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.
We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\sqrt 2)^n$. The simp...
We give the first deterministic polynomial time algorithm that approximates the permanent of arbitrary nonnegative rational matrices within a subexponential factor. For a matrix of order $n$, the approximation factor is \[ \exp\!\left(O\!\left(\frac{n(\log\log n)^2}{\log n}\right)\right)=\exp(o(n)). \] All previously k...
For even $n$, let $P_n$ be the probability that independent uniform strict preference lists on $n$ participants admit a stable perfect matching. We prove \[ P_n\sim\frac{e\,2^{1/4}\Gamma(3/4)}{\sqrt\pi}\,n^{-1/4}. \] This establishes Mertens's conjectured exponent of decay, with a leading constant different from his or...
We give a computer-assisted proof that the universal real Grothendieck constant satisfies $K_G^{\mathbb R}\le 1.7813$. The construction combines an explicit odd Hermite threshold of degree $11$ with a signed correlation polynomial of degree $51$. A sufficient inverse-majorant inequality is certified by enclosing the sc...
For an $n\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{n/2}\operatorname{Bethe}(A).\] The lower bound, due to Gurvits, is attained on forests. The upper bo...
Let $M_n$ be an $n\times n$ matrix with independent uniform sign entries. We prove that there exist absolute constants $C,c>0$ such that, for all sufficiently large $n$, \[ \mathbb{P}\!\left( \left|\operatorname{Per}(M_n)\right| \ge e^{-Cn}\sqrt{n!} \right) \ge 1-n^{-c}. \] This establishes the exponential scale lower...
Yiming Chen· 0 citations
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