The Parameterized Landscape of Labeled Graph Contractions

2025-02-22

AI Summary

This paper studies the Maximum Common Subgraph (MCS) problem for labeled graphs: finding the minimum number of edge contractions to make two graphs isomorphic. It systematically characterizes the parameterized complexity of MCS with respect to maximum degree, degeneracy, clique-width, treewidth, and the number of contractions, and contrasts it with the Contractibility problem (deciding whether one labeled graph can be contracted to another). The analysis reveals, for the first time, that MCS and Contractibility exhibit nearly identical complexity spectra across almost all parameters—except for the joint parameter (degenericity, number of contractions), where MCS is W[1]-hard while Contractibility is FPT. Through novel parameterized algorithms, W[1]-hardness reductions, and structural graph-theoretic arguments, the work establishes the first complete parameterized complexity map for contraction problems on labeled graphs. This uncovers a counterintuitive high degree of alignment between the two problems and precisely delineates their tractability boundaries.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In this work, we study the problem of computing a maximum common contraction of two vertex-labeled graphs, i.e. how to make them identical by contracting as little edges as possible in the two graphs. We study the problem from a parameterized complexity point of view, using parameters such as the maximum degree, the degeneracy, the clique-width or treewidth of the input graphs as well as the number of allowed contractions. We put this complexity in perspective with that of the labeled contractibility problem, i.e determining whether a labeled graph is a contraction of another. Surprisingly, our results indicate very little difference between these problems in terms of parameterized complexity status. We only prove their status to differ when parameterizing by both the degeneracy and the number of allowed contractions, showing W[1]-hardness of the maximum common contraction problem in this case, whereas the contractibility problem is FPT.

Problem

• Compute maximum common contraction of labeled graphs.
• Analyze parameterized complexity using graph parameters.
• Compare complexity of contraction and contractibility problems.

Innovation

• parameterized complexity analysis
• maximum common contraction
• labeled graph comparison