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Improved Explicit Hitting-Sets for ROABPs

Authors Zeyu Guo, Rohit Gurjar

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  • 16 pages

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Author Details

Zeyu Guo
  • Department of Computer Science, University of Haifa, Israel
Rohit Gurjar
  • Department of Computer Science and Engineering, IIT Bombay, India


We thank Nitin Saxena, Sumanta Ghosh, and Pranav Bisht for many helpful discussions about PIT for ROABPs and related models.

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Zeyu Guo and Rohit Gurjar. Improved Explicit Hitting-Sets for ROABPs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 4:1-4:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


We give improved explicit constructions of hitting-sets for read-once oblivious algebraic branching programs (ROABPs) and related models. For ROABPs in an unknown variable order, our hitting-set has size polynomial in (nr)^{(log n)/(max{1, log log n-log log r})}d over a field whose characteristic is zero or large enough, where n is the number of variables, d is the individual degree, and r is the width of the ROABP. A similar improved construction works over fields of arbitrary characteristic with a weaker size bound. Based on a result of Bisht and Saxena (2020), we also give an improved explicit construction of hitting-sets for sum of several ROABPs. In particular, when the characteristic of the field is zero or large enough, we give polynomial-size explicit hitting-sets for sum of constantly many log-variate ROABPs of width r = 2^{O(log d/log log d)}. Finally, we give improved explicit hitting-sets for polynomials computable by width-r ROABPs in any variable order, also known as any-order ROABPs. Our hitting-set has polynomial size for width r up to 2^{O(log(nd)/log log(nd))} or 2^{O(log^{1-ε} (nd))}, depending on the characteristic of the field. Previously, explicit hitting-sets of polynomial size are unknown for r = ω(1).

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Pseudorandomness and derandomization
  • polynomial identity testing
  • hitting-set
  • arithmetic branching programs
  • derandomization


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