Branch and Cut∗

Combinatorial optimization problems can often be formulated as mixed integer linear programming problems, as discussed in Section 1.4.1.1 in this encyclopedia. They can then be solved using branch-and-cut, which is an exact algorithm combining branch-and-bound (see Section 1.4.1.2 of this encyclopedia) and cutting planes (see Section 1.4.3 of this encyclopedia). The basic idea is to take a linear programming relaxation of the problem, solve the relaxation, and then either improve the relaxation by adding additional valid constraints, or split the problem into two or more subproblems and repeat the process. Gomory first proposed strengthening linear programming relaxations of integer programming problems by incorporating extra constraints (or cutting planes) in the 1950’s [28]. These cutting planes are derived from the optimal simplex tableau, so they are broadly applicable. However, they fell into disfavor for many years because they seemed to get stuck and run out of power. The cuts are now used in more sophisticated ways and are incorporated into the major commercial packages for integer programming. These packages also include several other families of general cutting planes. General cutting planes are discussed in section 1 of this entry and also in the entries in section 1.4.3 of the Encyclopedia. Land and Doig [42] proposed a branch-and-bound approach in 1960. In branch-andbound, the linear programming relaxation of the integer program is solved. If the solution is ∗To appear in the Wiley Encyclopedia of Operations Research and Management Science †Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12180-1590, U.S.A. Email:[email protected]. The work of this author was supported in part by the National Science Foundation under grant DMS-0715446.