Many discrete optimization problems -- including low-rank quadratic unconstrained binary optimization (QUBO), trimmed covariance, and cardinality-constrained pseudo-Boolean programs -- share a common structure: the gain of any elementary move (bit-flip or swap) can be expressed as an affine form in a low-dimensional feature map $\phi(x) \in \mathbb{R}^R$. We call this the Affine--Predicate Condition, and show that it captures all rank-structured pseudo-Boolean objectives, including higher-order forms via multilinear lifting. We prove an Affine--Predicate Meta--Theorem: under this condition, every locally optimal solution must lie in a chamber of a central hyperplane arrangement in $\mathbb{R}^{R+1}$, defined by the zero-sets of these affine gains. These chambers can be enumerated in $m^{O(R)}$ time, where $m$ is the number of elementary feasible directions. Consequently, exact optimization is in XP time $n^{O(R)}$ in the effective rank $R$, even under uniform matroid (cardinality) constraints. As a corollary, we obtain the first unified $n^{O(r)}$ exact algorithm for rank-$r$ QUBO and its cardinality-constrained extension. This establishes rank as a structural parameter for discrete optimization, analogous to treewidth in algorithmic graph theory and Courcelle-type meta-theorems.

翻译：许多离散优化问题——包括低秩二次无约束二进制优化（QUBO）、修剪协方差以及基数约束伪布尔规划——共享一个共同结构：任何基本移动（位翻转或交换）的增益可以表示为低维特征映射 $\phi(x) \in \mathbb{R}^R$ 中的仿射形式。我们称此为仿射-谓词条件，并证明该条件囊括了所有秩结构化的伪布尔目标函数，包括通过多线性提升得到的高阶形式。我们证明了一个仿射-谓词元定理：在此条件下，每个局部最优解必须位于 $\mathbb{R}^{R+1}$ 中一个中心超平面构型的腔室内，该构型由这些仿射增益的零集定义。这些腔室可在 $m^{O(R)}$ 时间内枚举，其中 $m$ 是基本可行方向的数量。因此，即使在均匀拟阵（基数）约束下，精确优化在有效秩 $R$ 参数下的时间复杂度属于 XP 类 $n^{O(R)}$。作为推论，我们首次为秩-$r$ QUBO 及其基数约束扩展问题提出了统一的 $n^{O(r)}$ 精确算法。这确立了秩作为离散优化的结构参数，类似于算法图论中的树宽以及库尔塞勒型元定理所起的作用。