Can You Link Up With Treewidth?

2024-10-03

Symposium on Theoretical Aspects of Computer Science

AI Summary

This paper addresses the colorful subgraph detection problem within the framework of parameterized complexity, introducing a novel graph parameter—*linkage capacity*—to uniformly characterize conditional time lower bounds. Methodologically, it pioneers the use of Beneš networks—replacing traditional expander-based arguments—combined with ETH-based reductions, random graph analysis, and treewidth correlation analysis. Key contributions are: (1) proving that graphs of treewidth $t$ have linkage capacity $Omega(t/log t)$, and establishing a tight $Theta(k)$ bound for most dense graphs; (2) deriving a conditional lower bound of $n^{o(gamma(H))}$ for detecting a colorful $H$-subgraph, where $gamma(H)$ is precisely characterized by the linkage capacity of $H$; and (3) obtaining new tight lower bounds for induced subgraph counting, thereby simplifying and generalizing Marx’s seminal result to a broader class of graphs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

In a fundamental paper in parameterized complexity theory, Marx [ToC '10] constructed $k$-vertex graphs $H$ of maximum degree $3$ such that $n^{o(k /log k)}$ time algorithms for detecting colorful $H$-subgraphs would refute the Exponential-Time Hypothesis (ETH). This result is widely used to obtain almost-tight conditional lower bounds for parameterized problems under ETH. We give a new and fully self-contained proof of this result that further simplifies a recent work by Karthik et al. [SOSA 2024]. In our proof, we introduce a novel graph parameter of independent interest, the linkage capacity $gamma(H)$, and show that detecting colorful $H$-subgraphs in time $n^{o(gamma(H))}$ refutes ETH. Then, we use a simple construction of communication networks credited to Benev{s} to obtain $k$-vertex graphs of maximum degree $3$ and linkage capacity $Omega(k / log k)$, avoiding arguments involving expander graphs, which were required in previous papers. We also show that every graph $H$ of treewidth $t$ has linkage capacity $Omega(t / log t)$, thus recovering a stronger result shown by Marx [ToC '10] with a simplified proof. Additionally, we obtain new tight lower bounds on the complexity of colorful subgraph detection for certain types of patterns by analyzing their linkage capacity: We prove that almost all $k$-vertex graphs of polynomial average degree $Omega(k^{eta})$ for $eta>0$ have linkage capacity $Theta(k)$, which implies tight lower bounds for finding such patterns $H$. As an application of these results, we also obtain tight lower bounds for counting small induced subgraphs having a fixed property $Phi$, improving bounds from, e.g., [Roth et al., FOCS 2020].

Problem

• Detecting colorful H-subgraphs refutes ETH
• Linkage capacity determines subgraph detection complexity
• Treewidth bounds linkage capacity in graphs

Innovation

• Introduces linkage capacity γ(H) for subgraph detection
• Uses Beneš networks to simplify expander graph arguments
• Links treewidth to linkage capacity Ω(t/log t)