Problem
Research questions and friction points this paper is trying to address.
permanent
deterministic approximation
nonnegative matrix
Bethe approximation
Beyond the Bethe Approximation of the Permanent
🤖 AI Summary
本文改进了非负矩阵永久值的确定性近似算法，通过新的证明方法将近似因子从$(\sqrt{2})^n$降低至$c^n$（其中$c<\sqrt{2}$），突破了经典Bethe近似的限制。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
deterministic approximation
permanent of nonnegative matrix
Bethe approximation improvement
new certificate
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2023-12-17Bulletin of the American Mathematical SocietyCitations: 59
