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Topic: Branch and bound

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Approximation algorithm

Traveling salesman problem

	Branch and bound - Wikipedia, the free encyclopedia
		Branch and bound is a general algorithmic method for finding optimal solutions of various optimization problems, especially in discrete and combinatorial optimization.
		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.
	en.wikipedia.org /wiki/Branch_and_bound (819 words)

	JOT: Journal of Object Technology - Branch and Bound Implementations for the Traveling Salesperson Problem
		When the algorithm branches, and after imposing the decision logic to include or exclude edges, a lower bound is computed for the node.
		Therefore, twice the lower bound for this unconstrained root node is therefore: 1 + 2 + 2 + 4 + 4 + 5 + 1 + 1 + 1 + 1 + 1 + 2 = 25.
		Twice the lower bound with the constraint 12 is therefore 9 + 10 + 4 + 5 + 1 + 1 + 1 + 1 + 1 + 2 = 35.
	www.jot.fm /issues/issue_2003_03/column7 (3446 words)

	JOT: Journal of Object Technology - Branch and Bound Implementations for the Traveling
		In the first column published in the March/April, 2003 issue, a branch and bound algorithm that involves generating nodes that represent partial tours with constraints was presented and implemented.
		The node with the smallest lower bound is used to generate nodes at level 3.
		After the lower bounds for each of these nodes are computed, level four nodes are generated from the level 3 node with smallest lower bound.
	www.jot.fm /issues/issue_2003_05/column7 (1678 words)

	Branch and bound algorithm in C# (Site not responding. Last check: 2007-11-02)
		The basis of branch and bound algorithms is a ranking function.
		Two branch and bound solutions to the TSP are described in this section.
		The branch and bound algorithm is implemented to solve the tsp problem within the given time bound.
	codersource.net /csharp_branch_and_bound_algorithm_implementation.aspx (1239 words)

	A parallel synchronized branch and bound algorithm
		The parallelization of the best-first branch and bound algorithm on distributed memory machines is relatively difficult.
		Although knowing good upper bounds does not diminish the overall execution time, they are important in order to reduce memory consumption, especially when using a best-first branch and bound strategy.
		The selection of the variable used in the branching of a subproblem is based on the estimated degradation of the objective function, the notion of pseudo-costs and the previously branched subproblems.
	sawww.epfl.ch /SIC/SA/publications/SCR94/6-94-page15.html (4190 words)

	[No title]
		The Bounding Procedure A lower bound on the total tardiness resulting from a completion of the partial sequence Qk consists of two components: (1) the actual tardiness due to the assigned jobs in Qk and (2) a lower bound on the tardiness due to the remaining Nk jobs.
		A lower bound for the problem is developed and used as a part of a branch and bound solution approach.
		The truncated branch and bound approach may provide a good practical approach to solving this type of problem, but it may have to be supplemented with other heuristics when there is a large difference between the current upper and lower bounds when the algorithm is truncated.
	www.msu.edu /~ragatz/sdsetup.doc (4308 words)

	Branch and Bound (Site not responding. Last check: 2007-11-02)
		It uses a novel Branch and Bound procedure in which the problem is partitioned by fixing the edge incidence of selected subsets of clustered customers.
		A branch and bound algorithm uses a divide and conquer strategy to partition the solution space into subproblems and then optimizes individually over each subproblem.
		Finally, if we cannot prune the subproblem, we are forced to branch and add the children of this subproblem to the list of active candidates.
	osiris.tuwien.ac.at /~wgarn/VehicleRouting/neo/algorithms/branchBound.html (375 words)

	Branch & Bound Algorithm
		In a branch and bound algorithm a given subproblem, which cannot be bounded, has to be divided into at least two new restricted subproblems.
		A crucial point for a good branch and bound algorithm is the branching strategy: In our application this means the decision, which pair of vertices should be used for the join and split in a given situation.
		This possibility for forcing a join is extremely helpfully if the bounds are close to a given feasible solution and the graph is somehow irregular such that there are edges which are not loaded with the maximal congestion.
	www.upb.de /cs/sensen/GraphPart/ESA_html/node4.html (1132 words)

	Branch-and-Bound Algorithms (Site not responding. Last check: 2007-11-02)
		If a cost criteria is available, the node to be expanded next (i.e., the branch) is the one with the best cost within the queue.
		In such a case, the cost function may also be used to discard (i.e., the bound) from the queue nodes that can be determined to be expensive.
		A function can be considered to be a tree generator, if when given a node X and index i it produces the i’th child of the node.
	www.cse.ohio-state.edu /~gurari/course/cis680/cis680Ch20.html (313 words)

