持久式和混合判别式的一种有效树分解方法
An efficient tree decomposition method for permanents and mixed discriminants
摘要：
我们提出了一个有效的算法来计算结构矩阵和多维数组（张量）的永久式，混合判别式和超行列式。我们描述的稀疏结构的数组中的一个图形，我们假设它的树宽，表示为$ω$，是小的。我们的算法需要$O(n 2^ω)$算术运算来计算持久式，而$O(n^2 + n 3^ω)$用于混合判别式和超行列式。我们最后表明，混合体积计算仍然是困难的有界树宽假设。
英文摘要：
We present an efficient algorithm to compute permanents, mixed discriminants and hyperdeterminants of structured matrices and multidimensional arrays (tensors). We describe the sparsity structure of an array in terms of a graph, and we assume that its treewidth, denoted as $ω$, is small. Our algorithm requires $O(n 2^ω)$ arithmetic operations to compute permanents, and $O(n^2 + n 3^ω)$ for mixed discriminants and hyperdeterminants. We finally show that mixed volume computation continues to be hard under bounded treewidth assumptions.
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