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On Finer Separations Between Subclasses of Read-Once Oblivious ABPs

Authors C. Ramya , Anamay Tengse



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Author Details

C. Ramya
  • Chennai Mathematical Institute, India
Anamay Tengse
  • Department of Computer Science, University of Haifa, Israel

Acknowledgements

We thank Ramprasad Saptharishi for numerous insightful discussions about the various structured models, which motivated this work. We also thank Mrinal Kumar for his helpful comments about our work which helped us in enchancing the presentation. We thank Manoj Gopalakrishan and the organisers of Thursday Theory Lunch at IIT Bombay for organising a talk by Debasattam Pal, where we first came across the work of {Möller} and Stetter (1995) that essentially led to the main results in this paper. We thank the anonymous reviewers for their valuable inputs on the earlier version of the paper.

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C. Ramya and Anamay Tengse. On Finer Separations Between Subclasses of Read-Once Oblivious ABPs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 53:1-53:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.STACS.2022.53

Abstract

Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials). Our main result concerns commutative ROABPs, where all coefficient matrices commute with each other, and diagonal ROABPs, where all the coefficient matrices are just diagonal matrices. In particular, we show a somewhat surprising connection between these models and the model of depth-three powering circuits that is related to the Waring rank of polynomials. We show that if the dimension of partial derivatives captures Waring rank up to polynomial factors, then the model of diagonal ROABPs efficiently simulates the seemingly more expressive model of commutative ROABPs. Further, a commutative ROABP that cannot be efficiently simulated by a diagonal ROABP will give an explicit polynomial that gives a super-polynomial separation between dimension of partial derivatives and Waring rank. Our proof of the above result builds on the results of Marinari, Möller and Mora (1993), and Möller and Stetter (1995), that characterise rings of commuting matrices in terms of polynomials that have small dimension of partial derivatives. The algebraic structure of the coefficient matrices of these ROABPs plays a crucial role in our proofs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebraic Complexity Theory
  • Algebraic Branching Programs
  • Commutative Matrices

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