Can You Link Up With Treewidth?
Radu Curticapean, Simon D\"oring, Daniel Neuen, Jiaheng Wang

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 -vertex graphs of maximum degree such that time algorithms for detecting colorful -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 , and show that detecting colorful -subgraphs in time refutes ETH. Then, we use a simple construction of communication networks credited to Bene\v{s} to obtain -vertex graphs of maximum degree and linkage capacity , avoiding arguments…
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