Lawler, E.L., Wood, D.E.: Branch-and-bound methods: A survey. Operations Research 14(4) (1966) 699--719  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in 1992, under the guidance of Barry Crabtree. Chris Voudouris is employed by the EPSRC funded project, ref GR H75275. I. Introduction Due to their combinatorial explosion nature, many real life constraint optimization problems are hard to solve using complete methods such as branch bound [17, 14, 21, 23]. One way to contain the combinatorial explosion problem is to sacrifice completeness. Some of the best known methods which use this strategy are local search methods, the basic form of which often referred to as hill climbing. The problem is seen as an optimization problem according to an ....

E.W. Lawler and D.E. Wood, "Branch-and-bound methods: a survey", 15 Operations Research 14, 699-719 (1966).

....domain values. The subproblems are solved applying recursively the same transformation rule. These algorithms generate a tree that is normally traversed in a depth first manner, which has the benefit of being space linear. In constraint optimization, search follows a branch and bound (BB) schema [21]; in constraint satisfaction, search can be specialized to handle and propagate relations [15, 23] In the worst case, search algorithms need to explore the whole search tree. Nevertheless, in practice they typically do much better. Dynamic programming algorithms solve a problem by a sequence of ....

....it to nodes lower in the ordering. The induced width of a graph, denoted w # (o) is the maximum width of nodes in the induced graph. Finding the ordering o with minimum w # (o) is an NP complete problem [1] 2.2. Branch and Bound Branch and Bound (BB) is a search schema for COP solving [21, 13]. It traverses the search tree defined by the problem, where internal nodes represent incomplete assignments and leaf nodes stand for complete ones, which may or may not be optimal. During the traversal, which is typically depth first, BB keeps the cost of the best solution found so far. Its cost ....

Lawler, E. L. and D. E. Wood: 1966, `Branch-and-bound methods: A survey'. Operations Research 14(4), 699--719.

....of time exponentially dependent on the size of the problem instance. If one can be sure to encounter only very small problem instances, it is practical to use potentially exponential time algorithms, e.g. A [113] dynamic programming [114] mixed integer linear programming [115] branch and bound [116], or one of the trivial exhaustive searches. However, if one might encounter large problem instances, the above conjecture implies that one must settle for an algorithm producing solutions that are not guaranteed to be optimal. Approximation algorithms may be used to produce solutions to ....

E. L. Lawler and D. E. Wood, "Branch-and-bound methods: A survey," Operations Research, pp. 699--719, July 1966.

.... in areas as diverse as network design [13, 19] chemical process synthesis [5, 9, 16] product marketing [6] and capital budgeting [17, 20, 23] Methods for solving mixed integer nonlinear programming problems are surveyed in [10, 12] Branch and bound algorithms such as the ones described in [11, 17, 18, 22] work by explicitly enumerating possible values of the zero one variables until an optimal integer solution has been found. A branch and bound algorithm begins by solving the continuous relaxation of the original problem. If a zero one variable is fractional at optimality, the algorithm ....

E. L. Lawler and D. E. Wood, Branch and Bound Methods: A Survey. Operations Research 14, 699--719 (1966).

....finding the global minimum is valuable, there are several heuristic methods for minimizing the likelihood of not finding the global solution, e.g. running the algorithm repeatedly from different starting points. There are also methods for global optimiza tion, e.g. branch and bound (see, e.g. [33, 9]) that are guaranteed to find the globally optimal design. These methods, however, are often orders of magnitude less efficient than the standard (local optimization) methods. The purpose of this paper is to show that by a change of variables, a wide variety of magnitude filter design problems ....

E. L. Lawler and D. E. Wood. Branch-and-bound methods: A survey. Operations Research, 14:699-719, 1966.

....often work well, it must be emphasized that these algorithms are worst case combinatoric. Thus they may require unacceptably long computation times on some simple problems. Traditionally, branch and bound algorithms have been used in discrete programming problems (see [6, 7] for early articles, [8, 9] for surveys and [10, 11, 3] for texts) In [12] Hansen combines interval analysis with a branch and bound scheme very similar to ours for global optimization. A more recent application of a branch and bound algorithm on a parametric robustness problem arising in control systems analysis is in ....

E. L. Lawler and D.E. Wood, "Branch-and-bound methods: A survey", Operations Research, 14, pp. 699--719, (1966).

....In 1968, Hart, Nilsson, and Raphael [Har68] introduced a generic heuristic search technique named A . Pohl [Poh77] and Pearl [Pea84] theoretically analyzed the procedure. Simultaneously, in the Operations Research community, another heuristic search technique was introduced, Branch andBound [Law66]. Held and Karp [Hel71] in 1970 introduced the idea of relaxation based heuristics. Prieditis described automatic generation of effective heuristic search techniques [Pri93] The most constrained variable heuristic was introduced by Bitner and Reingold [Bit75] The technique was systematically ....

