Contraction-based Separation and Lifting for Solving the Max-Cut Problem

Abstract

The max-cut problem is an NP-hard combinatorial optimization problem defined on undirected weighted graphs. It consists in finding a subset of the graph's nodes such that the aggregate weight of the edges between the subset and its complement is maximized. In this doctoral thesis we present a new separation approach to be used within a branch-and-cut algorithm for solving max-cut problems to optimality. This method is based on graph contraction and allows the fast separation of so-called odd-cycle inequalities. In addition, we describe techniques to add possibly missing edges to an already contracted graph. This allows solving max-cut problems on sparse graphs by using methods that were originally intended for complete graphs and could not have been applied otherwise. We investigate the theoretical aspects of this combined approach and also explain its realization within a branch-and-cut framework. Finally, we evaluate the performance of our separation procedure on a variety of test instances.