Knapsack-Based Cutting Planes for the Max-Cut Problem

We present a new procedure for generating cutting planes for the max-cut problem. The procedure consists of three steps. First, we generate a violated (or near-violated) linear inequality that is valid for the semidefinite programming (SDP) relaxation of the max-cut problem. This can be done by computing the minimum eigenvalue of a certain matrix. Second, we use this linear inequality to construct a ‘knapsack relaxation’ of the given max-cut instance. Third, we generate cutting planes that are valid for the knapsack relaxation, using existing techniques from the literature on the knapsack problem. The procedure enables us to obtain upper bounds that are comparable with those obtained with SDP, but without using an SDP solver.