E.L. Lawler, D.E. Wood. Branch-and-bound methods: A survey. Operations Research, Vol. 14, pp. 699-719, 1966.

....in C by the value in the same column in the first line (the cost of the G i des associated with column i) Resolution of the TSP. The actual solving of the TSP from matrix C # can be done by using several methods. The most common method is the branch and bound method (see Chapters 9 and 10 of [9] and [12, 7] An outline of the method can be found in Appendix B of [10] The worst case complexity for solving the TSP is 524 H. MARCHAND, O. BOIVINEAU, AND S. LAFORTUNE (n 1) However, the branch and bound method is expected to give a solution to the TSP in a tolerable amount of time. To ....

E. L. Lawler and D. E. Wood, Branch-and-bound methods: A survey, Oper. Res., 14 (1966), pp. 699--719.

....1 represents the search tree created when branching according to the example in Section 2. 1;1] 1; 0] 0; 1] Figure 1: Search Tree Now we mark paths along which all search domains contain a solution of the problem as solution paths (similar trees already have been used by Lawler and Wood [24] to visualize branch and bound methods) For example, in global 4 optimization and minimax optimization these are the paths containing optimizers. Since we can not follow in nite paths throughout, in practice one usually nishes after following a solution path to a certain depth. By using a ....

....ties. Berner [3] discusses the properties of various box choice stratagies, and Csendes [7] investigates the reject index. For general discussion see various books on the subject [19, 13, 38] Heuristics for branch and bound algorithms for discrete domains, already were discussed very early [24, 27] and have later been theoretically justi ed, for example using stochastic models [46] They are especially important for parallel branch and bound [5] Related research in Arti cial Intelligence [33, 30] studies search procedures for two major classes of objects: Locally nite) graphs [9, 12] ....

E. L. Lawler and D. E. Wood. Branch-and-bound methods: A survey. Operations Research, 14:699-719, 1966.

....that are not dominated by other solutions. A PCOP can be solved in a variety of ways, the most trivial (and ine#cient) is finding all the solutions of the CSP and a posteriori select only the solutions of the PCOP. A more e#cient approach is an extension of the Branch and Bound (B B) procedure [6]. B B is an e#cient, widely used method for solving COPs; it could be described as follows: first we find a solution (typically using some kind of tree search) then we add a further constraint whose informal meaning is any new solution must be better than the current best . The operational ....

E.L. Lawler and D.E. Wood. Branch-and-bound methods: a survey. Operations Research, 14(4):699--719, 1966.

.... The virtue of Branchand Bound is that it requires a limited amount of memory and can be used as an anytime 1 Branch and Bound was proposed originally as a general search paradigm that includes Depth First search and Best First search as special cases when the heuristic function is admissible [23, 14]. To put our work in this context, by Branch and Bound in this paper we mean Depth First Branch and Bound. 2 scheme; whenever interrupted, Branch and Bound outputs the best solution found so far. Best First explores the search space in uniform frontiers of partial instantiations, each having the ....

E. L. Lawler and D. E. Wood. Branch-and-bound methods: A survey. Operations Research, 14:699--719, 1966.

....meaning that they cannot be solved exactly with limited computation time and memory. In reality, we often do not need optimal solutions, but rather good ones that can be found quickly. Therefore, approximation algorithms are of considerable practical importance. Branch and bound (BnB) [15, 2, 13] is a general problem solving paradigm for finding optimal solutions to NP complete [6] combinatorial optimization problems. Although it was originally developed for optimal solutions, it can, in fact, be adapted for finding approximate solutions as well. The idea, which is well known in ....

Lawler, E.L., and D.E. Wood, "Branch-and-bound methods: A survey," Operations Research, 14 (1966) 699-719.

....all of the controllers built in the first step in order to visit each of the goals with least total cost. We solve this problem by defining the notion of a scheduler and then by mapping the problem of finding an optimal scheduler to an instance of the well known Traveling Salesman Problem (TSP) [7]. We finally suggest various strategies to reduce the complexity of the TSP resolution while still preserving global optimality. Keywords Discrete Event Systems, Optimal Control, Scheduler, Traveling Salesman Problem This research is supported in part by INRIA and by the Department of Defense ....

