Abstract
We study the finegrained 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 edge separators.
Let p,q ∈ ℕ such that p is a prime and q ≥ 3. We show:
 If p divides q1, there is a (q1)^{ctw}n^{O(1)} time algorithm for counting list qcolorings modulo p of nvertex graphs of cutwidth ctw. Furthermore, there is no ε > 0 for which there is a (q1ε)^{ctw} n^{O(1)} time algorithm that counts the number of list qcolorings modulo p of nvertex graphs of cutwidth ctw, assuming the Strong Exponential Time Hypothesis (SETH).
 If p does not divide q1, there is no ε > 0 for which there exists a (qε)^{ctw} n^{O(1)} time algorithm that counts the number of list qcolorings modulo p of nvertex graphs of cutwidth ctw, assuming SETH. The lower bounds are in stark contrast with the existing 2^{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 ε > 0 for which there is an algorithm that, given a graph G and a cutwidth ordering of cutwidth ctw, counts the number of spanning connected edge sets of G modulo p in time (p  ε)^{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(tw)}n^{O(1)} time algorithms for this problem.
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 nontrivial way.
BibTeX  Entry
@InProceedings{groenland_et_al:LIPIcs.STACS.2022.36,
author = {Groenland, Carla and Mannens, Isja and Nederlof, Jesper and Szil\'{a}gyi, Krisztina},
title = {{Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {36:136:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772228},
ISSN = {18688969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/15846},
URN = {urn:nbn:de:0030drops158464},
doi = {10.4230/LIPIcs.STACS.2022.36},
annote = {Keywords: connected edge sets, cutwidth, parameterized algorithms, colorings, counting modulo p}
}
Keywords: 

connected edge sets, cutwidth, parameterized algorithms, colorings, counting modulo p 
Collection: 

39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022) 
Issue Date: 

2022 
Date of publication: 

09.03.2022 