The Fine-Grained Complexity of Computing the Tutte Polynomial of a Linear Matroid

We show that computing the Tutte polynomial of a linear matroid of dimension $k$ on $k^{O(1)}$ points over a field of $k^{O(1)}$ elements requires $k^{\Omega(k)}$ time unless the #ETH---a counting extension of the Exponential Time Hypothesis of Impagliazzo and Paturi [CCC 1999] due to Dell et al. [ACM TALG 2014]---is false. This holds also for linear matroids that admit a representation where every point is associated to a vector with at most two nonzero coordinates. Moreover, we also show that the same is true for computing the Tutte polynomial of a binary matroid of dimension $k$ on $k^{O(1)}$ points with at most three nonzero coordinates in each point's vector. These two results stand in sharp contrast to computing the Tutte polynomial of a $k$-vertex graph (that is, the Tutte polynomial of a graphic matroid of dimension $k$---which is representable in dimension $k$ over the binary field so that every vector has exactly two nonzero coordinates), which is known to be computable in $2^k k^{O(1)}$ time [Bj\"orklund et al., FOCS 2008]. Our lower-bound proofs proceed in three steps: (i) a classic connection due to Crapo and Rota [1970] between the number of tuples of codewords of full support and the Tutte polynomial of the matroid associated with the code; (ii) an earlier-established \#ETH-hardness of counting the solutions to a bipartite $(d,2)$-CSP on $n$ vertices in $d^{o(n)}$ time; and (iii) new embeddings of such CSP instances as questions about codewords of full support in a linear code. We complement these lower bounds with a matching upper-bound algorithm design that computes the Tutte polynomial of a linear matroid of dimension $k$ on $k^{O(1)}$ points in $k^{O(k)}$ arithmetic operations in the base field.