正确 $e^{(\gamma+o(1))n}$ 近似计算正半定矩阵的永久值
数据结构与算法
2026-05-22 v1
摘要
我们确定在指数中下界阶梯情况下，永久值正半定矩阵的最佳确定性多项式时间近似比率。如果 A ⪰ 0 A\succeq 0 A ⪰ 0 d = rank ( A ) d=\operatorname{rank}(A) d = rank ( A ) A = V V † A=VV^\dagger A = V V † V ∈ C n × d V\in\mathbb{C}^{n\times d} V ∈ C n × d v 1 , … , v n v_1,\ldots,v_n v 1 , … , v n V V V Φ ( V ) = max X ≻ 0 { ∑ i = 1 n log ( v i † X v i ) + log det X − tr X + d } , P ^ ( A ) = e Φ ( V ) . \Phi(V)=\max_{X\succ 0} \left\{\sum_{i=1}^n \log(v_i^\dagger Xv_i)+\log\det X-\operatorname{tr} X+d\right\}, \qquad \widehat P(A)=e^{\Phi(V)}. Φ ( V ) = X ≻ 0 max { i = 1 ∑ n log ( v i † X v i ) + log det X − tr X + d } , P ( A ) = e Φ ( V ) . e − γ n P ^ ( A ) ≤ per ( A ) ≤ P ^ ( A ) . e^{-\gamma n}\widehat P(A)\le \operatorname{per}(A)\le \widehat P(A). e − γ n P ( A ) ≤ per ( A ) ≤ P ( A ) . γ \gamma γ ε > 0 \varepsilon>0 ε > 0 e ( γ + ε ) n e^{(\gamma+\varepsilon)n} e ( γ + ε ) n e ( γ − ε ) n e^{(\gamma-\varepsilon)n} e ( γ − ε ) n P ≠ N P \mathrm{P}\ne\mathrm{NP} P = NP e ( γ + o ( 1 ) ) n e^{(\gamma+o(1))n} e ( γ + o ( 1 )) n γ \gamma γ E [ log T ] = − γ \mathbb{E}[\log T]=-\gamma E [ log T ] = − γ T ∼ Exp ( 1 ) T\sim\operatorname{Exp}(1) T ∼ Exp ( 1 )
引用
@article{arxiv.2605.21946,
title = {Optimal $e^{(\gamma+o(1))n}$-Approximation of the Permanent of Positive Semidefinite Matrices},
author = {Nima Anari and Farzam Ebrahimnejad},
journal= {arXiv preprint arXiv:2605.21946},
year = {2026}
}