The goal of this paper is to investigate a family of optimization problems arising from list homomorphisms, and to understand what the best possible algorithms are if we restrict the problem to bounded-treewidth graphs. For a fixed $H$, the input of the optimization problem LHomVD($H$) is a graph $G$ with lists $L(v)$, and the task is to find a set $X$ of vertices having minimum size such that $(G-X,L)$ has a list homomorphism to $H$. We define analogously the edge-deletion variant LHomED($H$). This expressive family of problems includes members that are essentially equivalent to fundamental problems such as Vertex Cover, Max Cut, Odd Cycle Transversal, and Edge/Vertex Multiway Cut. For both variants, we first characterize those graphs $H$ that make the problem polynomial-time solvable and show that the problem is NP-hard for every other fixed $H$. Second, as our main result, we determine for every graph $H$ for which the problem is NP-hard, the smallest possible constant $c_H$ such that the problem can be solved in time $c^t_H\cdot n^{O(1)}$ if a tree decomposition of $G$ having width $t$ is given in the input.Let $i(H)$ be the maximum size of a set of vertices in $H$ that have pairwise incomparable neighborhoods. For the vertex-deletion variant LHomVD($H$), we show that the smallest possible constant is $i(H)+1$ for every $H$. The situation is more complex for the edge-deletion version. For every $H$, one can solve LHomED($H$) in time $i(H)^t\cdot n^{O(1)}$ if a tree decomposition of width $t$ is given. However, the existence of a specific type of decomposition of $H$ shows that there are graphs $H$ where LHomED($H$) can be solved significantly more efficiently and the best possible constant can be arbitrarily smaller than $i(H)$. Nevertheless, we determine this best possible constant and (assuming the SETH) prove tight bounds for every fixed $H$.

翻译：本文旨在研究一类源于列表同态的优化问题，并理解当问题限制在有界树宽图上时，可能达到的最优算法。对于固定的$H$，优化问题LHomVD($H$)的输入是一个图$G$及其列表$L(v)$，任务是最小化顶点集$X$的大小，使得$(G-X,L)$存在到$H$的列表同态。我们类似地定义边删除变体LHomED($H$)。这一表达力强的问题族包含的成员本质上等价于基本问题，如顶点覆盖、最大割、奇环横贯以及边/顶点多路割。对于两种变体，我们首先刻画了使问题多项式时间可解的图$H$，并证明对于每个其他固定的$H$，问题是NP难的。其次，作为主要结果，我们针对每个使问题NP难的图$H$，确定了最小可能常数$c_H$，使得在输入中给定$G$的宽度为$t$的树分解时，问题可在时间$c^t_H\cdot n^{O(1)}$内求解。令$i(H)$为$H$中具有两两不可比邻域的顶点集的最大大小。对于顶点删除变体LHomVD($H$)，我们证明每个$H$的最小可能常数为$i(H)+1$。边删除版本的情况更为复杂。对于每个$H$，若给定宽度为$t$的树分解，LHomED($H$)可在时间$i(H)^t\cdot n^{O(1)}$内求解。然而，$H$的特定分解类型的存在表明，存在图$H$使得LHomED($H$)可被更高效地求解，且最佳可能常数可任意小于$i(H)$。尽管如此，我们仍确定了这一最佳可能常数，并（假设SETH）为每个固定的$H$证明了紧界。