# Branch and bound

Branch and bound (BB) is a general algorithm for finding optimal solutions of various optimization problems, especially in discrete and combinatorial optimization. It consists of a systematic enumeration of all candidate solutions, where large subsets of fruitless candidates are discarded en masse, by using upper and lower estimated bounds of the quantity being optimized.

The method was first proposed by A. H. Land and A. G. Doig in 1960 for linear programming.

## General description

For definiteness, we assume that the goal is to find the minimum value of a function f(x) (e.g., the cost of manufacturing a certain product), where x ranges over some set S of admissible or candidate solutions (the search space or feasible region). Note that one can find the maximum value of f(x) by finding the minimum of g(x) = − f(x).

A branch-and-bound procedure requires two tools. The first one is a splitting procedure that, given a set S of candidates, returns two or more smaller sets $S_1, S_2, \ldots$ whose union covers S. Note that the minimum of f(x) over S is $\min\{v_1, v_2, \ldots\}$, where each vi is the minimum of f(x) within Si. This step is called branching, since its recursive application defines a tree structure (the search tree) whose nodes are the subsets of S.

Another tool is a procedure that computes upper and lower bounds for the minimum value of f(x) within a given subset S. This step is called bounding.

The key idea of the BB algorithm is: if the lower bound for some tree node (set of candidates) A is greater than the upper bound for some other node B, then A may be safely discarded from the search. This step is called pruning, and is usually implemented by maintaining a global variable m (shared among all nodes of the tree) that records the minimum upper bound seen among all subregions examined so far. Any node whose lower bound is greater than m can be discarded.

The recursion stops when the current candidate set S is reduced to a single element; or also when the upper bound for set S matches the lower bound. Either way, any element of S will be a minimum of the function within S.

### Effective subdivision

The efficiency of the method depends strongly on the node-splitting procedure and on the upper and lower bound estimators. All other things being equal, it is best to choose a splitting method that provides non-overlapping subsets.

Ideally the procedure stops when all nodes of the search tree are either pruned or solved. At that point, all non-pruned subregions will have their upper and lower bounds equal to the global minimum of the function. In practice the procedure is often terminated after a given time; at that point, the minimum lower bound and the minimum upper bound, among all non-pruned sections, define a range of values that contains the global minimum. Alternatively, within an overriding time constraint, the algorithm may be terminated when some error criterion, such as (max - min)/(min + max), falls below a specified value.

The efficiency of the method depends critically on the effectiveness of the branching and bounding algorithms used; bad choices could lead to repeated branching, without any pruning, until the sub-regions become very small. In that case the method would be reduced to an exhaustive enumeration of the domain, which is often impractically large. There is no universal bounding algorithm that works for all problems, and there is little hope that one will ever be found; therefore the general paradigm needs to be implemented separately for each application, with branching and bounding algorithms that are specially designed for it.

Branch and bound methods may be classified according to the bounding methods and according to the ways of creating/inspecting the search tree nodes.

The branch-and-bound design strategy is very similar to backtracking in that a state space tree is used to solve a problem. The differences are that the branch-and-bound method (1) does not limit us to any particular way of traversing the tree and (2) is used only for optimization problems.

This method naturally lends itself for parallel and distributed implementations, see, e.g., the traveling salesman problem article.

## Applications

This approach is used for a number of NP-hard problems, such as

It may also be a base of various heuristics. For example, one may wish to stop branching when the gap between the upper and lower bounds becomes smaller than a certain threshold. This is used when the solution is "good enough for practical purposes" and can greatly reduce the computations required. This type of solution is particularly applicable when the cost function used is noisy or is the result of statistical estimates and so is not known precisely but rather only known to lie within a range of values with a specific probability. An example of its application here is in biology when performing cladistic analysis to evaluate evolutionary relationships between organisms, where the data sets are often impractically large without heuristics.

For this reason, branch-and-bound techniques are often used in game tree search algorithms, most notably through the use of alpha-beta pruning.