**Branch and bound** (**BB**, **B&B**, or **BnB**) is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores *branches* of this tree, which represent subsets of the solution set. Before enumerating the candidate solutions of a branch, the branch is checked against upper and lower estimated *bounds* on the optimal solution, and is discarded if it cannot produce a better solution than the best one found so far by the algorithm.

The algorithm depends on the efficient estimation of the lower and upper bounds of a region/branch of the search space and approaches exhaustive enumeration as the size (n-dimensional volume) of the region tends to zero.

The method was first proposed by A. H. Land and A. G. Doig in 1960 for discrete programming, and has become the most commonly used tool for solving NP-hard optimization problems. The name "branch and bound" first occurred in the work of Little *et al.* on the traveling salesman problem.

The goal of a branch-and-bound algorithm is to find a value x that maximizes or minimizes the value of a real-valued function *f*(*x*), called an objective function, among some set S of admissible, or candidate solutions. The set S is called the search space, or feasible region. The rest of this section assumes that minimization of *f*(*x*) is desired; this assumption comes without loss of generality, since one can find the maximum value of *f*(*x*) by finding the minimum of *g*(*x*) = −*f*(*x*). A B&B algorithm operates according to two principles:

It recursively splits the search space into smaller spaces, then minimizing *f*(*x*) on these smaller spaces; the splitting is called *branching*.
Branching alone would amount to brute-force enumeration of candidate solutions and testing them all. To improve on the performance of brute-force search, a B&B algorithm keeps track of *bounds* on the minimum that it is trying to find, and uses these bounds to "prune" the search space, eliminating candidate solutions that it can prove will not contain an optimal solution.
Turning these principles into a concrete algorithm for a specific optimization problem requires some kind of data structure that represents sets of candidate solutions. Such a representation is called an *instance* of the problem. Denote the set of candidate solutions of an instance I by S_{I}. The instance representation has to come with three operations:

branch(*I*) produces two or more instances that each represent a subset of S_{I}. (Typically, the subsets are disjoint to prevent the algorithm from visiting the same candidate solution twice, but this is not required. The only requirement for a correct B&B algorithm is that the optimal solution among S_{I} is contained in at least one of the subsets.)
bound(*I*) computes a lower bound on the value of any candidate solution in the space represented by I, that is, bound(*I*) ≤ *f*(*x*) for all x in S_{I}.
solution(*I*) determines whether I represents a single candidate solution. (Optionally, if it does not, the operation may choose to return some feasible solution from among S_{I}.)
Using these operations, a B&B algorithm performs a top-down recursive search through the tree of instances formed by the branch operation. Upon visiting an instance I, it checks whether bound(*I*) is greater than the lower bound for some other instance that it already visited; if so, I may be safely discarded from the search and the recursion stops. This pruning step is usually implemented by maintaining a global variable that records the minimum lower bound seen among all instances examined so far.

The following is the skeleton of a generic branch and bound algorithm for minimizing an arbitrary objective function f. To obtain an actual algorithm from this, one requires a bounding function g, that computes lower bounds of f on nodes of the search tree, as well as a problem-specific branching rule.

- Using a heuristic, find a solution x
_{h} to the optimization problem. Store its value, *B* = *f*(*x*_{h}). (If no heuristic is available, set B to infinity.) B will denote the best solution found so far, and will be used as an upper bound on candidate solutions.
- Initialize a queue to hold a partial solution with none of the variables of the problem assigned.
- Loop until the queue is empty:
- Take a node N off the queue.
- If N represents a single candidate solution x and
*f*(*x*) < *B*, then x is the best solution so far. Record it and set *B* ← *f*(*x*).
- Else,
*branch* on N to produce new nodes N_{i}. For each of these:
- If
*g*(*N*_{i}) > *B*, do nothing; since the lower bound on this node is greater than the upper bound of the problem, it will never lead to the optimal solution, and can be discarded.
- Else, store N
_{i} on the queue.

Several different queue data structures can be used. A stack (LIFO queue) will yield a depth-first algorithm. A best-first branch and bound algorithm can be obtained by using a priority queue that sorts nodes on their g-value. The depth-first variant is recommended when no good heuristic is available for producing an initial solution, because it quickly produces full solutions, and therefore upper bounds.

When
x
is a vector of
R
n
, branch and bound algorithms can be combined with interval analysis and contractor techniques in order to provide guaranteed enclosures of the global minimum.

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

Integer programming
Nonlinear programming
Travelling salesman problem (TSP)
Quadratic assignment problem (QAP)
Maximum satisfiability problem (MAX-SAT)
Nearest neighbor search
Flow shop scheduling
Cutting stock problem
False noise analysis (FNA)
Computational phylogenetics
Set inversion
Parameter estimation
0/1 knapsack problem
Feature selection in machine learning
Structured prediction in computer vision
Branch-and-bound 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.

Nau *et al.* present a generalization of branch and bound that also subsumes the A*, B* and alpha-beta search algorithms from artificial intelligence.