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