Kumar, V. and Kanal, L. N., 1983, "A General Branch And Bound Formulation For Understanding And Synthesizing And/Or Tree Search Procedures," Artificial Intelligence, 21:179-198.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....has nished. In a asynchronous implementation, the function returns a Subproblem immediately as long as the subproblem pools are not empty. 6 Discussion We checked if our proposed interface did not lose generality of branch and bound algorithms by comparing with several general formulations [1, 4, 5, 8]. As a result, we found out that our implementation does not support pruning by dominance test[3] However, it can be supported only by adding a declaration of the public member function dominated(IntDataBase , SubproblemBase ) on the base class of the Subproblem and realized in the derived class. ....

V.Kumar and L.N.Kanal. A General Branch and Bound Formulation for Understanding and Synthesizing And/Or Tree Search Procedures. Articial Intelligence, 21:179-198, 1983.

....of new lower bounds and the discovery (and verification) of new combinatorial search strategies. 1 Introduction Many formalisms have been proposed over the years to capture combinatorial optimization algorithms such as dynamic programming, branch and bound, and greedy (see, for example, [6, 13, 16, 17, 18, 19]) These models tend be relatively narrow in that each captures one specific class of solution, and hence provides neither conceptual unity nor a common framework in which the techniques can be studied. In 1989 Helman [9] presented a common formalism that captures dynamic programming and branch ....

....15, 16, 17] are finite automata based and have been used successfully to derive the functional equations associated with dynamic programming and to study the decidability of the general solvability of these problems and questions of their minimal representations. Other models of branch and bound [16, 19] view the solution process in terms of subproblem decomposition, where the exact nature of a subproblem is left unspecified. By 1 abstracting out the nature of the subproblems, the model becomes more flexible, but it also becomes less computationally precise. Our model of combinatorial ....

Kumar, V. and Kanal, L., "A general branch and bound formulation for understanding and synthesizing and/or tree search procedures", Artificial Intelligence 21, 1983, 179--198.

....plays a key role in this version. The SSS algorithm originates from Stockmann [St] The explanation in this paper is rather opaque, the algorithm is presented as a (semi ) parallel search in the state space consisting of so called partial (min ) solution trees. Later papers by Kumar and Kanal [KK1, KK2] recognized this algorithm as a special case of Branch and Bound. In [KK2] the observation is made that there is a dual view on SSS , namely as a (sequential) search over (max ) solution trees. We will expand on this view and give it some formal underpinning. Several authors have studied SSS , ....

V. Kumar and L.N. Kanal, A General Branch and Bound Formulation for Understanding and Synthesizing And/Or Tree Search Procedures, Artificial Intelligence 21 (1983) 179-198

....quality assessment, basic selection strategies, pruning mechanisms, representation of search spaces, iterative searches, and approximation searches. 2.1.1. Preliminary A branch and bound (B B) search is a general search algorithm [49] shown to be a general formulation of many search algorithms [43]. In this thesis, without loss of generality, we discuss search in terms of the B B search. A B B search decomposes a problem into smaller subproblems and repeatedly decomposes them until a solution is found or infeasibility is proved. Each subproblem is represented by a node in the search tree. ....

V. Kumar and L. N. Kanal, "A General Branch and Bound Formulation for Understanding and Synthesizing And/Or Tree Search Procedures," Artificial Intelligence, vol. 21, no. 1-2, pp. 179-198, North-Holland, 1983.

....0.96 0.98 0.98 0.94 535 1.00 1.01 1.00 0.99 1.01 1.00 s444 101 0.71 0.16 0.13 0.72 0.72 0.43 535 0.97 1.02 1.02 0.99 1.02 1.00 s1196 101 1.00 1.02 1.02 0.99 1.02 1.01 535 1.00 1.00 1.01 0.99 1.01 1.00 Example 2. 2 [VC] The second example is based on applying a branch and bound (B B) search [49,50] to solve a vertex cover (VC) problem [17] The undirected graph representing each problem instance of VC is characterized by (1) the number of vertices in the graph (problem size) 2) the degree of connectivity (DC) that measures the probability that an edge exists between two vertices, and (3) ....

