Spanning Tree Constrained Determinantal Point Processes are Hard to (Approximately) Evaluate

We consider determinantal point processes (DPPs) constrained by spanning trees. Given a graph G=(V,E) and a positive semi-definite matrix A indexed by E, a spanning-tree DPP defines a distribution such that we draw Ssubseteq E with probability proportional to det(AS) only if S induces a spanning tree. We prove sharptextsfP-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.