A Courcelle-Type Metatheorem for Rank-Bounded Unconstrained Binary Optimization
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 $\psi: \{0,1\}^n \to \mathbb{R}^r$ such that each coordinate flip has finite gain $\Delta_{\pm e_i}f(x)=\langle v_{\pm e_i},\psi(x)\rangle+\beta_{\pm e_i}$. 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 $\operatorname{sign}\Delta_{+e_i}=-\operatorname{sign}\Delta_{-e_i}$ for all $i$, in which case $x_i^\star=1$ iff $\Delta_{+e_i}>0$. 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.