A Fine-Grained Classification of the Complexity of Evaluating the Tutte Polynomial on Integer Points Parameterized by Treewidth and Cutwidth.

We give a fine-grained classification of evaluating the Tutte polynomial $T(G;x,y)$ on all integer points on graphs with small treewidth and cutwidth. Specifically, we show for any point $(x,y) \in \mathbb{Z}^2$ that either - can be computed in polynomial time, - can be computed in $2^{O(tw)}n^{O(1)}$ time, but not in $2^{o(ctw)}n^{O(1)}$ time assuming the Exponential Time Hypothesis (ETH), - can be computed in $2^{O(tw \log tw)}n^{O(1)}$ time, but not in $2^{o(ctw \log ctw)}n^{O(1)}$ time assuming the ETH, where we assume tree decompositions of treewidth $tw$ and cutwidth decompositions of cutwidth $ctw$ are given as input along with the input graph on $n$ vertices and point $(x,y)$. To obtain these results, we refine the existing reductions that were instrumental for the seminal dichotomy by Jaeger, Welsh and Vertigan~[Math. Proc. Cambridge Philos. Soc'90]. One of our technical contributions is a new rank bound of a matrix that indicates whether the union of two forests is a forest itself, which we use to show that the number of forests of a graph can be counted in $2^{O(tw)}n^{O(1)}$ time.