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A review on distinct methods and approaches to perform triangulation for Bayesian networks
 Advances in Probabilistic Graphical Models
, 2007
"... Summary. Triangulation of a Bayesian network (BN) is somehow a necessary step in order to perform inference in a more efficient way, either if we use a secondary structure as the join tree (JT) or implicitly when we try to use other direct techniques on the network. If we focus on the first procedur ..."
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Summary. Triangulation of a Bayesian network (BN) is somehow a necessary step in order to perform inference in a more efficient way, either if we use a secondary structure as the join tree (JT) or implicitly when we try to use other direct techniques on the network. If we focus on the first procedure, the goodness of the triangulation will affect on the simplicity of the join tree and therefore on a quicker and easier inference process. The task of obtaining an optimal triangulation (in terms of producing the minimum number of triangulation links a.k.a. fillins) has been proved as an NPhard problem. That is why many methods of distinct nature have been used with the purpose of getting as good as possible triangulations for any given network, especially important for big structures, that is, with a large number of variables and links. In this chapter, we attempt to introduce the problem of triangulation, locating it in the compilation process and showing first its relevance for inference, and consequently for working with Bayesian networks. After this introduction, the most popular and used strategies to cope with the triangulation problem are reviewed,
ACSTS: TRAIN SCHEDULING USING ANT COLONY SYSTEM
, 2006
"... This paper develops an algorithm for the train scheduling problem using the ant colony system metaheuristic called ACSTS. At first, a mathematical model for a kind of train scheduling problem is developed and then the algorithm based on ACS is presented to solve the problem. The problem is consider ..."
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This paper develops an algorithm for the train scheduling problem using the ant colony system metaheuristic called ACSTS. At first, a mathematical model for a kind of train scheduling problem is developed and then the algorithm based on ACS is presented to solve the problem. The problem is considered as a traveling salesman problem (TSP) wherein cities represent the trains. ACS determines the sequence of trains dispatched on the graph of the TSP. Using the sequences obtained and removing the collisions incurred, train scheduling is determined. Numerical examples in small andmedium sizes are solved using ACSTS and compared to exact optimum solutions to check for quality and accuracy. Comparison of the solutions shows that ACSTS results in good quality and time savings. A case study is presented to illustrate the solution. Copyright © 2006 K. Ghoseiri and F. Morshedsolouk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
ACSTS: TRAIN SCHEDULING USING ANT COLONY SYSTEM
, 2006
"... This paper develops an algorithm for the train scheduling problem using the ant colony system metaheuristic called ACSTS. At first, a mathematical model for a kind of train scheduling problem is developed and then the algorithm based on ACS is presented to solve the problem. The problem is consider ..."
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This paper develops an algorithm for the train scheduling problem using the ant colony system metaheuristic called ACSTS. At first, a mathematical model for a kind of train scheduling problem is developed and then the algorithm based on ACS is presented to solve the problem. The problem is considered as a traveling salesman problem (TSP) wherein cities represent the trains. ACS determines the sequence of trains dispatched on the graph of the TSP. Using the sequences obtained and removing the collisions incurred, train scheduling is determined. Numerical examples in small andmedium sizes are solved using ACSTS and compared to exact optimum solutions to check for quality and accuracy. Comparison of the solutions shows that ACSTS results in good quality and time savings. A case study is presented to illustrate the solution.
Belief updating in Bayesian networks by using a criterion of minimum time
, 2008
"... Variable elimination (VE) and clustering algorithms (CAs) are two widely used algorithms for exact inference in Bayesian networks. Both the problem of finding an optimal variable elimination ordering in VE and the problem of finding an optimal graph triangulation in CAs are N Pcomplete, although gr ..."
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Variable elimination (VE) and clustering algorithms (CAs) are two widely used algorithms for exact inference in Bayesian networks. Both the problem of finding an optimal variable elimination ordering in VE and the problem of finding an optimal graph triangulation in CAs are N Pcomplete, although greedy algorithms work well in practice. Usually, VE selects the next variable to be eliminated such that a new potential of minimum size is generated during the elimination process. CAs create a variable elimination sequence in order to triangulate the moral graph; usually, the next variable to be eliminated is selected such that a new clique of minimum size is created during the elimination process. This paper presents an approach which makes use of a criterion of minimum time (CMT) for the selection of the next variable to be eliminated in VE or in CAs, and compares its performance with that of the traditional approaches using a criterion of minimum size. The results show that, in general, the CMT introduced in this paper allows inference time to be reduced. Results regarding memory requirements are also reported.