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Branch and cut [1] is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer values. [2] Branch and cut involves running a branch and bound algorithm and using cutting planes to tighten
The following is the skeleton of a generic branch and bound algorithm for minimizing an arbitrary objective function f. [3] To obtain an actual algorithm from this, one requires a bounding function bound, that computes lower bounds of f on nodes of the search tree, as well as a problem-specific branching rule.
Branch and bound; Branch and cut; Branch and price; if the problem has some extra structure, it may be possible to apply delayed column generation. Such integer-programming algorithms are discussed by Padberg and in Beasley.
The related branch and cut method combines the cutting plane and branch and bound methods. In any subproblem, it runs the cutting plane method until no more cutting planes can be found, and then branches on one of the remaining fractional variables.
Branch and price is a branch and bound method in which at each node of the search tree, columns may be added to the linear programming relaxation (LP relaxation). At the start of the algorithm, sets of columns are excluded from the LP relaxation in order to reduce the computational and memory requirements and then columns are added back to the LP relaxation as needed.
The knapsack problem has well-known methods to solve it, such as branch and bound and dynamic programming. The Delayed Column Generation method can be much more efficient than the original approach, particularly as the size of the problem grows.
Cutting planes were proposed by Ralph Gomory in the 1950s as a method for solving integer programming and mixed-integer programming problems. However, most experts, including Gomory himself, considered them to be impractical due to numerical instability, as well as ineffective because many rounds of cuts were needed to make progress towards the solution.
[a] Branch (or branching, branched) may also refer to the act of switching execution to a different instruction sequence as a result of executing a branch instruction. Branch instructions are used to implement control flow in program loops and conditionals (i.e., executing a particular sequence of instructions only if certain conditions are ...