Algebraic Branching Programs, Border Complexity, and Tangent Spaces

Authors Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, Nitin Saurabh

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Markus Bläser
  • Department of Computer Science, Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Christian Ikenmeyer
  • University of Liverpool, UK
Meena Mahajan
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Anurag Pandey
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Nitin Saurabh
  • Technion-IIT, Haifa, Israel


We thank Michael Forbes for illuminating discussions and for telling us about his (correct) intuition concerning Nisan’s result. We thank the Simons Institute for the Theory of Computing (Berkeley), Schloss Dagstuhl - Leibniz-Zentrum für Informatik (Dagstuhl), and the International Centre for Theoretical Sciences (Bengaluru), for hosting us during several phases of this research.

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Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, and Nitin Saurabh. Algebraic Branching Programs, Border Complexity, and Tangent Spaces. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices. This means that the set of noncommutative polynomials with ABP width complexity at most k is Zariski-closed, an important property in geometric complexity theory. It follows that approximations cannot help to reduce the required ABP width. It was mentioned by Forbes that this result would probably break when going from single-(source,sink) ABPs to trace ABPs. We prove that this is correct. Moreover, we study the commutative monotone setting and prove a result similar to Nisan, but concerning the analytic closure. We observe the same behavior here: The set of polynomials with ABP width complexity at most k is closed for single-(source,sink) ABPs and not closed for trace ABPs. The proofs reveal an intriguing connection between tangent spaces and the vector space of flows on the ABP. We close with additional observations on VQP and the closure of VNP which allows us to establish a separation between the two classes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Complexity classes
  • Algebraic Branching Programs
  • Border Complexity
  • Tangent Spaces
  • Lower Bounds
  • Geometric Complexity Theory
  • Flows
  • VQP
  • VNP


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