Complexity results for the gap inequalities for the max-cut problem

Abstract

In 1996, Laurent and Poljak introduced an extremely general class of cutting planes for the max-cut problem, called gap inequalities. We prove several results about them, including the following: (i) there must exist non-dominated gap inequalities with gap larger than 1, unless = co-; (ii) there must exist non-dominated gap inequalities with exponentially large coefficients, unless = co-; (iii) the separation problem for gap inequalities can be solved in finite time (specifically, doubly exponential time).

A guide to conic optimisation and its applications

“…For example, from a consideration of 3-by-3 principal submatrices of Z, it can be shown that a point in the projection cannot violate any of the triangle inequalities (10), (11) by more than 1/4. This provides a partial explanation for the strength of the SDP bound; see, e.g., [8,29,57,58] for more details.…”