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Computing permanents of complex diagonally dominant matrices and tensors

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Abstract

We prove that for any λ > 1, fixed in advance, the permanent of an n × n complex matrix, where the absolute value of each diagonal entry is at least λ times bigger than the sum of the absolute values of all other entries in the same row, can be approximated within any relative error 0 < ϵ < 1 in quasi-polynomial nO(lnn-lnϵ) time. We extend this result to multidimensional permanents of tensors and apply it to weighted counting of perfect matchings in hypergraphs.

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Acknowledgment

The author is grateful to Alex Samorodnitsky for suggesting several improvements and to Piyush Srivastava for pointing out [7] and for suggesting a simplification of the original proof of Lemma 2.1.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA

    Alexander Barvinok

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  1. Alexander Barvinok
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Correspondence to Alexander Barvinok.

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This research was partially supported by NSF Grant DMS 1361541.

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Barvinok, A. Computing permanents of complex diagonally dominant matrices and tensors. Isr. J. Math. 232, 931–945 (2019). https://doi.org/10.1007/s11856-019-1896-0

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  • DOI: https://doi.org/10.1007/s11856-019-1896-0

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