  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.
