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The zeta and Moebius transforms over the subset lattice of $n$ elements andthe so-called subset convolution are examples of unary and binary operations onset functions. While their direct computation requires $O(3^n)$ arithmeticoperations, less naive algorithms only use $2^n \mathrm{poly}(n)$ operations,nearly linear in the input size. Here, we investigate a related $n$-aryoperation that takes $n$ set functions as input and maps them to a new setfunction. This operation, we call multi-subset transform, is the coreingredient in the known inclusion--exclusion recurrence for weighted sums overacyclic digraphs, which extends Robinson's recurrence for the number oflabelled acyclic digraphs. Prior to this work the best known complexity boundwas the direct $O(3^n)$. By reducing the task to multiple instances ofrectangular matrix multiplication, we improve the complexity to $O(2.985^n)$.

Fast Multi-Subset Transform and Weighted Sums Over Acyclic Digraphs