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Counting the Number of Perfect Matchings, and Generalized Decision Trees

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Abstract

We consider a generalization of the Pólya–Kasteleyn approach to counting the number of perfect matchings in a graph based on computing the symbolic Pfaffian of a directed adjacency matrix of the graph. Complexity of algorithms based on this approach is related to the complexity of the sign function of a perfect matching in generalized decision tree models. We obtain lower bounds on the complexity of the sign of a perfect matching in terms of the deterministic communication complexity of the XOR sign function of a matching. These bounds demonstrate limitations of the symbolic Pfaffian method for both the general case and the case of sparse graphs.

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Notes

  1. Of course, we may use arbitrary polynomials as factors. However, no way to apply this more general construction is seen so far. Therefore, we confine ourselves with monomials here.

  2. Informally, the sense of the logarithmic factor is to reduce the general model to a model with queries admitting binary answers.

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Funding

The research was carried out at the expense of the Russian Science Foundation, project no. 20-11-20203.

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Authors and Affiliations

  1. National Research University—Higher School of Economics, Moscow, Russia

    M. N. Vyalyi

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Translated from Problemy Peredachi Informatsii, 2021, Vol. 57, No. 2, pp. 51–70 https://doi.org/10.31857/S0555292321020042.

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Vyalyi, M. Counting the Number of Perfect Matchings, and Generalized Decision Trees. Probl Inf Transm 57, 143–160 (2021). https://doi.org/10.1134/S0032946021020046

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