The Binary Polynomial Optimization (BPO) problem is defined as the problem of maximizing a given polynomial function over all binary points. The main contribution of this paper is to draw a novel connection between BPO and the field of Knowledge Compilation. This connection allows us to unify and significantly extend the state-of-the-art for BPO, both in terms of tractable classes, and in terms of existence of extended formulations. In particular, for instances of BPO with hypergraphs that are either $\beta$-acyclic or with bounded incidence treewidth, we obtain strongly polynomial algorithms for BPO, and extended formulations of polynomial size for the corresponding multilinear polytopes. The generality of our technique allows us to obtain the same type of results for extensions of BPO, where we enforce extended cardinality constraints on the set of binary points, and where variables are replaced by literals. We also obtain strongly polynomial algorithms for the variant of the above problems where we seek $k$ best feasible solutions, instead of only one optimal solution. Computational results show that the resulting algorithms can be significantly faster than current state-of-the-art.

翻译：二元多项式优化（BPO）问题定义为在所有二元点上最大化给定多项式函数的问题。本文的主要贡献在于建立BPO与知识编译领域之间的新颖联系。这一联系使我们能够统一并显著扩展BPO领域的最新技术成果，无论是在可处理类别方面，还是在扩展形式的存在性方面。具体而言，对于具有β-无环或有界关联树宽超图的BPO实例，我们获得了BPO的强多项式时间算法，以及对应多线性多胞形的多项式规模扩展形式。我们技术的普适性使我们能够在BPO的扩展问题上获得同类结果，包括对二元点集合施加扩展基数约束的情况，以及用文字替换变量的情况。对于上述问题的变体——寻求k个最优可行解而非单个最优解，我们也获得了强多项式时间算法。计算结果表明，所得算法相比当前最优技术可显著提升求解速度。