....with least total cost; we call this problem that of finding a Stepwise DP Optimal scheduler . We solve this problem by defining the notion of a scheduler and then by mapping the problem of finding a Stepwise DP Optimal scheduler to an instance of the well known Traveling Salesman Problem (TSP) [7]. We finally suggest strategies to reduce the computational complexity of this step while still preserving global optimality by taking advantage of particular properties of the structure of Stepwise DP Optimal schedulers. One of the differences between DP Optimality and Stepwise DP Optimality ....

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E. L. Lawler and D. E. Wood. Branch-and-bound methods: A survey. Operations Research, 14(4):699-- 719, 1966.

....of the shifted system x = A(q) ffI)x for some ff; this value of ff then serves as a lower bound for MSD(A;Q init ) An upper bound for MSD(A;Q init ) is obtained by one of the optimistic methods above. However, these bounds may be unsatisfactory, in which case, a branch and bound technique [LW66, Bal68] is used to systematically improve the bounds. At each stage of the algorithm, guaranteed upper and lower bounds are available for MSD(A;Q init ) The use of branch and bound algorithms for robustness analysis is not new. De Gaston and Safonov [GS88] use a branch and bound algorithm for ....

....would perform substantially better than a branch and bound algorithm on MSD computation. In the following section, we describe the basic branch and bound algorithm in detail; we then use it to compute the MSD in subsequent sections. 2 The branch and bound algorithm The branch and bound algorithm [LW66, Bal68] finds the (global) minimum of a function f : R m R over an m dimensional rectangle Q init . 6 For a rectangle Q Q init we define Phi min (Q) min q2Q f(q) Then, the algorithm computes Phi min (Q init ) to within an absolute accuracy of ffl 0, using two functions Phi lb ....

E. L. Lawler and D.E. Wood. Branch-and-bound methods: A survey. Operations Research, 14:699--719, 1966.

....c j ) q i ; d j ; q i d j )g (83) used in [10] is nothing else than (79) 80) 81) which has been also essentially established in [19] However, the form (79) 80) 81) which is an explicit description of (83) is more convenient for computational implementation. Using the general BB scheme [16], both [12, 13] and [10] don t take the bilinear charaterization of (69) into account for BB processes. The branching in their BB algorithms is performed in the (x; y) space (of dimension N 1 N 2 ) rather than in the x space (of dimension N 1 N 2 ) and thus can be not practical for the case of ....

E.L. Lawler, D.E. Wood, Branch and bound methods: a survey, Operation Research 14(1966), 699-719.

....requires extensive resources, as will be shown formally in Section 5. It is therefore essential that one designs carefully the algorithm and uses the available resources smartly. The method we propose to use is based on the branch and bound search technique developed by operations researchers [4]. The use of some branch and bound techniques for nding optimal solutions for combinatorial auctions has been considered 2 in [1] and more thoroughly in [7] The case of multi unit combinatorial auctions is considered in [6] where a wealth of heuristics are proposed. This paper concentrates on ....

E.L. Lawler and Wood D.E. Branch-and-bound methods: A survey. Operations Research, 14(4):699-719, 1966.

....requires extensive resources, as will be shown formally in Section 5. It is therefore essential that one designs carefully the algorithm and uses the available resources smartly. The method we propose to use is based on the branch and bound search technique developed by operations researchers [5]. The use of some branch and bound techniques for nding optimal solutions for combinatorial auctions has been considered in [1] and more thoroughly in [8] The case of multi unit combinatorial auctions is considered in [7] where a wealth of heuristics are proposed. This paper concentrates on the ....

E. Lawler and W. D.E. Branch-and-bound methods: A survey. Operations Research, 14(4):699-719, 1966.

....namely the quality of the original root solution, the method used to determine a new lower bound in the process of pruning all non optimal solutions, and the method used to determine which solution subproblem to expand at any point. A general survey of branch and bound approaches is provided in [Lawler and Wood, 1966] . The branch and bound approach described in [Stinson et al. 1978] generates the tree by progressively scheduling activities forward from the start of the schedule. Each node is expanded by creating a new node for each possible combination of activities that could be scheduled according to both ....

E.L. Lawler and D.E. Wood. Branch-and-bound methods: A survey. Operations Research, 14(4):699--719, July--August 1966.

....of [5] is not to study the parallelization of this method, but rather to develop an efficient load balancing strategy that uses priority of tasks. Therefore the chosen B B algorithm is sufficient for our purpose (for more information about B B algorithms and their parallelizations, refer to [4, 7, 3, 2, 11, 12]) The chosen B B algorithm uses a distributed queue organization to temporarily stock the generated subproblems. We chose to execute only one process per processor in the PVM machine. Each processor has a local queue in its local memory and executes the sequential B B algorithm using this ....

