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The Optimal Approximation Ratio for Permanents of PSD Matrices

Algorithms & optimization · posed by open in the approximation algorithms literature · solved

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Statement

What is the best deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix? Resolved up to lower-order terms in the exponent: an explicit concave maximisation P^(A)\widehat P(A) satisfies e−γnP^(A)≤per(A)≤P^(A)e^{-\gamma n}\widehat P(A) \le \mathrm{per}(A) \le \widehat P(A), giving a deterministic e(γ+ε)ne^{(\gamma+\varepsilon)n}-approximation for every ε>0\varepsilon > 0 and matching the known e(γ−ε)ne^{(\gamma-\varepsilon)n} hardness, where γ\gamma is the Euler-Mascheroni constant.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Nima Anari and Farzam Ebrahimnejad, using GPT 5.5 Pro Extended

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    ai co developed
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    The authors describe two different interaction styles converging on the same result: the first author's interaction was one-shot, the second author's involved high-level guidance. Both state they verified the theorem and proof themselves. Codex was used separately to assemble and typeset the manuscript, and the disclosure keeps that clerical use distinct from the mathematics.

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