Branch-and-Cut Algorithms for Combinatorial Optimization and Their Implementation in ABACUS

Branch-and-cut (-and-price) algorithms belong to the most successful techniques for solving mixed integer linear programs and combinatorial optimization problems to optimality (or, at least, with certified quality). In this unit, we concentrate on sequential branch-and-cut for hard combinatorial optimization problems, while branch-and-cut for general mixed integer linear programming is treated in [? Martin] and parallel branch-and-cut is treated in [? Ladanyi/Ralphs/Trotter]. After telling our most recent story ofa successful application of branch-and-cut in Section 1, we give in Section 2 a brief review ofthe history, including the contributions of pioneers with an emphasis on the computational aspects of their work. In Section 3, the components ofa generic branch-and-cut algorithm are described and illustrated on the traveling salesman problem. In Section 4, we first elaborate a bit on the important separation problem where we use the traveling salesman problem and the maximum cut problem as examples, then we show how branch-and-cut can be applied to problems with a very large number of variables (branch-and-cut-and-price). Section 5 is devoted to the design and applications of the ABACUS software framework for the implementation of branch-and-cut algorithms. Finally, in Section 6, we make a few remarks on the solution of the exercise consisting of the design of a simple TSP-solver in ABACUS.