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On Enumerating Monomials and Other Combinatorial Structures by Polynomial Interpolation

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We study the problem of generating the monomials of a black box polynomial in the context of enumeration complexity. We present three new randomized algorithms for restricted classes of polynomials with a polynomial or incremental delay, and the same global running time as the classical ones. We introduce TotalBPP, IncBPP and DelayBPP, which are probabilistic counterparts of the most common classes for enumeration problems. Our interpolation algorithms are applied to algebraic representations of several combinatorial enumeration problems, which are so proved to belong to the introduced complexity classes. In particular, the spanning hypertrees of a 3-uniform hypergraph can be enumerated with a polynomial delay. Finally, we study polynomials given by circuits and prove that we can derandomize the interpolation algorithms on classes of bounded-depth circuits. We also prove the hardness of some problems on polynomials of low degree and small circuit complexity, which suggests that our good interpolation algorithm for multilinear polynomials cannot be generalized to degree 2 polynomials. This article is an improved and extended version of Strozecki (Mathematical Foundations of Computer Science, pp. 629–640, 2010) and of the third chapter of the author’s Ph.D. Thesis (Strozecki, Ph.D. Thesis, 2010).

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Acknowledgements

Thanks to Hervé Fournier “l’astucieux”, Guillaume Malod, Sylvain Perifel and Arnaud Durand for their helpful comments about this article. The conversations I had with Meena Mahajan led to improvements of the last section of this article. I would also like to thank an anonymous referee of the conference version of this article for his good suggestions.

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    Yann Strozecki

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Strozecki, Y. On Enumerating Monomials and Other Combinatorial Structures by Polynomial Interpolation. Theory Comput Syst 53, 532–568 (2013). https://doi.org/10.1007/s00224-012-9442-z

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