While several classes of integer linear optimization problems are known to be solvable in polynomial time, far fewer tractability results exist for integer nonlinear optimization. In this work, we narrow this gap by identifying a broad class of discrete nonlinear optimization problems that admit polynomial-time algorithms. Central to our approach is the notion of projection-width, a structural parameter for systems of separable constraints, defined via branch decompositions of variables and constraints. We show that several fundamental discrete optimization and counting problems can be solved in polynomial time when the projection-width is polynomially bounded, including optimization, counting, top-k, and weighted constraint violation problems. Our results subsume and generalize some of the strongest known tractability results across multiple research areas: integer linear optimization, binary polynomial optimization, and Boolean satisfiability. Although these results originated independently within different communities and for seemingly distinct problem classes, our framework unifies and significantly generalizes them under a single structural perspective.

翻译：尽管已知多类整数线性优化问题可在多项式时间内求解，但整数非线性优化的可处理性结果却少得多。本研究通过识别一类可接受多项式时间算法的离散非线性优化问题，缩小了这一差距。我们方法的核心是投影宽度的概念，这是一种针对可分离约束系统的结构参数，通过变量与约束的分支分解来定义。我们证明，当投影宽度具有多项式上界时，若干基本离散优化与计数问题可在多项式时间内求解，包括优化、计数、top-k及加权约束违反问题。我们的结果涵盖并推广了多个研究领域（整数线性优化、二元多项式优化及布尔可满足性）中已知的最强可处理性结果。尽管这些结果最初在不同研究领域针对看似不同的问题类别独立提出，但我们的框架在单一结构视角下将它们统一并显著推广。