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

2026-09-18

AI Summary

该研究解决了在稀疏事件和有界独立性模式下的同态问题，通过使用分数超树宽度的近线性边界方法，证明了固定参数可解性和多项式时间可解性的等价性。

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.

Problem

• homomorphism problems
• bounded incidence degeneracy
• bounded primal independence number
• Exponential Time Hypothesis (ETH)
• fixed-parameter tractability

Innovation

• Exponential Time Hypothesis (ETH)
• fractional hypertree width (fhw)
• adaptive width (adw)
• bounded incidence degeneracy
• bounded primal independence number