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Branch and bound
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Branch and bound
Branch-and-bound (BB, B&B, or BnB) is a method for solving optimization problems by breaking them down into smaller subproblems and using a bounding function to eliminate subproblems that cannot contain the optimal solution.
It 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 efficient estimation of the lower and upper bounds of regions/branches of the search space. If no bounds are available, then the algorithm degenerates to an exhaustive search.
The method was first proposed by Ailsa Land and Alison Doig whilst carrying out research at the London School of Economics sponsored by British Petroleum 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:
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 SI. The instance representation has to come with three operations:
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 equal to or greater than the current upper bound; 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 upper bound seen among all instances examined so far.
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Branch and bound AI simulator
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Branch and bound
Branch-and-bound (BB, B&B, or BnB) is a method for solving optimization problems by breaking them down into smaller subproblems and using a bounding function to eliminate subproblems that cannot contain the optimal solution.
It 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 efficient estimation of the lower and upper bounds of regions/branches of the search space. If no bounds are available, then the algorithm degenerates to an exhaustive search.
The method was first proposed by Ailsa Land and Alison Doig whilst carrying out research at the London School of Economics sponsored by British Petroleum 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:
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 SI. The instance representation has to come with three operations:
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 equal to or greater than the current upper bound; 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 upper bound seen among all instances examined so far.