A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Let $H$ be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom($H$), we are given a graph $G$, whose every vertex $v$ is assigned with a list $L(v)$ of vertices of $H$. We ask whether there exists a homomorphism $h$ from $G$ to $H$, which respects lists $L$, i.e., for every $v \in V(G)$ it holds that $h(v) \in L(v)$. The complexity dichotomy for LHom($H$) was proven by Feder, Hell, and Huang [JGT 2003]. We are interested in the complexity of the problem, parameterized by the treewidth of the input graph. This problem was investigated by Egri, Marx, and Rz\k{a}\.zewski [STACS 2018], who obtained tight complexity bounds for the special case of reflexive graphs $H$. In this paper we extend and generalize their results for \emph{all} relevant graphs $H$, i.e., those, for which the LHom{H} problem is NP-hard. For every such $H$ we find a constant $k = k(H)$, such that LHom($H$) on instances with $n$ vertices and treewidth $t$ * can be solved in time $k^{t} \cdot n^{\mathcal{O}(1)}$, provided that the input graph is given along with a tree decomposition of width $t$, * cannot be solved in time $(k-\varepsilon)^{t} \cdot n^{\mathcal{O}(1)}$, for any $\varepsilon >0$, unless the SETH fails. For some graphs $H$ the value of $k(H)$ is much smaller than the trivial upper bound, i.e., $|V(H)|$. Obtaining matching upper and lower bounds shows that the set of algorithmic tools we have discovered cannot be extended in order to obtain faster algorithms for LHom($H$) in bounded-treewidth graphs. Furthermore, neither the algorithm, nor the proof of the lower bound, is very specific to treewidth. We believe that they can be used for other variants of LHom($H$), e.g. with different parameterizations.

中文翻译：

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有界树宽图的列表同态问题的全复杂度分类

从图 $G$ 到图 $H$ 的同态是从 $V(G)$ 到 $V(H)$ 的边保持映射。令 $H$ 是一个可能有循环的固定图。在由 LHom($H$) 表示的列表同态问题中，我们给出了一个图 $G$，其每个顶点 $v$ 都被分配了一个 $H$ 顶点的列表 $L(v)$。我们问是否存在从 $G$ 到 $H$ 的同态 $h$，它尊重列表 $L$，即对于每一个 $v \in V(G)$ 它都持有 $h(v) \in L (五)$。Feder、Hell 和 Huang [JGT 2003] 证明了 LHom($H$) 的复杂性二分法。我们对问题的复杂性感兴趣，由输入图的树宽参数化。Egri、Marx 和 Rz\k{a}\.zewski [STACS 2018] 研究了这个问题，他们为自反图 $H$ 的特殊情况获得了严格的复杂性界限。在本文中，我们扩展并概括了他们对 \emph{all} 相关图 $H$ 的结果，即那些 LHom{H} 问题是 NP-hard 的。对于每一个这样的 $H$，我们找到一个常数 $k = k(H)$，这样 LHom($H$) 在具有 $n$ 个顶点和树宽 $t$ * 的实例上可以及时求解 $k^{t } \cdot n^{\mathcal{O}(1)}$，假设输入图与宽度$t$的树分解一起给出，*无法及时求解$(k-\varepsilon)^{ t} \cdot n^{\mathcal{O}(1)}$，对于任何 $\varepsilon >0$，除非 SETH 失败。对于某些图形 $H$，$k(H)$ 的值远小于平凡的上限，即 $|V(H)|$。获得匹配的上界和下界表明我们发现的算法工具集无法扩展以在有界树宽图中获得更快的 LHom($H$) 算法。此外，无论是算法还是下界的证明，都不是非常特定于树宽的。我们相信它们可以用于 LHom($H$) 的其他变体，例如具有不同的参数化。