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)


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.


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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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