branchandboundprosess

branchandboundprosess, often written as the branch-and-bound process, is a family of exact algorithms used to solve discrete optimization problems, especially those with integer variables. It systematically explores a search tree and uses bounds to prune branches that cannot contain an optimal solution. It is widely used in operations research and computer science for problems such as knapsack, traveling salesman, scheduling, and network design.

The method starts with a root node representing the original problem. Each node corresponds to a subproblem with some decisions fixed. For every node, a bound is computed: a lower bound for minimization (or an upper bound for maximization) on the objective value achievable within that subproblem. If this bound is worse than the best known feasible solution, the node is pruned. If the subproblem is feasible, a feasible solution can be obtained and may improve the best bound. If not pruned, the node is expanded by branching on a decision variable or constraint, producing child subproblems. A node selection strategy, such as best-bound first or depth-first, determines the order of exploration.

Relaxations, such as linear programming relaxations or other convex relaxations, are commonly used to compute tight bounds. In many implementations, branch-and-bound is augmented with cutting planes (branch-and-cut) to strengthen relaxations or with column generation (branch-and-price) to handle large numbers of variables.

Applications include vehicle routing, personnel scheduling, facility location, production planning, and various graph and set-partitioning problems. Although branchandboundprosess provides exact optima, its worst-case complexity is exponential and memory usage can be significant. Effective bounding strategies, heuristics, and problem-specific insights are essential for practical performance in modern solvers.