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Blackbox identity testing for sum of special ROABPs and its border class

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Abstract

We look at the problem of blackbox polynomial identity testing (PIT) for the model of read-once oblivious algebraic branching programs (ROABP), where the number of variables is logarithmic to the input size of ROABP. We restrict width of ROABP to a constant and study the more general sum-of-ROABPs model. This model is nontrivial due to the arbitrary individual-degree. We give the first poly(\(s\))-time blackbox PIT for sum of constant-many, size-\(s\), \(O(log s)\)-variate constant-width ROABPs. The previous best for this model was quasi-polynomial time (Gurjar et al, CCC'15; Computational Complexity'17) which is comparable to brute-force in the log-variate setting. We also show that we can work with unbounded-many such ROABPs if each ROABP computes a homogeneous polynomial (or more generally for degree-preserving sums). We also give poly-time PIT for the border. We introduce two new techniques, both of which also work for the border version of the stated models. (1) The leading-degree-part of an ROABP can be made syntactically homogeneous in the same width. (2) There is a direct reduction from PIT of sum-of-ROABPs to PIT of single ROABP (over any field). Our methods improve the time complexity for PIT of sum-of-ROABPs in the log-variate regime.

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Acknowledgements

We thank Subhayan Saha, Sumanta Ghosh and Zeyu Guo for various discussions related to the algebraic models studied here. We thank the anonymous reviewers for their suggestions to improve the writing. We especially thank the anonymous reviewer who gave an alternate proof of one of the lemmas here using border complexity, which inspired us to think about the border PIT algorithms. N.S. thanks the funding support from DST (DST/SJF/MSA-01/2013- 14) and N. Rama Rao Chair.

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  1. Department of Computer Science and Engineering, IIT Kanpur, Kanpur, Uttar Pradesh, 208016, India

    Pranav Bisht & Nitin Saxena

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  1. Pranav Bisht
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  2. Nitin Saxena
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Correspondence to Pranav Bisht.

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Bisht, P., Saxena, N. Blackbox identity testing for sum of special ROABPs and its border class. comput. complex. 30, 8 (2021). https://doi.org/10.1007/s00037-021-00209-y

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