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Kronecker Scaling of Tensors with Applications to Arithmetic Circuits and Algorithms

Authors Andreas Björklund , Petteri Kaski , Tomohiro Koana , Jesper Nederlof

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LIPIcs.ICALP.2026.36.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • tensor rank
  • Kronecker powers
  • arithmetic circuits
  • permanent
  • parameterized algorithms

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Abstract

We show that sufficiently low tensor rank for the balanced tripartitioning tensor P_d(x,y,z) = ∑_{A,B,C ∈ binom([3d],d):A∪ B∪ C = [3d]} x_A y_B z_C for a large enough constant d implies uniform arithmetic circuits for the matrix permanent that are exponentially smaller than circuits obtainable from Ryser’s formula.
Under the same low-rank assumption, we obtain exponential-time improvements over the state of the art for a wide variety of related counting and decision problems.
Our main methodological contribution is that the tensors P_n have a desirable Kronecker scaling property: They can be decomposed efficiently into a small sum of restrictions of Kronecker powers of P_d for constant d. We prove this with a new technique relying on Steinitz’s lemma, which we hence call Steinitz balancing.
As a consequence of our methods, we show that the mentioned low-rank assumption (and hence the improved algorithms) is implied by Strassen’s asymptotic rank conjecture [Progr. Math. 120 (1994)], a bold conjecture that has recently seen intriguing progress.

Cite As Get BibTex

Andreas Björklund, Petteri Kaski, Tomohiro Koana, and Jesper Nederlof. Kronecker Scaling of Tensors with Applications to Arithmetic Circuits and Algorithms. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 36:1-36:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.36

Author Details

Andreas Björklund
  • Lund, Sweden
Petteri Kaski
  • Aalto University, Espoo, Finland
Tomohiro Koana
  • The University of Tokyo, Japan
Jesper Nederlof
  • Utrecht University, The Netherlands

Funding

  • Koana, Tomohiro: Supported by JST CREST Grant Number JPMJCR24Q2 and JST ERATO Grant Number JPMJER2301.
  • Nederlof, Jesper: Supported by the European Research Council (ERC), grant agreement No. 853234 (project COALESCE).

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