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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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References

  • Aaronson S., van Melkebeek D. (2011) On Circuit Lower Bounds from Derandomization. Theory of Computing 7(1): 177–184

    Article  MathSciNet  Google Scholar 

  • M. Agrawal (2003). On derandomizing tests for certain polynomial identities. In Proceedings of the 18th Annual IEEE Conference on Computational Complexity, 355–359. ISBN 0769518796. ISSN 1093-0159.

  • Agrawal M., Biswas S. (2003) Primality and identity testing via chinese remaindering. Journal of the ACM 50(4): 429–443

    Article  MathSciNet  Google Scholar 

  • M. Agrawal, Saha, R. Saptharishi & N. Saxena (2012). Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, 599–614.

  • M. Agrawal & V. Vinay (2008). Arithmetic circuits: A chasm at depth four. In Proceedings of the 49th Annual Symposium on Foundations of Computer Science, 67–75.

  • M. Anderson, D. van Melkebeek & I. Volkovich (2011). Derandomizing Polynomial Identity Testing for Multilinear Constant-Read Formulae. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity, 273–282.

  • Arvind V., Mukhopadhyay P. (2010) The ideal membership problem and polynomial identity testing. Information and Computation 208(4): 351–363

    Article  MATH  MathSciNet  Google Scholar 

  • Baur W., Strassen V. (1983) The complexity of partial derivatives. Theoretical Computer Science 22: 317–330

    Article  MATH  MathSciNet  Google Scholar 

  • M. Ben-Or & P. Tiwari (1988). A deterministic algorithm for sparse multivariate polynomial interpolation. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing, 301–309. ISBN 0897912640.

  • Bläser M., Hardt M., Lipton R., Vishnoi N. (2009) Deterministically testing sparse polynomial identities of unbounded degree. Information Processing Letters 109(3): 187–192

    Article  MATH  MathSciNet  Google Scholar 

  • DeMillo R., Lipton R. (1978) A probabilistic remark on algebraic program testing. Information Processing Letters 7(4): 193–195

    Article  MATH  Google Scholar 

  • Z. Dvir & A. Shpilka (2007). Locally decodable codes with two queries and polynomial identity testing for depth 3 circuits. SIAM Journal on Computing 36(5), 1404–1434. ISSN 0097-5397.

  • Dvir Z., Shpilka A., Yehudayoff A. (2009) Hardness-randomness tradeoffs for bounded depth arithmetic circuits. SIAM Journal on Computing 39(4): 1279–1293

    Article  MATH  MathSciNet  Google Scholar 

  • A. Gupta, P. Kamath, N. Kayal & R. Saptharishi (2013). Arithmetic circuits: A chasm at depth three. Technical Report 26, Electronic Colloquium on Computational Complexity.

  • M. Jansen, Y. Qiao & J. Sarma (2009). Deterministic Identity Testing of Read-Once Algebraic Branching Programs. Technical Report abs/0912.2565, CoRR.

  • Kabanets V., Impagliazzo R. (2004) Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity 13(1): 1–46

    Article  MATH  MathSciNet  Google Scholar 

  • Karnin Z., Mukhopadhyay P., Shpilka A., Volkovich I. (2013) Deterministic Identity Testing of Depth 4 Multilinear Circuits with Bounded Top Fan-In. SIAM Journal on Computing 42(6): 2114–2131

    Article  MATH  MathSciNet  Google Scholar 

  • Z. Karnin & A. Shpilka (2008). Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in. In Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, 280–291.

  • N. Kayal & S. Saraf (2009). Blackbox polynomial identity testing for depth 3 circuits. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, 198–207.

  • Kinne J., van Melkebeek D., Shaltiel R. (2012) Pseudorandom Generators, Typically-Correct Derandomization, and Circuit Lower Bounds. Computational Complexity 21(1): 3–61

    Article  MATH  MathSciNet  Google Scholar 

  • Klivans A., Shpilka A. (2006) Learning restricted models of arithmetic circuits. Theory of computing 2(10): 185–206

    Article  MathSciNet  Google Scholar 

  • A. Klivans & D. Spielman (2001). Randomness efficient identity testing of multivariate polynomials. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, 216–223.

  • Lovász L. (1979) On determinants, matchings and random algorithms. In Fundamentals of Computation Theory, volume 79: 565–574

    Google Scholar 

  • Mignon T., Ressayre N. (2004) A Quadratic Bound for the Determinant and Permanent Problem. International Mathematics Research Notices 79: 4241–4253

    Article  MathSciNet  Google Scholar 

  • Nisan N., Wigderson A. (1996) Lower bound on arithmetic circuits via partial derivatives. Computational Complexity 6: 217–234

    Article  MathSciNet  Google Scholar 

  • R. Raz & A. Yehudayoff (2008). Lower Bounds and Separations for Constant Depth Multilinear Circuits. In Proceedings of the 23rd Annual IEEE Conference on Computational Complexity, 128–139.

  • S. Saraf & I. Volkovich (2011). Black-Box Identity Testing of Depth-4 Multilinear Circuits. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, 421–430.

  • N. Saxena (2008). Diagonal circuit identity testing and lower bounds. In Proceedings of the 35th International Colloquium on Automata, Languages and Programming, 60–71.

  • Saxena N. (2009) Progress on polynomial identity testing. Bulletin of the EATCS 99: 49–79

    MATH  Google Scholar 

  • N. Saxena & C. Seshadhri (2009). An almost optimal rank bound for depth-3 identities. In Proceedings of the 24th Annual IEEE Conference on Computational Complexity, 137–148.

  • N. Saxena & C. Seshadhri (2010). From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-Box Identity Test for Depth-3 Circuits. In Proceedings of the 51st Annual Symposium on Foundations of Computer Science, 21–29.

  • Saxena N., Seshadhri C. (2012) Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn’t Matter. SIAM Journal on Computing 41(5): 1285–1298

    Article  MATH  MathSciNet  Google Scholar 

  • Schwartz J. (1980) Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM 27(4): 701–717

    Article  MATH  Google Scholar 

  • A. Shpilka & I. Volkovich (2008). Read-once polynomial identity testing. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 507–516.

  • A. Shpilka & I. Volkovich (2009). Improved polynomial identity testing for read-once formulas. In Proceedings of the 13th International Workshop on Randomization and Computation, 700–713.

  • Shpilka A., Wigderson A. (2001) Depth-3 Arithmetic Circuits over Fields of Characteristic Zero. Computational Complexity 10(1): 1–27

    Article  MATH  MathSciNet  Google Scholar 

  • A. Shpilka & A. Yehudayoff (2010). Arithmetic Circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science 5(3–4), 207–388.

  • R. Zippel (1979). Probabilistic algorithms for sparse polynomials. Symbolic and Algebraic Computation 216–226.

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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

Authors
  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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