稀疏关联和有界独立模式上的同态问题的FPT=PTIME
FPT=PTIME for Homomorphism Problems on Sparse-Incidence and Bounded-Independence Patterns
- Institute of Logic and Computation, TU Wien（逻辑与计算研究所，维也纳科技大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究证明在ETH下，对有界关联退化或有界原始独立数的模式超图，同态问题的固定参数可处理性与多项式时间可解性等价，由分数超树宽刻画，并给出近线性上界及计数坍缩结果。
AI中文摘要：
假设指数时间假说（ETH）成立，对于由有界关联退化度或有界原始独立数的模式超图类所指定的同态问题，固定参数可处理性与多项式时间可解性是一致的。在这两种情况下，可处理性由有界分数超树宽刻画。Grohe（JACM 2007）在有界元数下建立了相应的FPT-PTIME等价性。我们的结果允许无界元数，并覆盖了重要情形，如有界度模式及其关联图排除固定子式的模式。基于Korchemna等人（FOCS 2024）最近的分数平衡分离器框架和舍入定理，我们证明了分数超树宽（$\mathsf{fhw}$）关于自适应宽度（$\mathsf{adw}$）的近线性上界。对于每个满足$\mathsf{adw}(H)\geq 2$的超图$H$，有$\mathsf{fhw}(H)=O\bigl(\lambda(H)\mathsf{adw}(H)\log\mathsf{adw}(H)\bigr)$，其中$\lambda(H)=\min\{\mu(H),\max\{1,\log\alpha(H)\}\}$，$\mu(H)$表示关联退化度，$\alpha(H)$是原始图的独立数。作为进一步推论，我们获得了在每个有界$\lambda$类上精确同态计数的相应FPT-PTIME坍缩。更一般地，对于每个递归可枚举的模式超图类，假设ETH，参数化同态问题的固定参数可处理性意味着相应非参数化问题的拟多项式时间可解性。
英文摘要：
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.
