In the general AntiFactor problem, a graph $G$ is given with a set $X_v\subseteq \mathbb{N}$ of forbidden degrees for every vertex $v$ and the task is to find a set $S$ of edges such that the degree of $v$ in $S$ is not in the set $X_v$. Standard techniques (dynamic programming + fast convolution) can be used to show that if $M$ is the largest forbidden degree, then the problem can be solved in time $(M+2)^k\cdot n^{O(1)}$ if a tree decomposition of width $k$ is given. However, significantly faster algorithms are possible if the sets $X_v$ are sparse: our main algorithmic result shows that if every vertex has at most $x$ forbidden degrees (we call this special case AntiFactor$_x$), then the problem can be solved in time $(x+1)^{O(k)}\cdot n^{O(1)}$. That is, the AntiFactor$_x$ is fixed-parameter tractable parameterized by treewidth $k$ and the maximum number $x$ of excluded degrees. Our algorithm uses the technique of representative sets, which can be generalized to the optimization version, but (as expected) not to the counting version of the problem. In fact, we show that #AntiFactor$_1$ is already #W[1]-hard parameterized by the width of the given decomposition. Moreover, we show that, unlike for the decision version, the standard dynamic programming algorithm is essentially optimal for the counting version. Formally, for a fixed nonempty set $X$, we denote by $X$-AntiFactor the special case where every vertex $v$ has the same set $X_v=X$ of forbidden degrees. We show the following lower bound for every fixed set $X$: if there is an $\epsilon>0$ such that #$X$-AntiFactor can be solved in time $(\max X+2-\epsilon)^k\cdot n^{O(1)}$ on a tree decomposition of width $k$, then the Counting Strong Exponential-Time Hypothesis (#SETH) fails.

翻译：在一般 AntiFactor 问题中，给定图 $G$ 和每个顶点 $v$ 的禁止度数集合 $X_v\subseteq \mathbb{N}$，任务是找到一个边集 $S$，使得 $v$ 在 $S$ 中的度数不在集合 $X_v$ 中。标准技术（动态规划 + 快速卷积）可用于证明：若 $M$ 为最大禁止度数，则在给定宽度为 $k$ 的树分解后，该问题可在时间 $(M+2)^k\cdot n^{O(1)}$ 内求解。然而，当集合 $X_v$ 稀疏时，可能存在显著更快的算法：我们的主要算法结果表明，若每个顶点至多有 $x$ 个禁止度数（称此特例为 AntiFactor$_x$），则问题可在时间 $(x+1)^{O(k)}\cdot n^{O(1)}$ 内求解。即，AntiFactor$_x$ 关于树宽 $k$ 和排除度数最大值 $x$ 是固定参数可处理的。我们的算法使用了代表集技术，该技术可推广到优化版本，但（如预期）不能推广到计数版本。事实上，我们证明 #AntiFactor$_1$ 关于给定分解的宽度已经是 #W[1]-困难的。此外，我们表明，与判定版本不同，标准动态规划算法对于计数版本本质上是最优的。形式化地，对于固定非空集合 $X$，记 $X$-AntiFactor 为每个顶点 $v$ 具有相同禁止度数集合 $X_v=X$ 的特例。我们对每个固定集合 $X$ 证明如下下界：若存在 $\epsilon>0$，使得 #$X$-AntiFactor 在宽度为 $k$ 的树分解上可在时间 $(\max X+2-\epsilon)^k\cdot n^{O(1)}$ 内求解，则计数强指数时间假设 (#SETH) 不成立。