AI 中文总结
研究二元多项式优化问题，通过专注于一般变量消除方案，开发等效问题的显式多线性多项式形式，刻画新特殊类别，为广泛问题类别提供有效解决方案，概括多种可处理特殊情况。
AI 中文摘要
在二元多项式优化问题（BPO）中，我们要最大化一个依赖于n个二元变量的多线性多项式表达式。这是一个困难的优化类别，包含许多NP难问题，如无约束二次二元优化。文献中考虑了几个可处理的特殊类别。我们专注于BPO的一般变量消除方案，为消除给定变量子集后得到的等效BPO问题开发独特的显式多线性多项式形式。这种等效BPO问题的封闭形式表示使我们能够刻画新的特殊类别，递归应用消除方法可提供高效计算解决方案。我们的方法是基础的、代数的，并为广泛的问题类别提供有效解决方案，恰当地概括了上述所有可处理的特殊情况。
英文摘要
In a binary polynomial optimization problem (BPO, in short) we are maximizing a multilinear polynomial expression depending on n binary variables. This is a hard optimization class, containing many NP-hard problems, including unconstrained quadratic binary optimization. Several tractable special classes were considered in the literature, including problems with bounded tree-width (Crama, Hansen, Jaumard, 1990), Berge-acyclic problems (Buchheim, Crama, and Heck, 2019), $β$-acyclic problems (Del Pia and Di Gregorio, 2022, 2023), limited reach problems (Clausen, Crama, Lusby, Rodríguez, and Ropke, 2024), and $α$-acyclic problems with bounded rank (Del Pia and Khajavirad, 2025). We focus on a general variable elimination scheme for BPO, and develop the unique explicit multi-linear polynomial form for the equivalent BPO problem obtained after the elimination of a given subset of the variables. The obtained closed form representation of such an equivalent BPO problem allows us to characterize new special classes for which this elimination method, when applied recursively, provides a computationally efficient solution. Our approach is elementary1, algebraic, and provides efficient solution to a wide problem class that properly generalizes all of the above mentioned tractable special cases.
