Fast Multi-Subset Transform and Weighted Sums Over Acyclic Digraphs

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