- Branch and bound
Branch and bound (BB) is a general
algorithmfor 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
1960for linear programming.
For definiteness, we assume that the goal is to find the minimum value of a function (e.g., the cost of manufacturing a certain product), where ranges over some set of "admissible" or "candidate solutions" (the "
search space" or "feasible region").
A branch-and-bound procedure requires two tools. The first one is a "splitting" procedure that, given a set of candidates, returns two or more smaller sets whose union covers . Note that the minimum of over is , where each is the minimum of within . This step is called branching, since its recursive application defines a
tree structure(the "search tree") whose "nodes" are the subsets of .
Another tool is a procedure that computes upper and lower bounds for the minimum value of within a given subset . This step is called bounding.
The key idea of the BB algorithm is: if the "lower" bound for some tree node (set of candidates) is greater than the "upper" bound for some other node , then A may be safely discarded from the search. This step is called pruning, and is usually implemented by maintaining a shared variable that records the minimum upper bound seen among all subregions examined so far. Any node whose lower bound is greater than can be discarded.
The recursion stops when the current candidate set is reduced to a single element; or also when the upper bound for set matches the lower bound. Either way, any element of will be a minimum of the function within .
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 maximum 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 approach is used for a number of
NP-hardproblems, such as
Traveling salesman problem(TSP)
Quadratic assignment problem(QAP)
Maximum satisfiability problem(MAX-SAT)
Nearest neighbor search(NNS)
False noise analysis(FNA)
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 biologywhen 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.
A* search algorithm
* Classes of algorithms by design paradigm
* [http://plagiata.net.ru/?p=100 Branch and bound on Delphi]
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