FPT=PTIME for Homomorphism Problems on Sparse-Incidence and Bounded-Independence Patterns

Abstract: Assuming the Exponential Time Hypothesis (ETH), fixed-parameter tractability and polynomial-time solvability coincide for homomorphism problems specified by classes of pattern hypergraphs of bounded incidence degeneracy or bounded primal independence number. In both cases, tractability is characterised by bounded fractional hypertree width. Grohe (JACM 2007) established the corresponding FPT-PTIME equivalence under bounded arity. Our result allows unbounded arity and covers important cases such as bounded-degree patterns and patterns whose incidence graphs exclude a fixed minor. Building on the recent fractional balanced-separator framework and rounding theorem of Korchemna et al. (FOCS 2024), we prove a near-linear bound on fractional hypertree width ($\mathsf{fhw}$) in terms of adaptive width ($\mathsf{adw}$). For every hypergraph $H$ with $\mathsf{adw}(H)\geq 2$, [ \mathsf{fhw}(H)=O\bigl(λ(H)\mathsf{adw}(H)\log\mathsf{adw}(H)\bigr), ] where $λ(H)=\min{μ(H),\max{1,\logα(H)}}$, with $μ(H)$ denoting incidence degeneracy and $α(H)$ the independence number of the primal graph. As a further consequence, we obtain a corresponding FPT-PTIME collapse for exact homomorphism counting on every bounded-$λ$ class. More generally, for every recursively enumerable class of pattern hypergraphs, fixed-parameter tractability of the parameterised homomorphism problem implies quasipolynomial-time solvability of the corresponding unparameterised problem, assuming ETH.