Tight bounds for counting colorings and connected edge sets parameterized by cutwidth

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. Let $p,q \in \mathbb{N}$ such that $p$ is a prime and $q \geq 3$. We show: - 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 ${\text{ctw}}$. Furthermore, no algorithm can count the number of distinct $q$-colorings modulo $p$ in time $(q-1-\varepsilon)^{\text{ctw}} n^{O(1)}$ for some $\varepsilon>0$, assuming the Strong Exponential Time Hypothesis (SETH). - If $p$ does not divide $q-1$, no algorithm can count the number of distinct $q$-colorings modulo $p$ in time $(q-\varepsilon)^{\text{ctw}} n^{O(1)}$ for some $\varepsilon>0$, 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]. Furthermore, by building upon the above lower bounds, we obtain the following lower bound for counting connected spanning edge sets: there is no $\varepsilon>0$ for which there is an algorithm that, given a graph $G$ and a cutwidth ordering of cutwidth ${\text{ctw}}$, counts the number of spanning connected edge sets of $G$ modulo $p$ in time $(p - \varepsilon)^{\text{ctw}} n^{O(1)}$, assuming SETH. We also give an algorithm with matching running time for this problem. Before our work, even for the treewidth parameterization, the best conditional lower bound by Dell et al.~[ACM Trans. Algorithms'14] only excluded $2^{o({\text{tw}})}n^{O(1)}$ time algorithms for this problem. Both our algorithms and lower bounds employ use of the matrix rank method.