发表机构
Carnegie Mellon University（卡内基梅隆大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对两个稠密计数问题提出确定性FPTAS，采用高斯截断原理改进算法效率，将Cuckler-Kahn误差优化并实现多项式枚举，解决了原有算法拟多项式时间的问题。
AI 中文摘要
我们针对两个稠密计数问题给出了确定性FPTAS（完全多项式时间近似方案），而已知的基于无零点插值的确定性算法在这些问题上的运行时间为拟多项式时间。对于固定的0<γ<1/2和0<θ≤1，第一个问题是：当对称矩阵A的支撑图G的最小度至少为(1/2+γ)n，且其非零元素位于[θ,1]区间内时，近似计算haf(A)；还在类似的二分图条件下近似计算行列式，包括支撑完全在[θ,1]区间内的矩阵。对于固定的β>0和0<κ≤1，第二个问题是：当零对角实对称矩阵J满足max_{i,j}|J_{ij}|≤β/n且λ_max(J)≤1−κ时，近似计算零场伊辛配分函数Z(J)，无需单独施加低特征值条件。我们进一步证明log haf(A)=h_A(G)−n/2+O_{γ,θ}(1)，以及Z(J)=2^n det(I−J)^{-1/2}(1+O_{β,κ}(1/n))，其中h_A(G)是最大加权分数匹配熵。对于无权图，第一个公式在固定边际类上将Cuckler-Kahn误差从o(n)改进为O_γ(1)，并将其扩展到[θ,1]区间内的权重。两种算法均采用共同的高斯截断原理：每个问题变为固定整函数在高斯坐标上的乘积积分，其矩矩阵元素可能为阶1/n的不定项；抵消线性项并精确重求和二次项后，剩余坐标余项至少为三阶消失；复伸缩处理小支撑，大支撑时通过大偏差率界定重组尾部，该速率优于子集熵；截断误差至多为(CR/n)^{R/2}+e^{-cn}，这种快于几何的衰减允许R log(en/R)=O(log n + log(1/ε))，从而实现多项式枚举。
英文摘要
We give deterministic FPTASes for two dense counting problems on which the known deterministic algorithms, based on zero-free interpolation, run in quasipolynomial time. For fixed $0<γ<1/2$ and $0<θ\leq1$, the first approximates $\mathrm{haf}(A)$ for a symmetric matrix $A$ when its support graph $G$ has minimum degree at least $(1/2+γ)n$ and its nonzero entries lie in $[θ,1]$. It also approximates permanents under the analogous bipartite condition, including full-support matrices in $[θ,1]$. For fixed $β>0$ and $0<κ\leq1$, the second approximates the zero-field Ising partition function $Z(J)$ for zero-diagonal real symmetric matrices $J$ satisfying $\max_{i,j}|J_{ij}|\leqβ/n$ and $λ_{\max}(J)\leq1-κ$. No separate lower-eigenvalue condition is imposed. We further prove $\log\mathrm{haf}(A)=h_A(G)-n/2+O_{γ,θ}(1)$ and $Z(J)=2^n\det(I-J)^{-1/2}(1+O_{β,κ}(1/n))$. Here $h_A(G)$ is the maximum weighted fractional-matching entropy. For unweighted graphs, the first formula improves the Cuckler--Kahn error from $o(n)$ to $O_γ(1)$ on the fixed-margin class and extends it to weights in $[θ,1]$. Both algorithms use a common Gaussian truncation principle. Each problem becomes an integral of a product of a fixed entire function over Gaussian coordinates, with possibly indefinite moment matrix entries of order $1/n$. Cancelling the linear term and exactly resumming the quadratic term leaves a coordinate remainder vanishing to order at least three. Complex dilation handles small supports. For large supports, we bound the recombined tail by a large-deviation rate that beats the entropy of the subsets. The truncation error is at most $(CR/n)^{R/2}+e^{-cn}$. This faster-than-geometric decay permits $R\log(en/R)=O(\log n+\log(1/ε))$ and hence polynomial enumeration.
