Can You Link Up With Treewidth?
Radu Curticapean, Simon Döring, Daniel Neuen, Jiaheng Wang
TL;DR
The paper addresses the ETH-based hardness of detecting colorful fixed-pattern subgraphs, introducing the linkage capacity $oldsymbol{gamma}(H)$ to capture how well endpoints can be connected in blowups $Hoxtimes K_t$. By showing $oldsymbol{gamma}(H)$ controls the time complexity of ColSub$(H)$ and linking it to treewidth, the authors derive new self-contained proofs of Marx’s sparse-bound results, construct patterns with large $oldsymbol{gamma}(H)$ via Bene networks, and establish tight lower bounds for dense and average-case patterns. They also connect these lower bounds to counting problems for small induced subgraphs through the framework of $oldsymbol{gamma}$ and concurrent-flow arguments, with implications for uncolored subgraph counting and induced-subgraph invariants. The work unifies several strands—expander-free constructions, treewidth-based bounds, and random-graph density analyses—into a single toolkit for ETH-based lower bounds on subgraph detection and counting. The resulting results sharpen our understanding of when near-optimal conditional lower bounds are achievable and highlight the central role of treewidth and routing capacity in subgraph problems.
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 $γ(H)$, and show that detecting colorful $H$-subgraphs in time $n^{o(γ(H))}$ refutes ETH. Then, we use a simple construction of communication networks credited to Beneš to obtain $k$-vertex graphs of maximum degree $3$ and linkage capacity $Ω(k / \log k)$, avoiding arguments involving expander graphs, which were required in previous papers. We also show that every graph $H$ of treewidth $t$ has linkage capacity $Ω(t / \log t)$, 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 $Ω(k^β)$ for $β> 0$ have linkage capacity $Θ(k)$, 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 $Φ$, improving bounds from, e.g., [Roth et al., FOCS 2020].