E. L. Lawler and D. E. Wood. Branch and bound methods: A survey. Operations Research, (14):670--719, 1966.

....[97] Our results for static load balancing also apply to certain problems where the load is dynamic, that is, the total load varies with time 1 . Dynamic load balancing is required in a wide variety of applications, including operating systems [48, 83] combinatorial optimization problems [80], adaptive mesh partitioning [68, 118] and fine grain functional programming [59] In certain problems arising in these applications, it is possible to divide the overall computation into phases, where the distributed system alternates between static load balancing and executing a portion of the ....

E. L. Lawler and D. E. Wood. Branch and bound methods: a survey. Operations Research, 14:699--719, 1966.

....EDF) and those that optimize a global performance measure (e.g. makespan) Objective functions of the latter type are indicated by a star ( in the table. GAST currently provides support for two task allocation strategies: The Exhaustive option is based on the branch andbound (B B) strategy [38]. The Greedy option selects an implementation of a greedy search strategy. A large set of parameters from the B B model of Kohler and Steiglitz [35] is available for the Exhaustive option, which allows for the emulation of both the B B and the greedy search strategy. A more in depth description of ....

E. L. Lawler and D. E. Wood. Branch-and-bound methods: A survey. Operations Research, 14(4):699--719, July/August 1966.

....methods are of great importance. These 1 For simplicity, we assume P is a minimization problem; B B can equally well be used to solve maximization problems. 1 methods fall into two categories. Methods in the first category use solely depth first search (DFS) and include depth first B B (DFBB) [21, 20], iterative deepening A (IDA ) 13] DFS [37] MIDA [38] and IDA CR [35] To perform DFS, these methods employ a stack which stores information regarding nodes on some root leaf subpath in the search space, where level 0 of the stack corresponds to the root node, and level i to the ith ....

E.L. Lawler and D.E. Wood, "Branch-and-bound methods: A survey," Operations Research, Vol. 14, pp. 699-719, 1966.

.... A Constraint Satisfaction Optimisation Problem (CSOP) consist of a standard CSP and an optimisation function which maps every solution (complete labelling of variables) to a numerical value [24] The most widely used algorithm for finding optimal solutions is called Branch and Bound (B B) [14] and it can be applied to CSOP as well. The B B needs a heuristic function that maps the partial labelling to a numerical value. It represents an under estimate (in case of minimisation) of the objective function for the best complete labelling obtained from the partial labelling. The algorithm ....

Lawler, E.W., Wood, D.E.: Branch-and-bound methods: a survey, in: Operations Research 14:699-719, 1966

....the bounding rule) Otherwise, the worker checks whether the length of its partial match is already a satisfactory match but one. If so, it changes the optimal match in RESULT to its own evaluated match (extended to a satisfactory match) This algorithm is essentially a branch and bound algorithm [19]. For a detailed discussion of the task evaluation refer to [16] 5.2 Parallel Best First Search Tree Algorithm An alternative parallel best first search tree algorithm terminates at the first satisfactory match. The central data structure is a distributed priority queue of entries of the ....

E.L. Lawler and D.E. Wood. Branch-and-bound methods: a survey. Operations Research, 14(4):699--719, July 1966.

.... A Constraint Satisfaction Optimisation Problem (CSOP) consists of a standard CSP and an optimisation function that maps every solution (complete labelling of variables) to a numerical value [37] The most widely used algorithm for finding optimal solutions is called Branch and Bound (B B) [24] and it can be applied to CSOP as well. The B B needs a heuristic function that maps the partial labelling to a numerical value. It represents an under estimate (in case of minimisation) of the objective function for the best complete labelling obtained from the partial labelling. The algorithm ....

Lawler, E.W., Wood, D.E.: Branch-and-bound methods: a survey, in: Operations Research 14:699-719, 1966

....once. We want to compute an optimal route for some cities in the Ruhrgebiet, an area in Germany named after the river Ruhr. The selected cities with their connections and distances are displayed in Figure 2. The salesman should start in Essen. The problem can be solved using branch and bound (Lawler and Wood, 1966). It uses a tree to structure the search space of possible solutions. The root of the tree is the city in which the salesman should start. Each path from the root to a node represents a partial tour for the salesman. Leaf nodes represent either partial tours without connections to not yet visited ....

Lawler, E., and Wood, D., Branch-and-bound methods: a survey, Operations Research 14, 699--719 (1966).

