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

TL;DR

This paper provides a detailed classification of the computational complexity for evaluating the Tutte polynomial at integer points on graphs with small treewidth and cutwidth, revealing precise time bounds under ETH.

Contribution

It refines existing reductions to classify evaluation complexity of the Tutte polynomial based on treewidth and cutwidth, introducing new bounds and a novel rank argument.

Findings

Polynomial-time computability for some points

Time bounds based on treewidth and cutwidth assumptions

New rank bound for counting forests

Abstract

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.