We present the first uniform XP exact algorithm for unconstrained binary optimization of quadratic, polynomial, fractional, and other objectives under a single parameter, the differentially affine (DA) rank $r$. An objective $f: \{0,1\}^n \to \mathbb{R}$ has DA rank $r$ if there is a feature map $ψ: \{0,1\}^n \to \mathbb{R}^r$ such that each coordinate flip has finite gain $Δ_{\pm e_i}f(x)=\langle v_{\pm e_i},ψ(x)\rangle+β_{\pm e_i}$. Our algorithm enumerates the $O((2n)^r)$ chambers of the induced hyperplane arrangement and applies a two-sided local-optimality test: a solution exists on a chamber and is unique iff $\operatorname{sign}Δ_{+e_i}=-\operatorname{sign}Δ_{-e_i}$ for all $i$, in which case $x_i^\star=1$ iff $Δ_{+e_i}>0$. This yields $n^{O(r)}$ time with $O(n)$ decoding per chamber. The framework uniformly covers a wide range of nonlinear functions, including all rank-$r$ quadratics, low-Waring-rank pseudo-Boolean polynomials, finite products/ratios on positive domains, finite-basis separable sums via explicit lifts, Taylor-series approximations of analytic functions, and compositions of all the foregoing. Applications include Ising spin models, optimal experimental design, portfolio optimization, and robust statistics. Prior to our work, only specialized subcases involving sparsity, convexity, submodularity, etc. were known to be tractable. Analogous in spirit to Courcelle's theorem (MSO on bounded treewidth graphs) and Grohe's meta-theorems for constraint satisfaction, our result replaces logical width with analytic rank for nonlinear pseudo-Boolean optimization.

翻译：我们提出了首个统一的XP精确算法，用于在单一参数——微分仿射（DA）秩$r$下，对二次、多项式、分式及其他目标函数进行无约束二元优化。目标函数$f: \\{0,1\\}^n \\to \\mathbb{R}$具有DA秩$r$，若存在特征映射$\\psi: \\{0,1\\}^n \\to \\mathbb{R}^r$，使得每个坐标翻转具有有限增益$\\Delta_{\\pm e_i}f(x)=\\langle v_{\\pm e_i},\\psi(x)\\rangle+\\beta_{\\pm e_i}$。我们的算法枚举诱导超平面排列的$O((2n)^r)$个腔室，并应用双侧局部最优性检验：解存在于某个腔室且唯一当且仅当对所有$i$有$\\operatorname{sign}\\Delta_{+e_i}=-\\operatorname{sign}\\Delta_{-e_i}$，此时$x_i^\\star=1$当且仅当$\\Delta_{+e_i}>0$。这实现了$n^{O(r)}$时间复杂度和每个腔室$O(n)$解码开销。该框架统一覆盖了广泛的非线性函数类别，包括所有秩$r$二次型、低华林秩伪布尔多项式、正域上的有限乘积/比值、通过显式提升的有限基可分离和、解析函数的泰勒级数逼近，以及前述所有类型的复合。应用领域包括伊辛自旋模型、最优实验设计、投资组合优化和稳健统计。在我们的工作之前，仅已知涉及稀疏性、凸性、次模性等特例是可处理的。与库尔切勒定理（有界树宽图上的MSO逻辑）和格罗赫针对约束满足的元定理在精神上类似，我们的结果用解析秩替代了逻辑宽度，以处理非线性伪布尔优化问题。