论文标题
计算线性曲线的TUTTE多项式的细粒度复杂性
The Fine-Grained Complexity of Computing the Tutte Polynomial of a Linear Matroid
论文作者
论文摘要
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 {\ em等人} [ACM TALG 2014] ---是错误的。这也适用于线性矩阵，该线性矩阵允许一个表示每个点与最多两个非零坐标的向量相关联。我们还表明，计算$ k^{o（1）} $点的二进制二进制基质的Tutte多项式，每个点的向量中最多三个非零坐标。这与计算计算$ k $ vertex图的tutte多项式（即，是dimension $ k $的{\ em graphic}矩阵的tutte多项式，这是二进制$ k $的代表，在二进制中可以表示二进制中的二进制范围，以使每个载体都有两个非零均值，以便在$ nonnozer coordinate中均具有$} $} $} $ k^k^k^k. [Björklund{\ em等人}，焦点2008]。我们的较低的证明是通过（i）由于Crapo和Rota [1970]引起的（i）在全面支撑的代码字的数量与与代码相关的Matroid的Tutte多项式之间的连接； （ii）在$ d^{o（n）} $ time上计数$ n $ vertices上的$ n $ vertices上的$ n $ vertices上的$ n $ vertices上的解决方案（d，2）$ - csp; （iii）此类CSP实例的新嵌入方式，即有关线性代码中完全支持的代码字问题。我们通过两种算法设计对这些下限进行补充。第一个设计计算$ k^{o（1）} $ points $ k^{o（k）} $操作的尺寸线性矩阵的tutte多项式。第二个设计概括了björklund〜 {\ em等}算法，并在$ q^{k+1} k^{o（1）} $ time $ k^{o（1）$ q^e（1）$ q^e（o（1）$ q $ k^{o（1）$ quote的线性$ k $的线性矩阵中的时间$ k $ a，nonnonate nonyzere concore nonzere。
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 {\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 [Björklund {\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 Björklund~{\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.