	Branch & Bound Tree Search (Site not responding. Last check: 2007-11-02)
		Inspection of B1 establishes an upper bound for the four-taxon tree: B2 has the same length, but B3 exceeds this bound, and therefore this tree and all derivative trees can be exluded from further examination.
		The branch and bound process continues with the addition of a sixth taxon F to any of the seven branches in the five remaining trees, establishment of a new upper bound, and so on.
		Branch and Bound is a computationally efficient method for analyses involving up to ~20 taxa, for which there are 8.2 x 10
	www.mun.ca /biology/scarr/Branch_&_Bound_Tree_Search.html (238 words)

	Branch and Bound
		Branch and bound is the broad class of algorithms from which branch, cut, and price is descended.
		Branch and bound algorithms use a divide and conquer strategy to partition the solution space into subproblems and then optimizes individually over each of them.
		Finally, if all else fails, we are forced to branch and add the children of this subproblem to the list of active candidates.
	www.branchandcut.org /SYMPHONY/intro/node3.html (309 words)

	Branch and Bound Algorithm -- from Wolfram MathWorld
		Branch and bound algorithms are a variety of adaptive partition strategies have been proposed to solve global optimization models.
		Branch and bound subsumes many specific approaches, and allows for a variety of implementations.
		The general branch and bound methodology is applicable to broad classes of global optimization problems, e.g., in combinatorial optimization, concave minimization, reverse convex programs, DC programming, and Lipschitz optimization (Neumaier 1990, Hansen 1992, Ratschek and Rokne 1995, Kearfott 1996, Horst and Tuy 1996, Pintér 1996).
	mathworld.wolfram.com /BranchandBoundAlgorithm.html (268 words)

	branch and bound
		The Branch and Bound (BB) algorithm is one of the successful algorithms to use selective/partial enumeration rather than complete enumeration.
		Each branch represents leads to a tier in which some variable is held at its prescribed discrete value.
		This leads to 4 branches and the optimal values are indicated on the tree.
	www.rit.edu /~pnveme/EMEM820n/Mod7_Discrete/Mod7_content/mod7_sec2_branchbound.html (1003 words)

	JOT: Journal of Object Technology - A Java Implementation of the Branch and Bound Algorithm: The Asymmetric Traveling ...
		Branch and bound (BnB) is a set of enumerative methods applied to solving discrete optimization problems.
		In general, the better (more constrained) the relaxation R and the branching method, the better is the performance of the branch and bound method.
		Because the branching scheme used in this implementation forces some arcs to the solution and excludes others, the original assignment problem is converted to the Modified Assignment Problem (M.A.P.), which can be solved with the Hungarian algorithm provided some changes are (temporarily) applied to the cost matrix.
	www.jot.fm /issues/issue_2005_01/article5 (1792 words)

	8.4.2 Optimal Solution for TSP using Branch and Bound (Site not responding. Last check: 2007-11-02)
		Each time we branch, by considering the two children of a node, we try to infer additional decisions regarding which edges must be included or excluded from tours represented by those nodes.
		If the lower bound for a child is as high or higher than the lowest cost found so far, we can ``prune'' that child and need not consider or construct its descendants.
		Interestingly, there are situations where the lower bound for a node n is lower than the best solution so far, yet both children of n can be pruned because their lower bounds exceed the cost of the best solution so far.
	lcm.csa.iisc.ernet.in /dsa/node187.html (553 words)

	Branch & Bound
		Note that, I am not ignoring those solutions in the leaves of the branches that I have pruned, I have left them out of consideration after I have made sure that the optimal solution cannot be at any one of these nodes.
		To be able to have a tighter bound, one has to solve the problem with minimal relaxation.
		On the other hand, if our bounding scheme results in not very tight bounds, we would not be able to prune much of the tree, thus having have to evaluate large number of the nodes -at the extreme all of the nodes in the tree, which is worse than total enumeration of all alternatives.
	www.csulb.edu /~obenli/Research/IE-encyc/bb.html (1992 words)

	branch and bound (Site not responding. Last check: 2007-11-02)
		The key principle of bounding is that if the upper bound of a sub-problem (the best it can possibly be) is worse than the lower bound of the overall problem (the worst it can possibly be) then I can prune the sub-problem and not enumerate any of its members any further.
		To find a lower bound on a problem (the worst it can be) then ANY solution to the problem will do (clearly any feasible solution is at most as bad as I can do).
		The branch and bound algorithm alternates between computing the bounds and splitting a given problem or sub-problem into smaller pieces.
	coweb.math.gatech.edu:8888 /model/1179 (314 words)

	BoundBranch - Bazaar Version Control (Site not responding. Last check: 2007-11-02)
		A bound branch is a branch that does not commit a new revision until after the revision has been applied to another branch and first.
		A bound branch thusly requires lockstep mechanisms to ensure that if a revision can not be sent to both branches,
		Assuming that branch A is bound to branch B
	bazaar-vcs.org /BoundBranch (160 words)

