Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci