Answer First
Branch and bound solves integer programming efficiently by exploring a search tree while using LP relaxations to eliminate large portions of the search space. It guarantees optimality without checking every possible combination.
Problem Setup
An integer program restricts some or all variables to integer or binary values. The LP relaxation removes these restrictions: \[ \max c^\top x \quad \text{s.t. } Ax \le b, \; x \ge 0. \] If the LP relaxation gives an integer solution, it is optimal. If not, the algorithm branches on a fractional variable and bounds each branch using LP relaxations.
Step-by-Step Explanation
1. Solve the LP relaxation
This gives an upper bound (for maximization) on the best possible integer solution.
2. Check integrality
If all variables are integers, the solution is optimal.
3. Branch on a fractional variable
Create two subproblems:
- \(x_k \le \lfloor x_k^* \rfloor\)
- \(x_k \ge \lceil x_k^* \rceil\)
4. Bound each branch
If the LP relaxation of a branch is worse than the current best integer solution, prune it.
5. Continue until all branches are pruned or solved
The best integer solution found is guaranteed optimal.
Intuition
Branch and bound is like searching a decision tree with a flashlight: LP relaxations illuminate which branches are promising and which can be ignored.
Common Exam Mistakes
- Incorrect branching constraints.
- Misinterpreting LP relaxation bounds.
- Failing to prune dominated branches.
- Confusing feasible integer solutions with bounds.
Final Summary
Branch and bound solves integer programming by combining LP relaxations with systematic search. It eliminates impossible or unpromising regions and guarantees an optimal integer solution.
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