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The Complexity of Explicit Constructions

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Abstract

The existence of extremal combinatorial objects, such as Ramsey graphs and expanders, is often shown using the probabilistic method. It is folklore that pseudo-random generators can be used to obtain explicit constructions of these objects, if the test that the object is extremal can be implemented in polynomial time. In this paper, we pose several questions geared towards initiating a structural approach to the relationship between extremal combinatorics and computational complexity. One motivation for such an approach is to understand better why circuit lower bounds are hard. Another is to formalize connections between the two areas, so that progress in one leads automatically to progress in the other.

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Acknowledgements

I am grateful to Lance Fortnow and Srikanth Srinivasan for several productive discussions. Lance kindly allowed me to include Theorem 16 in this paper. Thanks also to Tony Tan for pointing me to the paper by Spencer [18].

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  1. University of Edinburgh, Edinburgh, UK

    Rahul Santhanam

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  1. Rahul Santhanam
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Correspondence to Rahul Santhanam.

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Santhanam, R. The Complexity of Explicit Constructions. Theory Comput Syst 51, 297–312 (2012). https://doi.org/10.1007/s00224-011-9368-x

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