towards optimal treewidth computation