Gap Inequalities for the Cut Polytope

We introduce a new class of inequalities valid for the cut polytope, which we call gap inequalities. Each gap inequality is given by a nite sequence of integers, whose \gap" is deened as the smallest discrepancy arising when decomposing the sequence into two parts as equal as possible. Gap inequalities include the hypermetric inequalities and the negative type inequalities, which have been extensively studied in the literature. They are also related to a positive semideenite relaxation of the max-cut problem. A natural question is to decide for which integer sequences the corresponding gap inequalities deene facets of the cut polytope. For this property, we present a set of necessary and suucient conditions in terms of the root patterns and of the rank of an associated matrix. We also prove that there is no facet deening inequality with gap greater than one and which is induced by a sequence of integers using only two distinct values.