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 finite sequence of integers, the ‘gap’ being defined as the smallest discrepancy arising when decomposing the sequence into two parts that are as equal as possible. Gap inequalities include hypermetric inequalities and negative type inequalities, which have been extensively studied in the literature. They are also related to a positive semidefinite relaxation of the max-cut problem. ral question is to decide for which integer sequences the corresponding gap inequalities define facets of the cut polytope. For this property, we present a set of necessary and sufficient conditions in terms of the root patterns and of the rank of an associated matrix. We also prove that there is no facet defining inequality with gap greater than one and which is induced by a sequence of integers using only two distinct values.