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On Closure Properties of Read-Once Oblivious Algebraic Branching Programs

Authors Robert Andrews , Jules Armand , Prateek Dwivedi , Magnus Rahbek Dalgaard Hansen , Nutan Limaye , Srikanth Srinivasan , Sébastien Tavenas

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LIPIcs.ITCS.2026.9.pdf
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ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Circuit complexity
  • Theory of computation → Algebraic complexity theory
Keywords
  • Factoring
  • Closure Properties
  • Sparsity Bounds
  • Symmetric Polynomials
  • roABP
  • Expander Graphs

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Abstract

We investigate the closure properties of read-once oblivious Algebraic Branching Programs (roABPs) under various natural algebraic operations and prove the following.  
- Non-closure under factoring: There is a sequence of explicit polynomials (f_n(x₁,…, x_n))_n that have poly(n)-sized roABPs such that some irreducible factor of f_n requires roABPs of superpolynomial size in any order. 
- Non-closure under powering: There is a sequence of polynomials (f_n(x₁,…, x_n))_n with poly(n)-sized roABPs such that any super-constant power of f_n does not have roABPs of polynomial size in any order (and f_nⁿ requires exponential size in any order). 
- Non-closure under symmetric operations: There are symmetric polynomials (f_n(e₁,…, e_n))_n that have roABPs of polynomial size such that f_n(x₁,…, x_n) do not have roABPs of subexponential size. (Here, e₁,…, e_n denote the elementary symmetric polynomials in n variables.)  These results should be viewed in light of known results on models such as algebraic circuits, (general) algebraic branching programs, formulas and constant-depth circuits, all of which are known to be closed under these operations. 
To prove non-closure under factoring, we construct hard polynomials based on expander graphs using gadgets that lift their hardness from sparse polynomials to roABPs. For symmetric compositions, we show that the circulant polynomial requires roABPs of exponential size in every variable order.

Cite As Get BibTex

Robert Andrews, Jules Armand, Prateek Dwivedi, Magnus Rahbek Dalgaard Hansen, Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. On Closure Properties of Read-Once Oblivious Algebraic Branching Programs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.9

Author Details

Robert Andrews
  • Cheriton School of Computer Science, University of Waterloo, Canada
Jules Armand
  • Université Savoie Mont Blanc, CNRS, LAMA, France
Prateek Dwivedi
  • IT University of Copenhagen, Denmark
Magnus Rahbek Dalgaard Hansen
  • IT University of Copenhagen, Denmark
Nutan Limaye
  • IT University of Copenhagen, Denmark
Srikanth Srinivasan
  • Department of Computer Science, University of Copenhagen, Denmark
Sébastien Tavenas
  • Université Savoie Mont Blanc, CNRS, LAMA, France

Funding

PD, MH, and NL thank Independent Research Fund Denmark (grant agreement No. 10.46540/3103-00116B) and the support of Basic Algorithms Research Copenhagen (BARC), funded by VILLUM Foundation Grant 54451.
  • Srinivasan, Srikanth: Supported by European Research Council (ERC) under grant agreement no. 101125652 (ALBA).
  • Tavenas, Sébastien: Supported by ANR project VONBICA (ANR-22-CE48-0007).

Acknowledgements

This research in this paper was carried out during a visit of some of the authors to Université Savoie Mont Blanc. The authors thank the Laboratoire de Mathématiques (LAMA) at Université Savoie Mont Blanc, which is supported by the AAP GAFA project, for their hospitality and the conducive environment for research. NL would like to thank Université Savoie Mont Blanc for hosting her as a visiting researcher.

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