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Some observations on holographic algorithms

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Abstract

We define the notion of diversity for families of finite functions and express the limitations of a simple class of holographic algorithms, called elementary algorithms, in terms of limitations on diversity. We show that this class of elementary algorithms is too weak to solve the Boolean circuit value problem, or Boolean satisfiability, or the permanent. The lower bound argument is a natural but apparently novel combination of counting and algebraic dependence arguments that is viable in the holographic framework. We go on to describe polynomial time holographic algorithms that go beyond the elementarity restriction in the two respects that they use exponential size fields, and multiple oracle calls in the form of polynomial interpolation. These new algorithms, which use bases of three components, compute the parity of the following quantities for degree three planar undirected graphs: the number of 3-colorings up to permutation of colors, the number of connected vertex covers, and the number of induced forests or feedback vertex sets. In each case, the parity can also be computed for any one slice of the problem, in particular for colorings where the first color is used a certain number of times, or where the connected vertex cover, feedback set or induced forest has a certain number of nodes.

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  1. Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, 02138, USA

    Leslie G. Valiant

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  1. Leslie G. Valiant
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Correspondence to Leslie G. Valiant.

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Valiant, L.G. Some observations on holographic algorithms. comput. complex. 27, 351–374 (2018). https://doi.org/10.1007/s00037-017-0160-4

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  • DOI: https://doi.org/10.1007/s00037-017-0160-4

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