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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Agrawal, M., Vinay, V.: Arithmetic circuits: a chasm at depth four. In: FOCS, pp. 67–75 (2008)
Amini, O., Fomin, F.V., Saurabh, S.: Counting subgraphs via homomorphisms. SIAM J. Discrete Math. 26(2), 695–717 (2012)
Arvind, V., Köbler, J., Kuhnert, S., Torán, J.: Solving linear equations parameterized by hamming weight. Algorithmica 75(2), 322–338 (2016)
Björklund, A.: Exact covers via determinants. In: STACS, pp. 95–106 (2010)
Björklund, A., Husfeldt, T., Taslaman, N.: Shortest cycle through specified elements. In: SODA, pp. 1747–1753 (2012)
Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory, vol. 7. Springer Science & Business Media, Heidelberg (2013)
Chauhan, A., Rao, B.V.R.: Parameterized analogues of probabilistic computation. In: CALDAM, pp. 181–192 (2015)
Chen, Z., Fu, B., Liu, Y., Schweller, R.T.: On testing monomials in multivariate polynomials. Theor. Comput. Sci. 497, 39–54 (2013)
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
Engels, C.: Why are certain polynomials hard?: a look at non-commutative, parameterized and homomorphism polynomials. Ph.D. thesis, Saarland University (2016)
Fischer, I.: Sums of like powers of multivariate linear forms. Mathemat. Mag. 67(1), 59–61 (1994)
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)
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
Gupta, A., Kamath, P., Kayal, N., Saptharishi, R.: Arithmetic circuits: a chasm at depth three. In: FOCS 2013, pp. 578–587. IEEE (2013)
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
Müller, M.: Parameterized randomization. Ph.D. thesis, Albert-Ludwigs-Universität Freiburg im Breisgau (2008)
Nisan, N.: Lower bounds for non-commutative computation. In: STOC, pp. 410–418. ACM (1991)
Raz, R.: Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM 56(2) (2009)
Raz, R., Shpilka, A.: Deterministic polynomial identity testing in non-commutative models. Comput. Complex. 14(1), 1–19 (2005)
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
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
Schwartz, J.T.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM (JACM) 27(4), 701–717 (1980)
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
Tavenas, S.: Improved bounds for reduction to depth 4 and depth 3. Inf. Comput. 240, 2–11 (2015)
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)
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
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Ghosal, P., Prakash, O., Rao, B.V.R. (2017). On Constant Depth Circuits Parameterized by Degree: Identity Testing and Depth Reduction. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-62389-4_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62388-7
Online ISBN: 978-3-319-62389-4
eBook Packages: Computer ScienceComputer Science (R0)