V. Kumar and L. N. Kanal, "A general branch and bound formulation for understanding and synthesizing and/or tree search procedures," Artif. Intell., vol. 21, no. 1-2, pp. 179--198, 1983.

....The Branch and Bound (BB) algorithm [11, 14] has been used to solve many problems in science and engineering that otherwise would have no efficient methods of solution. Examples include the well known traveling salesman problem [12] the integer programming problem [3] and state space search [7]. The algorithm has been employed in many applications such as floor planning of VLSI circuits [17] placement of electronic components [2] and robot path planning [5] The BB algorithm is computationally intensive, and can benefit form the high performance computing potential offered by parallel ....

V. Kumar and L. Kanal, "A general branch-andbound formulation for understanding and synthesizing AND/OR tree search procedures," Artificial Intelligence, vol. 21, pp. 179--198, 1983.

....of min. Strategies or solution trees To gain more insight into the minimax function with related properties, we consider the concept of a strategy. This notion is equivalent to the notion of a solution tree, introduced in [28] The idea of viewing a solution tree as a strategy originates from [11]. A strategy of Max consists of a subtree, including in each Max position exactly one continuation and in each min node all continuations (all countermoves to Max) Since the choice of Max in each position is known in a max strategy, Min is able to calculate the outcome for each series of ....

....open and closed solution trees. We mentioned, that every open max solution tree T has at least one leaf p with f (p) 1 and hence g(T ) 1. Such a solution tree can be given another provisional value, defined as the maximum of the game values of the closed leaves. Following the notation in [11], this value is denoted by c(T ) for a max solution tree T . So a new function on the set of max solution trees is defined. Of course, this function has a dual counterpart. Definition 4.1 For a max tree T and a min tree T Gamma in a search tree S, a function c is defined as: c(T ) ....

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V. Kumar and L.N. Kanal, A General Branch and Bound Formulation for Understanding and Synthesizing And/Or Tree Search Procedures, Artificial Intelligence 21 (1983), pp. 179-198.

.... which in the case of limited availability of parallel processing capacity chooses nodes deepest in the tree, a priority scheme which used heuristics would give a parallel version of SSS search [Stockman 79] That alpha beta search is simply a variation of branch andbound search is shown in [Kumar Kanal 83] In [Huntbach Burton 88] we give a parallel alpha beta search algorithm which, although not expressed in a concurrent logic language, similar to that of figure 6 uses a tree of prioritised concurrent processes which pass solutions up the tree and narrower bounds down the tree. The program in ....

. V.Kumar and L.N.Kanal. A general branch-and-bound formulation for understanding and synthesising and/or search problems. Artficial Intelligence 21, pp.179-198.

.... : 17 8 Conclusion and perspectives 17 Laboratoire PRiSM II PNN team Laboratoire PRiSM BOB : a Unified Platform for Implementing Branch and Bound like Algorithms 1 1 The need for a Branch and Bound library All Branch and Bound applications use the same components [4, 15, 19, 22, 26, 28, 29, 33]. When someone writes Branch and Bound algorithms to resolve real life problems, often he rewrites (implements) these components : Management of the priority queue, management of the upper bound, etc. He can not concentrate his work for the essential parts of the application (generation, ....

Kumar (V.) et Kanal (L. N.). -- A general branch and bound formulation for understanding and synthesizing and/or tree search procedures. Artificial Intelligence, vol. 21, n 1, 2, 1983, pp. 179--198. -- Reprinted in the book Search and Heuristics edited by Judea Pearl.

....amount of space, against TCGD, which uses polynomial amount of space, we modify Lawler and Wood s algorithm so that GDFS is carried out instead of BFS. 2 Approximate Branch and Bound Algorithms A branch and bound (B B) search [5, 7] is a general formulation of a wide range of heuristic searches [8], such as A [9] AO , B , game tree, and dynamic programming algorithms. A B B search decomposes a problem into smaller subproblems and repeatedly decomposes them until either a solution is found or infeasibility is proved. Each subproblem is represented by a node in the search tree. The ....

