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.
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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).
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2
Depth o(loglogn), poly − log(n) bits of non-uniformity, and size n O(1).
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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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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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