Can You Link Up With Treewidth?

TL;DR

This paper introduces a new graph parameter called linkage capacity to simplify and strengthen lower bounds in parameterized complexity, particularly for detecting colorful subgraphs and related problems, by providing more accessible proofs and broader applicability.

Contribution

The paper presents a simplified, self-contained proof of a key lower bound result using the linkage capacity parameter, and extends the analysis to graphs of bounded treewidth and dense graphs, improving existing bounds.

Findings

Linkage capacity $oldsymbol{ ext{γ}(H)}$ refutes ETH when detecting colorful $H$-subgraphs in $n^{o( ext{γ}(H))}$ time.

Graphs of treewidth $t$ have linkage capacity $oldsymbol{ ext{Ω}(t / ext{log} t)}$, recovering and strengthening previous results.

Almost all $k$-vertex graphs with polynomial average degree have linkage capacity $oldsymbol{ ext{Θ}(k)}$, leading to tight lower bounds for pattern detection.

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 Bene\v{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 $\Omega(k^{\beta})$ for $\beta > 0$ 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].