V. Kumar and L. N. Kanal, A general branch and bound formulation for understanding and synthesizing and/or tree search procedures, Artificial Intelligence 21, no. 1-2 North-Holland (1983) 179--198.

....tree search. Many researchers, particularly in AI, have modeled branch and bound as search in an AND OR graph (also known as hypergraphs) 18] AND nodes correspond to our decomposition relations and OR nodes correspond to the alternative decompositions obtained for each input. Kumar and Kanal [15] present such a model of branch and bound and show how algorithms such as AO , B , SSS , and alpha beta are special cases of the model. In particular, the alpha beta mechanism for searching game trees is revealed as depth first search for an optimal solution with respect to a cost function based ....

Kumar, V., and Kanal, L. A general branch and bound formulation for understanding and synthesizing and/or tree search procedures. Artificial Intelligence 21, 1-2 (March 1983), 179--198.

....tree search in terms of solution trees enables us to discover relations between two algorithms which where hitherto considered to be quite unrelated, viz. alphabeta, PVS and SSS [Sto79, PdB90] 2 Solution Trees and Bounds Solution trees occur in game tree literature mostly in relation to SSS [Sto79, Kuma83]. In this section we will show that there exists a relation between solution trees and bounds on the game value. Further, we use solution trees to define the notion of minimal or critical tree. Given a game tree, it generally takes a lot of effort to compute f(n) for a node n. However, ....

.... 6 1 m 1 6 fi fi L L 2 9 2 2 m 3 2 fi fi , l l l 7 2 7 3 m 2 8 fi fi L L 2 7 fi fi Phi Phi Phi Phi Phi H H H H H 4 2 m 4 3 4 2 m 3 4 fi fi S S 3 2 m 3 3 fi fi fi fi Figure 2: A Critical Tree with node types, values (f ) and bounds (f ; f Gamma ) [Kuma83]. Given a strategy of MIN or a max solution tree T , the pay off attainable for MAX is equal to the greatest game value in the terminals of T . By definition, this value is equal to g(T ) Stockman s theorem actually states that the game value f(G) the guaranteed pay off for MAX, is equal to the ....

V. Kumar and L.N. Kanal, A General Branch and Bound Formulation for Understanding and Synthesizing And/Or Tree Search Procedures, Artificial Intelligence 21 (1983), 179--198.

....affect the upper bound of the search are considered first. This is the so called best first behavior of the algorithm. Convergence occurs when no further progress can be made on lowering the upper bound. To better understand the behavior of SSS , the concept of a solution tree must be introduced [9, 13, 18, 24, 26, 30]. Figure 2 illustrates a max and min solution tree. In a max solution tree T , all children of max nodes (square nodes in the figure) of a game tree are considered, while just one child of each min node is included (circled nodes) In a min solution tree T Gamma , all children of min nodes ....

Vipin Kumar and Laveen N. Kanal. A general branch and bound formulation for understanding and synthesizing and/or tree search procedures. Artificial Intelligence, 21:179--198, 1983.

....Strategies or solution trees To gain more insight into the minimax function with related properties, we consider the concept of a strategy. This notion is equivalent to the notion of a solution tree, introduced in [31] The idea of viewing a solution tree as a strategy originates from [14]. A strategy of MAX consists of a subtree, including in each max position exactly one continuation and in each min node all continuations (all countermoves to MAX) Since the choice of MAX in each position is known in a strategy of MAX, MIN is able to calculate the outcome for each series of ....

...., see the cases 4, 5 and 6. Bounding is computing the c value. In the cases 5 and 6 this c value is not affected. Selecting a triple with maximal merit means selecting a subproblem with optimal c value. So, the best first selection criterion is applied. The most interesting point is dominance. In [14] the following definition was presented. A min solution tree T 1 dominates T 2 , if the set of open leaves of T 1 is a subset of T 2 s open set and c(T 1 ) c(T 2 ) If T 1 dominates T 2 , then every enhancement of the search tree that makes T 2 an optimal solution tree, also makes T 1 optimal. ....

