On Constant Depth Circuits Parameterized by Degree: Identity Testing and Depth Reduction

  • Purnata Ghosal
  • Om Prakash
  • B. V. Raghavendra Rao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10392)

Abstract

In this article we initiate the study of polynomials parameterized by degree by arithmetic circuits of small syntactic degree. We define the notion of fixed parameter tractability and show that there are families of polynomials of degree k that cannot be computed by homogeneous depth four \(\varSigma \varPi ^{\sqrt{k}}\varSigma \varPi ^{\sqrt{k}}\) circuits. Our result implies that there is no parameterized depth reduction for circuits of size \(f(k)n^{O(1)}\) such that the resulting depth four circuit is homogeneous.

We show that testing identity of depth three circuits with syntactic degree k is fixed parameter tractable with k as the parameter. Our algorithm involves an application of the hitting set generator given by Shpilka and Volkovich [APPROX-RANDOM 2009]. Further, we show that our techniques do not generalize to higher depth circuits by proving certain rank-preserving properties of the generator by Shpilka and Volkovich.

References

  1. 1.
    Agrawal, M., Vinay, V.: Arithmetic circuits: a chasm at depth four. In: FOCS, pp. 67–75 (2008)Google Scholar
  2. 2.
    Amini, O., Fomin, F.V., Saurabh, S.: Counting subgraphs via homomorphisms. SIAM J. Discrete Math. 26(2), 695–717 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arvind, V., Köbler, J., Kuhnert, S., Torán, J.: Solving linear equations parameterized by hamming weight. Algorithmica 75(2), 322–338 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Björklund, A.: Exact covers via determinants. In: STACS, pp. 95–106 (2010)Google Scholar
  5. 5.
    Björklund, A., Husfeldt, T., Taslaman, N.: Shortest cycle through specified elements. In: SODA, pp. 1747–1753 (2012)Google Scholar
  6. 6.
    Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory, vol. 7. Springer Science & Business Media, Heidelberg (2013)MATHGoogle Scholar
  7. 7.
    Chauhan, A., Rao, B.V.R.: Parameterized analogues of probabilistic computation. In: CALDAM, pp. 181–192 (2015)Google Scholar
  8. 8.
    Chen, Z., Fu, B., Liu, Y., Schweller, R.T.: On testing monomials in multivariate polynomials. Theor. Comput. Sci. 497, 39–54 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London (2013). http://dx.doi.org/10.1007/978-1-4471-5559-1 CrossRefMATHGoogle Scholar
  10. 10.
    Engels, C.: Why are certain polynomials hard?: a look at non-commutative, parameterized and homomorphism polynomials. Ph.D. thesis, Saarland University (2016)Google Scholar
  11. 11.
    Fischer, I.: Sums of like powers of multivariate linear forms. Mathemat. Mag. 67(1), 59–61 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM 63(4), 29:1–29:60 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S., Rao, B.V.R.: Faster algorithms for finding and counting subgraphs. J. Comput. Syst. Sci. 78(3), 698–706 (2012). http://dx.doi.org/10.1016/j.jcss.2011.10.001 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gupta, A., Kamath, P., Kayal, N., Saptharishi, R.: Arithmetic circuits: a chasm at depth three. In: FOCS 2013, pp. 578–587. IEEE (2013)Google Scholar
  15. 15.
    Kumar, M., Maheshwari, G., Sarma M.N., J.: Arithmetic circuit lower bounds via maxrank. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7965, pp. 661–672. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39206-1_56 CrossRefGoogle Scholar
  16. 16.
    Müller, M.: Parameterized randomization. Ph.D. thesis, Albert-Ludwigs-Universität Freiburg im Breisgau (2008)Google Scholar
  17. 17.
    Nisan, N.: Lower bounds for non-commutative computation. In: STOC, pp. 410–418. ACM (1991)Google Scholar
  18. 18.
    Raz, R.: Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM 56(2) (2009)Google Scholar
  19. 19.
    Raz, R., Shpilka, A.: Deterministic polynomial identity testing in non-commutative models. Comput. Complex. 14(1), 1–19 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Saptharishi, R., Chillara, S., Kumar, M.: A survey of lower bounds in arithmetic circuit complexity. Technical report (2016). https://github.com/dasarpmar/lowerbounds-survey/releases
  21. 21.
    Saxena, N.: Diagonal circuit identity testing and lower bounds. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5125, pp. 60–71. Springer, Heidelberg (2008). doi:10.1007/978-3-540-70575-8_6 CrossRefGoogle Scholar
  22. 22.
    Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM (JACM) 27(4), 701–717 (1980)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Shpilka, A., Volkovich, I.: Improved polynomial identity testing for read-once formulas. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX/RANDOM -2009. LNCS, vol. 5687, pp. 700–713. Springer, Heidelberg (2009). doi:10.1007/978-3-642-03685-9_52 CrossRefGoogle Scholar
  24. 24.
    Tavenas, S.: Improved bounds for reduction to depth 4 and depth 3. Inf. Comput. 240, 2–11 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Valiant, L.G., Skyum, S., Berkowitz, S., Rackoff, C.: Fast parallel computation of polynomials using few processors. SIAM J. Comput. 12(4), 641–644 (1983)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Zippel, R.: Probabilistic algorithms for sparse polynomials. In: Ng, E.W. (ed.) Symbolic and Algebraic Computation. LNCS, vol. 72, pp. 216–226. Springer, Heidelberg (1979). doi:10.1007/3-540-09519-5_73 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Purnata Ghosal
    • 1
  • Om Prakash
    • 1
  • B. V. Raghavendra Rao
    • 1
  1. 1.Department of Computer Science and EngineeringIIT MadrasChennaiIndia

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