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On TC0 Lower Bounds for the Permanent

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Computing and Combinatorics (COCOON 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7434))

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Abstract

In this paper we consider the problem of proving lower bounds for the permanent. An ongoing line of research has shown super-polynomial lower bounds for slightly-non-uniform small-depth threshold and arithmetic circuits [1,2,3,4]. We prove a new parameterized lower bound that includes each of the previous results as sub-cases. Our main result implies that the permanent does not have Boolean threshold circuits of the following kinds.

  1. 1

    Depth O(1), poly − log(n) bits of non-uniformity, and size s(n) such that for all constants c, s (c)(n) < 2n. The size s must satisfy another technical condition that is true of functions normally dealt with (such as compositions of polynomials, logarithms, and exponentials).

  2. 2

    Depth o(loglogn), poly − log(n) bits of non-uniformity, and size n O(1).

  3. 3

    Depth O(1), n o(1) bits of non-uniformity, and size n O(1).

Our proof yields a new “either or” hardness result. One instantiation is that either NP does not have polynomial-size constant-depth threshold circuits that use n o(1) bits of non-uniformity, or the permanent does not have polynomial-size general circuits.

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Authors and Affiliations

  1. Indiana State University, USA

    Jeff Kinne

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  1. Jeff Kinne
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Editors and Affiliations

  1. School of IT, University of Sydney, Building J12, 2006, Sydney, NSW, Australia

    Joachim Gudmundsson , Julián Mestre  & Taso Viglas ,  & 

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Kinne, J. (2012). On TC0 Lower Bounds for the Permanent. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_36

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  • DOI: https://doi.org/10.1007/978-3-642-32241-9_36

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32240-2

  • Online ISBN: 978-3-642-32241-9

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