A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. In the graph homomorphism problem, denoted by $Hom(H)$, the graph $H$ is fixed and we need to determine if there exists a homomorphism from an instance graph $G$ to $H$. We study the complexity of the problem parameterized by the cutwidth of $G$. We aim, for each $H$, for algorithms for $Hom(H)$ running in time $c_H^k n^{\mathcal{O}(1)}$ and matching lower bounds that exclude $c_H^{k \cdot o(1)}n^{\mathcal{O}(1)}$ or $c_H^{k(1-\Omega(1))}n^{\mathcal{O}(1)}$ time algorithms under the (Strong) Exponential Time Hypothesis. In the paper we introduce a new parameter that we call $\mathrm{mimsup}(H)$. Our main contribution is strong evidence of a close connection between $c_H$ and $\mathrm{mimsup}(H)$: * an information-theoretic argument that the number of states needed in a natural dynamic programming algorithm is at most $\mathrm{mimsup}(H)^k$, * lower bounds that show that for almost all graphs $H$ indeed we have $c_H \geq \mathrm{mimsup}(H)$, assuming the (Strong) Exponential-Time Hypothesis, and * an algorithm with running time $\exp ( {\mathcal{O}( \mathrm{mimsup}(H) \cdot k \log k)}) n^{\mathcal{O}(1)}$. The parameter $\mathrm{mimsup}(H)$ can be thought of as the $p$-th root of the maximum induced matching number in the graph obtained by multiplying $p$ copies of $H$ via certain graph product, where $p$ tends to infinity. It can also be defined as an asymptotic rank parameter of the adjacency matrix of $H$. Our results tightly link the parameterized complexity of a problem to such an asymptotic rank parameter for the first time.

翻译：图$G$到图$H$的同态是从$V(G)$到$V(H)$的边保持映射。在图同态问题$Hom(H)$中，图$H$是固定的，我们需要判断实例图$G$到$H$是否存在同态。我们研究该问题以$G$的切割宽为参数时的复杂度。针对每个$H$，我们的目标是设计运行时间为$c_H^k n^{\mathcal{O}(1)}$的$Hom(H)$算法，并在（强）指数时间假设下排除运行时间为$c_H^{k \cdot o(1)}n^{\mathcal{O}(1)}$或$c_H^{k(1-\Omega(1))}n^{\mathcal{O}(1)}$的算法。本文引入了一个新参数$\mathrm{mimsup}(H)$。主要贡献在于揭示了$c_H$与$\mathrm{mimsup}(H)$之间的紧密关联：* 信息论论证表明，自然动态规划算法所需的状态数至多为$\mathrm{mimsup}(H)^k$；* 下界证明，在（强）指数时间假设下，几乎所有图$H$确实满足$c_H \geq \mathrm{mimsup}(H)$；* 运行时间为$\exp ( {\mathcal{O}( \mathrm{mimsup}(H) \cdot k \log k)}) n^{\mathcal{O}(1)}$的算法。参数$\mathrm{mimsup}(H)$可视为通过特定图乘积将$p$份$H$相乘后所得图中最大诱导匹配数的$p$次根（$p$趋于无穷），也可定义为$H$邻接矩阵的渐近秩参数。我们的结果首次将问题的参数化复杂度与渐近秩参数紧密联系起来。