Can You Link Up With Treewidth?
<|reference_start|>Can You Link Up With Treewidth?: A central result of Marx [ToC '10] proves that there are $k$-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.   Our first contribution is a new and fully self-contained proof of this result that further simplifies a recent work by Karthik et al. [SOSA 2024]. Towards this end, we introduce a novel graph parameter, the linkage capacity $\gamma(H)$, and show with an elementary proof 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 the use of expander graphs. We also show that every graph $H$ of treewidth $t$ has linkage capacity $\Omega(t / \log t)$, thus recovering the stronger result of Marx [ToC '10] with a simplified proof.   Additionally, we obtain new tight lower bounds for certain types of patterns by analyzing their linkage capacity. For example, we prove that almost all $k$-vertex graphs of polynomial average degree $\Omega(k^{\beta})$ for some $\beta > 0$ have linkage capacity $\Theta(k)$, which implies tight lower bounds for such patterns $H$. As an application of these results, we also obtain tight lower bounds for counting small induced subgraphs having a certain property $\Phi$, improving bounds from [Roth et al., FOCS 2020].<|reference_end|>
@article{curticapean2024can,
    title={Can You Link Up With Treewidth?},
    author={Radu Curticapean, Simon D"oring, Daniel Neuen, Jiaheng Wang},
    journal={arXiv preprint arXiv:2410.02606},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.02606},
    primaryClass={cs.DS cs.CC}
}