V. Kumar and L.N. Kanal, A General Branch and Bound Formulation for Understanding and Synthesizing And/Or Tree Search Procedures, Artificial Intelligence 21 (1983), pp. 179-198.

No context found.

V. Kumar and L. Kanal. A general branch and bound formulation for understanding and synthesizing and/or tree search procedures. In J. Pearl, editor, Search and Heuristics. North-Holland, 1983.

.... the merit of a node N (up to our knowledge the only algorithms from AI that use such bounds are the ones by Ibaraki [18] and Berliner s B [2, 31] There have been several e orts to unify heuristics in discrete branch andbound with heuristic search algorithms from Arti cial Intelligence [16, 17, 22, 23]. There is also a bibliography on heuristic search from the branch and bound viewpoint [44] that covers literature up to the year 1992. Search heuristics for Collin s algorithm [6] the symbolic analogon to the quanti ed constraint solving method of the author, have been developed by H. Hong ....

V. Kumar and L. Kanal. A general branch and bound formulation for understanding and synthesizing and/or tree search procedures. In J. Pearl, editor, Search and Heuristics. North-Holland, 1983.

....are based on iterative deepening, which guides search based on results from previous iterations. Min max techniques present considerable challenges for parallel processing. Conceptually, ff Gamma fi game tree search algorithm can also be viewed as a depth first branch and bound algorithm [23] that searches for a maximum payoff strategy among all strategies represented in the game tree. However, game trees tend to be strongly ordered. Therefore, naive parallel formulations may expand a large number of nodes that are not expanded by the serial formulation. Second, the information about ....

V. Kumar and L. N. Kanal. A general branch-and-bound formulations for understanding and synthesizing and/or tree search procedures. Artificial Intelligence, 21:179--198, 1983.

....(depth first B B) 14, 19] Although depth first B B would usually perform much more work than best first B B, it is (like any other depth first search strategy) highly space efficient. Note that the alpha beta game tree search algorithm can be viewed as a depth first B B algorithm (see [12, 15]) 3 Parallel Depth First Search 3.1 A Parallel Formulation of Depth First Search) We parallelize DFS by sharing the work to be done among a number of processors. Each processor searches a disjoint part of the search space in a depth first fashion. When a processor has finished searching its ....

V. Kumar and L. N. Kanal. A general branch-and-bound formulations for understanding and synthesizing and/or tree search procedures. Artificial Intelligence, 21:179--198, 1983.

....speedups on a 1024 processor Ncube TM , a 128 processor Symult TM and a network of 16 SUN workstations in the context of the floorplan optimization problem for VLSI circuits. The classical ff Gamma fi game tree search algorithm can also be viewed as a depth first branch andbound algorithm[72, 78]. Unfortunately, the information about the current best solution (which in this case is the current best winning strategy) cannot be captured in a single number. Hence the simple solution discussed above cannot be used to develop an effective parallel formulation of ff Gamma fi. A number of ....

V. Kumar and L. N. Kanal. A general branch-and-bound formulations for understanding and synthesizing and/or tree search procedures. Artificial Intelligence, 21:179--198, 1983.

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Kumar, V. and Kanal, L. N., 1983, "A General Branch And Bound Formulation For Understanding And Synthesizing And/Or Tree Search Procedures," Artificial Intelligence, 21:179-198.

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V. Kumar and L.N. Kanal, A General Branch and Bound Formulation for Understanding and Synthesizing And#Or Tree Search Procedures, Arti#cial Intelligence 21 #1983#, 179#198.

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V. Kumar, L.N. Kanal (1983). A general branch and bound formulation for understanding and synthesizing And/Or tree search procedures. Art. Intelligence 21, 179--198.

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V. Kumar and L. N. Kanal, "A General Branch and Bound Formulation for Understanding and Synthesizing And/Or Tree Search Procedures," Artificial Intelligence, vol. 21, no. 1-2, pp. 179-198, North-Holland, 1983.

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