We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd $\beta$-cycle inequalities valid for this polytope, showed that these generally have Chv{\'a}tal rank 2 with respect to the standard relaxation and that, together with flower inequalities, they yield a perfect formulation for cycle hypergraph instances. Moreover, they describe a separation algorithm in case the instance is a cycle hypergraph. We introduce a weaker version, called simple odd $\beta$-cycle inequalities, for which we establish a strongly polynomial-time separation algorithm for arbitrary instances. These inequalities still have Chv{\'a}tal rank 2 in general and still suffice to describe the multilinear polytope for cycle hypergraphs. Finally, we report about computational results of our prototype implementation. The simple odd $\beta$-cycle inequalities sometimes help to close more of the integrality gap in the experiments; however, the preliminary implementation has substantial computational cost, suggesting room for improvement in the separation algorithm.

翻译：我们考虑二元多项式优化中自然出现的多线性多面体。Del Pia与Di Gregorio引入了对该多面体有效的一类奇$β$-环不等式，证明了这些不等式相对于标准松弛通常具有Chvátal秩2，并且与花形不等式结合可对超循环图实例给出完美刻画。此外，他们描述了当实例为超循环图时的分离算法。我们引入一种弱化版本，称为简单奇$β$-环不等式，并针对任意实例建立了强多项式时间分离算法。这些不等式在一般情况下仍具有Chvátal秩2，且仍足以描述超循环图的多线性多面体。最后，我们报告了原型实现的计算结果。在实验中，简单奇$β$-环不等式有时有助于缩小更多整数性差距；然而，初步实现的计算成本较高，表明分离算法仍有改进空间。