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, assuming the SETH. 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:本文的目的是要调查一组由列表同质性引发的优化问题, 并了解如果我们将问题限制在有约束的树叶图形中, 最可能的算法是什么。 对于固定的 $H, 优化问题LHomVD($H$) 的投入是一个G$, 列表为$L(v) 美元, 而任务在于找到一个固定的 美元( x$), 其最小大小为 $( G- X, L) 的 美元( 美元), 其一美元( 美元, 美元) 的同价( 美元) 。 我们类似地定义了 边际- 美元变量 LhomED( 美元) 。 对于固定的 美元( 美元 美元), 这个表示问题包括基本上相当于基本问题的成员, 如VertexCovernex Coverformalalalalalalalalalalalal, 如果每张的LHDRO( 美元) 问题, 我们确定每张恒定的 问题是每张的常价( 美元) 问题是每平的, 可能的。