	Branch-and-Bound
		In this test, a node is pruned when its optimal (lower bounding) solution is guaranteed to be worse (greater) than some known current best value (an upper bound on the global minimum).
		For example, when solving global minimization problems using interval analysis, the best upper bound value may be generated and updated by some heuristic combination of an interval extension of the objective function, a point objective function evaluation with interval arithmetic, and a local minimization with a verification by interval analysis.
		The idea is that the most important subproblems stored in the stack are examined with higher priority, in the hope that the global optimum be found early in the search process, thus allowing other later subproblems that do not possess the global optimum to be quickly pruned before they generate new nodes.
	www.nd.edu /~cgau/parallel-paper/node3.html (639 words)

	Branch and Bound (Site not responding. Last check: 2007-11-02)
		To apply branch and bound, one must have a means of computing a lower bound on an instance of the optimization problem and a means of dividing the feasible region of a problem to create smaller subproblems.
		There must also be a way to compute an upper bound (feasible solution) for at least some instances; for practical purposes, it should be possible to compute upper bounds for some set of nontrivial feasible regions.
		Since it is feasible, it can be used to prune the rest of the tree: if the lower bound for a node exceeds the best known feasible solution, no globally optimal solution can exist in the subspace of the feasible region represented by the node.
	www.cs.sandia.gov /opt/survey/branch-and-bound.html (318 words)

	Solver Technology - Mixed-Integer and Constraint Programming
		Its Branch and Bound method uses pseudocosts, degradation factors and strong branching, and it implements a number of cuts.
		Strong Branching is a method used to estimate the impact of branching on each integer variable on the objective function (its pseudocost), by performing a few iterations of the Dual Simplex method.
		Such pseudocosts are used to guide the choice of the next subproblem to explore, and the next integer variable to branch upon, throughout the Branch and Bound process.
	www.solver.com /technology3.htm (924 words)

	BB branch and bound frame
		The BB frame is a composed frame implementing a generic parallel branch and bound application with load balancing.
		Set this to the maximal number of branches the branch function is allowed to generate.
		This happens only if the "worse" bound is changed, so if you use a maximising algorithm, this will only happen for the lower_bound function; and if minimising is chosen, it happens for the upper_bound function.
	wwwcs.uni-paderborn.de /~alcom-it/Frames/BB/BB.html (1276 words)

	Branch-and-bound
		A cut-off or bounding function is required to bound the possible outcomes of the subtree.
		Since a parallel implementation may already have a processor search within a subtree, the search computations are wasted if it is found (subsequently but in parallel) that the subtree cannot possibly lead to a better solution.
		Branch and bound uses a queue of live nodes and proceeds to expand each node, adding new nodes to queue as they are found.
	www.cs.mu.oz.au /498/notes/node65.html (322 words)

	Branch and Bound
		This branching process can be carried out recursively; each of the two new problems will give rise to two more problems when we branch on one of the noninteger components of their solution.
		The branch emanating from such a node cannot give rise to a better integer solution than the one obtained so far, so we consider it no further; that is, we prune it.
		The branching variable is often chosen to be the one for which the degradation is greatest.
	www-fp.mcs.anl.gov /otc/guide/optweb/discrete/integerprog/section2_1_1.html (830 words)

	Knapsack Problem with Branch and Bound Pruning
		So, we will be able to prune branches when their total cost gets too big and when their total benefit will not get any bigger than the best benefit we have seen so far.
		Rather than thinking about branches, it may be useful to think about being at particular nodes and deciding which node to "expand" first.
		Getting to a good answer sooner means that we can more often ignore branches with bounds less than the best benefit we have found so far, i.e., we get to prune the tree more often.
	www.cs.uni.edu /~east/teaching/153/branch_bound/knapsack/overview_algorithm.html (965 words)

	Global Optimization of Nonconvex Nonlinear Programs Using Parallel Branch and Bound - Epperly (ResearchIndex) (Site not responding. Last check: 2007-11-02)
		Abstract: A branch and bound algorithm for computing globally optimal solutions to nonconvex nonlinear programs in continuous variables is presented.
		11 An interval branch and bound algorithm for bound constrained..
		4 Assuring a global optimum by the use of an upper bound on th..
	citeseer.ist.psu.edu /epperly95global.html (1771 words)

	Branch and Bound
		The branch and bound strategy divides a problem to be solved into a number of subproblems, similar to the strategy backtracking.
		All following calculations are terminated as soon as their costs exceed the costs of the upper bound.
		If a new cheaper solution was found, its value is used as the new upper bound.
	www.animal.ahrgr.de /showAnimationDetails.php3?lang=en&anim=9 (202 words)

	Other Observables. A Parallel Branch and Bound Example (Site not responding. Last check: 2007-11-02)
		An example of the use of this qualifier could be a branch and bound algorithm.
		The execution time depends on the number of visited nodes (but you don't know that number until you have finished your branch and bound).
		The experiment is a master-slave parallel branch and bound that was executed on an heterogeneous clusters of PCs (a so called beowulf).
	nereida.deioc.ull.es /~call/node34.html (1044 words)

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