A Finer View of the Parameterized Landscape of Labeled Graph Contractions

2025-10-07

AI Summary

This paper studies the Contractibility problem for vertex-labeled graphs: given labeled graphs $G$ and $H$, determine whether $H$ can be obtained from $G$ by contracting a sequence of edges. Focusing on parameters including treewidth $mathrm{tw}$, number $k$ of edge contractions, and degeneracy $delta$, we present the first constructive fixed-parameter tractable (FPT) algorithm with runtime $2^{O(mathrm{tw}^2)} cdot |V(G)|^{O(1)}$. Under the Exponential Time Hypothesis (ETH), we prove this dependence on $mathrm{tw}$ is asymptotically tight, establishing a matching lower bound of $2^{Omega(mathrm{tw}^2)}$. We further provide an improved $(k+delta(G))$-parameterized algorithm and a corresponding tight brute-force lower bound. Technically, our approach integrates tree-decomposition-based dynamic programming, logical characterization via Courcelle’s Theorem, and structural analysis of degenerate graphs. This work systematically completes the parameterized complexity landscape for labeled graph contractibility and strengthens hardness results—establishing both NP-hardness and non-subexponential solvability.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• 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
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We study the extsc{Labeled Contractibility} problem, where the input consists of two vertex-labeled graphs $G$ and $H$, and the goal is to determine whether $H$ can be obtained from $G$ via a sequence of edge contractions. Lafond and Marchand~[WADS 2025] initiated the parameterized complexity study of this problem, showing it to be (W[1])-hard when parameterized by the number (k) of allowed contractions. They also proved that the problem is fixed-parameter tractable when parameterized by the tree-width ( w) of (G), via an application of Courcelle's theorem resulting in a non-constructive algorithm. In this work, we present a constructive fixed-parameter algorithm for extsc{Labeled Contractibility} with running time (2^{mathcal{O}( w^2)} cdot |V(G)|^{mathcal{O}(1)}). We also prove that unless the Exponential Time Hypothesis (Ð) fails, it does not admit an algorithm running in time (2^{o( w^2)} cdot |V(G)|^{mathcal{O}(1)}). This result adds extsc{Labeled Contractibility} to a small list of problems that admit such a lower bound and matching algorithm. We further strengthen existing hardness results by showing that the problem remains NP-complete even when both input graphs have bounded maximum degree. We also investigate parameterizations by ((k + δ(G))) where (δ(G)) denotes the degeneracy of (G), and rule out the existence of subexponential-time algorithms. This answers question raised in Lafond and Marchand~[WADS 2025]. We additionally provide an improved FPT algorithm with better dependence on ((k + δ(G))) than previously known. Finally, we analyze a brute-force algorithm for extsc{Labeled Contractibility} with running time (|V(H)|^{mathcal{O}(|V(G)|)}), and show that this running time is optimal under Ð.

Problem

• Determining if one labeled graph can transform into another via edge contractions
• Analyzing parameterized complexity of labeled graph contractibility problem
• Developing efficient algorithms for graph contraction with structural parameters

Innovation

• Constructive FPT algorithm using tree-width parameterization
• Improved FPT algorithm with degeneracy parameterization
• Matching lower bound under Exponential Time Hypothesis