Branch-and-cut Algorithms for Integer Programming, Branch-and-cut

Branch-and-cut methods are exact algorithms for integer programming problems. They consist of a combination of a cutting plane method with a branch-and-bound algorithm. These methods work by solving a sequence of linear programming relaxations of the integer programming problem. Cutting plane methods improve the relaxation of the problem to more closely approximate the integer programming problem, and branch-and-bound algorithms proceed by a sophisticated divide and conquer approach to solve problems. The material in this entry builds on the material contained in the entries on cutting plane and branch-and-bound methods. Perhaps the best known branch-and-cut algorithms are those that have been used to solve traveling salesman problems. This approach is able to solve and prove optimality of far larger instances than other methods. Two papers that describe some of this research and also serve as good introductions to the area of branch-and-cut algorithms are [21, 32]. A more recent work on the branch-and-cut approach to the traveling salesman problem is [1]. Branch-and-cut methods have also been used to solve other combina-torial optimization problems; recent references include [8, 10, 13, 23, 24, 26]. For these problems, the cutting planes are typically derived from studies of the polyhedral combinatorics of the corresponding integer program. This enables the addition of strong cutting planes (usually facet defining inequalities), which make it possible to considerably reduce the size of the branch-and-bound tree. Far more detail about these strong cutting planes can be found elsewhere in this encyclopedia , for example in the entry on cutting plane methods for integer programming. Branch-and-cut methods for general integer programming problems are also of great interest It is usually not possible to efficiently solve a general integer programming problem using just a cutting plane approach, and it is therefore necessary to also to branch, resulting in a branch-and-cut approach. A pure branch-and-bound approach can be sped up considerably by the employment of a cutting plane scheme, either just at the top of the tree, or at every node of the tree. For general problems, the specialized facets used when solving a specific combinatorial optimization problem are not available. Some useful families of general inequalities have been developed; these include cuts based on knap-and Fenchel cutting planes [9]. All of these families of cutting planes are discussed in more detail later in this entry. The software packages MINTO [30] and ABACUS [28] implement branch-and-cut algorithms to solve integer programming problems. The …