The Fine-Grained Complexity of Computing the Tutte Polynomial of a Linear Matroid
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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^Ω(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 [Björklund 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.
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