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 {\em 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. 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. This is in sharp contrast to computing the Tutte polynomial of a $k$-vertex graph (that is, the Tutte polynomial of a {\em graphic} matroid of dimension $k$---which is representable in dimension $k$ over the binary field so that every vector has two nonzero coordinates), which is known to be computable in $2^k k^{O(1)}$ time [Bj\"orklund {\em et al.}, FOCS 2008]. Our lower-bound proofs proceed via (i) a 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 two algorithm designs. The first design computes the Tutte polynomial of a linear matroid of dimension~$k$ on $k^{O(1)}$ points in $k^{O(k)}$ operations. The second design generalizes the Bj\"orklund~{\em et al.} algorithm and runs in $q^{k+1}k^{O(1)}$ time for linear matroids of dimension $k$ defined over the $q$-element field by $k^{O(1)}$ points with at most two nonzero coordinates each.Comment: This version adds Theorem

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Ramified rectilinear polygons: coordinatization by dendrons

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group $D_4$), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.Comment: 27 pages, 6 figure