Branch and bound is a powerful method for solving integer programming problems. It systematically explores solution spaces, using bounds to prune unpromising branches and efficiently find optimal solutions for complex optimization challenges.

This approach is crucial in various applications, from facility location to production planning. By combining relaxation, branching, and bounding techniques, it tackles problems that are intractable through simple enumeration, making it a cornerstone of optimization theory.

Branch and Bound Method Fundamentals

Principles of branch and bound

• Core concept systematically enumerates candidate solutions while pruning subsets using upper and lower bounds (knapsack problem)
• Key steps involve relaxation solving LP relaxation, branching creating subproblems, bounding calculating bounds, pruning eliminating suboptimal subproblems
• Optimality criteria require integrality of solution and comparison of best integer solution to global upper bound

Application to integer programming

• Problem formulation includes objective function, constraints, integer restrictions on variables (facility location)
• Initial LP relaxation removes integrality constraints and solves relaxed problem
• Branching strategies select variables for branching and create two new subproblems
• Bounding techniques use LP relaxation solutions as bounds and update incumbent solution
• Pruning rules eliminate subproblems due to infeasibility, worse bound than incumbent, or optimality when integer solution equals bound

Branch and Bound Analysis

Components of branch and bound trees

• Tree structure consists of root node (initial LP relaxation), internal nodes (subproblems), leaf nodes (final solutions or pruned subproblems)
• Branches represent added constraints and disjunctive constraints for integer variables
• Node information includes objective function value, variable values, bound value
• Tree traversal strategies include depth-first search, breadth-first search, best-bound search

Efficiency vs limitations of method

• Computational complexity involves worst-case exponential time and average-case performance
• Factors affecting efficiency include problem size and structure, branching strategy effectiveness, strength of bounds
• Compares to cutting plane methods and heuristic approaches
• Limitations involve memory requirements for large problems, difficulty with highly symmetric problems, challenges in finding good initial bounds
• Enhancements and variations include branch and cut, branch and price, hybrid algorithms (Benders decomposition)