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

TL;DR

This paper establishes tight bounds for counting graph colorings and connected edge sets based on cutwidth, revealing complexity differences depending on algebraic properties, and employs advanced matrix rank techniques.

Contribution

It provides tight parameterized complexity bounds for counting colorings and edge sets with respect to cutwidth, using novel matrix rank methods and extending results to treewidth.

Findings

Algorithms match lower bounds under SETH assumptions.

Complexity varies depending on divisibility conditions of parameters.

Extends bounds to counting connected edge sets modulo p.

Abstract

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.