超越永久值的贝特近似
Beyond the Bethe Approximation of the Permanent
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中文总结 AI 辅助
该研究改进了永久值确定性近似的指数误差因子底数，证明经典贝特近似并非障碍，其证明借助了ChatGPT 5.6 Sol Pro完成，Codex协助了相关工作。
中文摘要 AI 辅助
经典贝特近似可对所有非负矩阵的永久值给出确定性近似，误差因子为$(\root\from{2})^n$。我们改进了该指数因子的底数：存在某个绝对常数$c<\root\from{2}$，对所有非负矩阵的永久值，存在确定性多项式时间的$c^n$近似算法。这表明经典贝特近似的保证并非永久值确定性近似的障碍。该证明针对贝特下界损失近全部因子的矩阵，补充了新的证明依据。作者提供了高层攻击方案，证明是在与ChatGPT 5.6 Sol Pro的交互中完成的，作者随后验证了结果，Codex协助了证明检查、手稿整理和排版。
英文摘要
The canonical Bethe approximation gives a deterministic approximation to the permanent of every nonnegative matrix within a factor of $(\sqrt{2})^n$. We improve the base of this exponential factor: for some absolute constant $c<\sqrt{2}$, there is a deterministic polynomial-time $c^n$-approximation for the permanent of every nonnegative matrix. This shows that the canonical Bethe guarantee is not a barrier for deterministic approximation of the permanent. The proof augments the Bethe lower bound with a new certificate tailored to matrices on which that lower bound loses nearly the full factor. The author supplied the high-level plan of attack, and the proof was developed in an interaction with ChatGPT 5.6 Sol Pro. The author subsequently verified the results. Codex assisted with proof checking, manuscript assembly, and typesetting.
发表机构
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
