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Optimal e^((γ+o(1))n)-Approximation of the Permanent of Positive Semidefinite Matrices

Nima Anari, Farzam Ebrahimnejad

For an n×n Hermitian positive semidefinite matrix A, an entropy-corrected concave Gram-factor relaxation P̂ satisfies e^(−γn)P̂≤per(A)≤P̂ and gives a deterministic polynomial-time e^((γ+ε)n)-approximation for every ε>0. The exponent is optimal up to lower-order terms under the cited NP-hardness, assuming P≠NP.

Sources for this record

What was already known

Anari and coauthors had an e^((γ+1)n) deterministic approximation. Ebrahimnejad, Nagda and Oveis Gharan reduced the upper exponent to γ+0.9999 and showed hardness below γ−ε.

AI involvement AI-autonomous

In a one-shot exchange without mathematical hints, GPT-5.5 Pro Extended produced the theorem, relaxation and proof for Nima Anari. Farzam Ebrahimnejad independently reached the same result in a guided exchange. Both authors report verifying the mathematics and explain the method; Codex typeset the manuscript.

The first interaction supplied only prior work and the target ratio; no hints or follow-up mathematical input preceded the model’s full result.

Importance Notable

Reports closure of the known approximation-exponent gap for PSD permanents.

Limitations

The algorithm attains γ+ε for any fixed ε>0, not an exact γ exponent at finite n. Optimality is conditional on P≠NP; it concerns PSD permanents, not arbitrary matrices.

Technical verification Formal (human)

Both authors explicitly report verifying the complete theorem and proof. This is a source-reported human check; no machine proof or independent replay is documented.

Formal checking

Both authors verified the mathematics after separate model interactions

Checker
Nima Anari and Farzam Ebrahimnejad, paper authors
Scope
Theorem 1 PSD permanent sandwich and deterministic approximation
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

The human authors explain the entropy-corrected relaxation through the Gaussian moment, Gibbs variational formula and matching upper/lower bounds.

Correction and publication history