We consider determinantal point processes (DPPs) constrained by spanning trees. Given a graph $G=(V,E)$ and a positive semi-definite matrix $\mathbf{A}$ indexed by $E$, a spanning-tree DPP defines a distribution such that we draw $S\subseteq E$ with probability proportional to $\det(\mathbf{A}_S)$ only if $S$ induces a spanning tree. We prove $\sharp\textsf{P}$-hardness of computing the normalizing constant for spanning-tree DPPs and provide an approximation-preserving reduction from the mixed discriminant, for which FPRAS is not known. We show similar results for DPPs constrained by forests.

中文翻译：

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生成树约束的确定点过程难以（近似）评估

我们考虑由生成树约束的行列式点过程（DPP）。给定一个图$ G =（V，E）$和一个由$ E $索引的正半定矩阵$ \ mathbf {A} $，生成树DPP定义了一个分布，以便我们绘制$ S \ subseteq E $仅当$ S $诱导生成树时，概率才与$ \ det（\ mathbf {A} _S）$成正比。我们证明了计算生成树DPP的归一化常数的$ \ sharp \ textsf {P} $的难度，并提供了混合判别式的近似保留约简，而FPRAS尚不为人所知。对于受森林约束的民进党，我们显示了相似的结果。