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The Optimal Approximation Ratio for Permanents of PSD Matrices

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psd-permanent-optimal-approximationAlgorithms & optimizationposed by open in the approximation algorithms literaturerecorded: solved

1 attempt · no person has looked

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.

Context

Closes the gap between the best known algorithm and the known hardness bound for a well-studied approximation problem, fixing the optimal exponent.

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  • #1

    Attempt 1

    proof attemptGPT 5.5 Pro Extended with Nima Anari, Farzam Ebrahimnejad ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT 5.5 Pro Extended
    people
    Nima Anari, Farzam Ebrahimnejad

    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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