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Deterministic polynomial identity tests for multilinear bounded-read formulae

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Abstract

We present a polynomial-time deterministic algorithm for testing whether constant-read multilinear arithmetic formulae are identically zero. In such a formula, each variable occurs only a constant number of times, and each subformula computes a multilinear polynomial. Our algorithm runs in time \({s^{O(1)}\cdot n^{k^{O(k)}}}\) , where s denotes the size of the formula, n denotes the number of variables, and k bounds the number of occurrences of each variable. Before our work, no subexponential-time deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in time \({n^{k^{O(k)} + O(k \log n)}}\) in general, and time \({n^{k^{O(k^2)} + O(kD)}}\) for depth D. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae and for multilinear depth-four formulae.

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Authors and Affiliations

  1. University of Wisconsin-Madison, Madison, WI, USA

    Matthew Anderson & Dieter van Melkebeek

  2. Technion, Haifa, Israel

    Ilya Volkovich

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  1. Matthew Anderson
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  2. Dieter van Melkebeek
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  3. Ilya Volkovich
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Correspondence to Matthew Anderson.

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Anderson, M., van Melkebeek, D. & Volkovich, I. Deterministic polynomial identity tests for multilinear bounded-read formulae. comput. complex. 24, 695–776 (2015). https://doi.org/10.1007/s00037-015-0097-4

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  • DOI: https://doi.org/10.1007/s00037-015-0097-4

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