Fast Multi-Subset Transform and Weighted Sums over Acyclic Digraphs

Authors Mikko Koivisto, Antti Röyskö

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Mikko Koivisto
  • Department of Computer Science, University of Helsinki, Finland
Antti Röyskö
  • Department of Computer Science, University of Helsinki, Finland


We thank Petteri Kaski for valuable discussions about the topic of the paper.

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Mikko Koivisto and Antti Röyskö. Fast Multi-Subset Transform and Weighted Sums over Acyclic Digraphs. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 29:1-29:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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(3ⁿ) arithmetic operations, less naive algorithms only use 2ⁿ 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 for computing the multi-subset transform was the direct O(3ⁿ). By reducing the task to rectangular matrix multiplication, we improve the complexity to O(2.985ⁿ).

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Bayesian networks
  • Moebius transform
  • Rectangular matrix multiplication
  • Subset convolution
  • Weighted counting of acyclic digraphs
  • Zeta transform


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