We study the fine-grained complexity of counting the number of colorings and connected spanning edge sets parameterized by the cutwidth and treewidth of the graph. While decompositions of small treewidth decompose the graph with small vertex separators, decompositions with small cutwidth decompose the graph with small \emph{edge} separators. Let $p,q \in \mathbb{N}$ such that $p$ is a prime and $q \geq 3$. - If $p$ divides $q-1$, there is a $(q-1)^{\text{ctw}}n^{O(1)}$ time algorithm for counting list $q$-colorings modulo $p$ of $n$-vertex graphs of cutwidth $\text{ctw}$ and for all $\varepsilon>0$ there is no algorithm running in time $(q-1-\varepsilon)^{\text{ctw}} n^{O(1)}$, assuming the Strong Exponential Time Hypothesis (SETH). - If $p$ does not divide $q-1$, there is a (folklore) $q^{\text{ctw}}n^{O(1)}$ time algorithm for counting list $q$-colorings modulo $p$ of $n$-vertex graphs of cutwidth $\text{ctw}$ and for all $\varepsilon>0$ there is no algorithm running in time $(q-\varepsilon)^{\text{ctw}} n^{O(1)}$, assuming SETH. The lower bounds are in stark contrast with the existing $2^{\text{ctw}}n^{O(1)}$ time algorithm to compute the chromatic number of a graph by Jansen and Nederlof~[Theor. Comput. Sci.'18]. Both our algorithms and lower bounds employ use of the matrix rank method, by relating the complexity of the problem to the rank of a certain `compatibility matrix' in a non-trivial way. We extend our lower bounds to counting connected spanning edge sets modulo $p$ and give an algorithm with matching running time for both treewidth and cutwidth.

翻译：我们研究了以图的割宽和树宽为参数，计数着色和连通生成边集数量的细粒度复杂度。尽管小树宽分解通过小顶点分割器分解图，但小割宽分解则通过小边分割器分解图。设 $p,q \in \mathbb{N}$，其中 $p$ 为素数且 $q \geq 3$。 - 若 $p$ 整除 $q-1$，则存在一个时间复杂度为 $(q-1)^{\text{ctw}}n^{O(1)}$ 的算法，用于对割宽为 $\text{ctw}$ 的 $n$ 顶点图计数模 $p$ 的列表 $q$-着色；并且对于所有 $\varepsilon>0$，在强指数时间假设（SETH）下，不存在时间复杂度为 $(q-1-\varepsilon)^{\text{ctw}} n^{O(1)}$ 的算法。 - 若 $p$ 不整除 $q-1$，则存在一个（众所周知的）时间复杂度为 $q^{\text{ctw}}n^{O(1)}$ 的算法，用于对割宽为 $\text{ctw}$ 的 $n$ 顶点图计数模 $p$ 的列表 $q$-着色；并且对于所有 $\varepsilon>0$，在SETH下，不存在时间复杂度为 $(q-\varepsilon)^{\text{ctw}} n^{O(1)}$ 的算法。 这些下界与Jansen和Nederlof~[Theor. Comput. Sci.'18]提出的计算图色数的 $2^{\text{ctw}}n^{O(1)}$ 时间算法形成鲜明对比。我们的算法和下界均利用了矩阵秩方法，通过将问题的复杂度与某个“兼容性矩阵”的秩以非平凡方式关联起来。我们将下界扩展至模 $p$ 的连通生成边集计数，并针对树宽和割宽给出了一个运行时间匹配的算法。