....states, IDA will degenerate to a series of depth limited depth rst searches. Depth First Branch and Bound IDA starts with a lower bound on the solution length and increases this lower bound each time it proves that no solution with this lower bound exists. Depth rst branch and bound (DFBB) LW66] starts with an upper bound on the solution length. The upper bound is used to prune parts of the search space that cannot contain a solution better than the current best solution. That means that whenever DFBB encounters a node s in the search space that has an f value (f(s) g(s) h(s) equal ....

E.L. Lawler and D. Woods. Branch-and-bound methods: A survey. Operations Research, 14, 1966.

....load balancing in fact study static load balancing. In dynamic load balancing, the load is dynamic, that is, the total workload may vary with time 1 . Dynamic load balancing is required in a wide variety of applications, including operating systems [11, 19] combinatorial optimization problems [18], and adaptive mesh partitioning [15, 29] The results and techniques of static load balancing are applicable for certain problems in which the computation can be divided into alternating phases of balancing and processing. For most applications, however, it is desired to have a continuous process ....

E. L. Lawler and D. E. Wood. Branch and bound methods: a survey. Operations Research, 14:699--719, 1966.

....whether a state is pruned requires a single bit. Thus, 2 135 bits would be required to represent the required information for this problem, a requirement well beyond the capacity of computational machinery into the foreseeable future. INSERT FIGURE 5 ABOUT HERE OPUS uses a branch and bound [9] search strategy similar to that illustrated in Figure 4, to guarantee that no two equivalent nodes in the search space are visited. However, it organises the search space so as to maximise the effect of pruning, achieving close to the effect illustrated in Figure 3 without any significant ....

Lawler, E.L. and D.E. Wood, Branch and bound methods: A survey. Operations Research, 1966. 149: p. 699-719.

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Lawler, E.L., Wood, D.E.: Branch-and-bound methods: A survey. Operations Research 14(4) (1966) 699--719

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E.W. Lawler and D.E. Wood. 1966. Branch-and-Bound Methods: a Survey. Operations Research 14: 699-719.

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E. L. Lawler and D. E. Wood, "Branch-and-bound methods: A survey," Oper. Res., vol. 14, no. 4, pp. 699--719, Apr. 1966.

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Lawler, E. L. and Wood, D. E., 1966, "Branch-and-bound Methods: A Survey," Operations Research, 14(4):699-719.

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Lawler, E.L. and Wood, D.E. 1966. Branch-and-bound methods: A survey. Operations Research 14(4), 699-719.

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E. L. Lawler and D.E. Wood, #Branch-and-bound methods: A survey", Operations Research, 14, pp. 699#719, #1966#.

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E. L. Lawler and D. E. Wood. Branch and bound methods: A survey,. Operations Research, 14(3):699--719, 1966.

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E. Lawler and D. Wood. Branch and bound methods: A survey. Operations Research, 14(291):699--719, 1966.

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Lawler, E., Wood, D.: Branch-and-bound methods: a survey. Operations Research 14 (1966) 699--719

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E. Lawler and D. Wood. Branch and bound methods: A survey. Operations Research, 14(291):699--719, 1966.

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E.L. Lawler and D.E. Wood. Branch and Bound methods: A survey. Operations Research 14 #1966#, 600#719.

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E. Lawler and D. Wood. Branch and bound methods: A survey. Operations Research, 14(291):699--719, 1966.

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E. L. Lawler and D. E. Wood. Branch-and-bound methods: A survey. Operations Research, 14:699--719, 1966. 13

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E.L. Lawler and D.E. Wood. Branch-and-bound methods: a survey. Operations Research, 14:699-719, 1966.

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E. L. Lawler and D. E. Wood, `Branch-and-bound methods: a survey', Operations Research, 14, 699--719 (1966).

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E. Lawler and D. Wood, "Branch-and bound methods: a survey," Oper. Res., 14, pp. 699719, 1966.

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E. Lawler and D. Wood, Branch and bound methods: a survey. Operations Research, 14, 699-719, 1966.

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E.L. Lawler, and D.E. Wood. Branch and Bound methods: A survey. Operations Research 14 #1966#, 600#719.

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E.L. Lawler, and D.E. Wood. Branch and Bound methods: A survey. Operations Research 14 (1966), 600--719.

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Lawler, E. and D. Wood (1966), "Branch and bound methods: a survey," Operations Research, 14, 699-719.

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E.L. Lawler, D.E. Wood (1966). Branch-and-bound methods: a survey. Oper. Res. 14, 699--719.

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