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Exact structure discovery in Bayesian networks with less space
 In Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence (UAI
, 2009
"... The fastest known exact algorithms for scorebased structure discovery in Bayesian networks on n nodes run in time and space 2 n n O(1). The usage of these algorithms is limited to networks on at most around 25 nodes mainly due to the space requirement. Here, we study space–time tradeoffs for finding ..."
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Cited by 11 (5 self)
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The fastest known exact algorithms for scorebased structure discovery in Bayesian networks on n nodes run in time and space 2 n n O(1). The usage of these algorithms is limited to networks on at most around 25 nodes mainly due to the space requirement. Here, we study space–time tradeoffs for finding an optimal network structure. When little space is available, we apply the Gurevich– Shelah recurrence—originally proposed for the Hamiltonian path problem—and obtain time 2 2n−s n O(1) in space 2 s n O(1) for any s = n/2,n/4,n/8,...; we assume the indegree of each node is bounded by a constant. For the more practical setting with moderate amounts of space, we present a novel scheme. It yields running time 2 n (3/2) p n O(1) in space 2 n (3/4) p n O(1) for any p = 0,1,...,n/2; these bounds hold as long as the indegrees are at most 0.238n. Furthermore, the latter scheme allows easy and efficient parallelization beyond previous algorithms. We also explore empirically the potential of the presented techniques. 1
A space–time tradeoff for permutation problems
 In Proceedings of the ACMSIAM Symposium on Discrete Algorithms (SODA
, 2010
"... Many combinatorial problems—such as the traveling salesman, feedback arcset, cutwidth, and treewidth problem— can be formulated as finding a feasible permutation of n elements. Typically, such problems can be solved by dynamic programming in time and space O ∗ (2 n), by divide and conquer in time O ..."
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Cited by 6 (3 self)
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Many combinatorial problems—such as the traveling salesman, feedback arcset, cutwidth, and treewidth problem— can be formulated as finding a feasible permutation of n elements. Typically, such problems can be solved by dynamic programming in time and space O ∗ (2 n), by divide and conquer in time O ∗ (4 n) and polynomial space, or by a combination of the two in time O ∗ (4 n 2 −s) and space O ∗ (2 s) for s = n, n/2, n/4,.... Here, we show that one can improve the tradeoff to time O ∗ (T n) and space O ∗ (S n) with T S < 4 at any √ 2 < S < 2. The idea is to find a small family of “thin ” partial orders on the n elements such that every linear order is an extension of one member of the family. Our construction is optimal within a natural class of partial order families. 1
Algorithms and Complexity Results for Exact Bayesian Structure Learning
"... Bayesian structure learning is the NPhard problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worstcase complexity of exact Bayesian structure learning under graph theoretic restrictions on the superstructure. ..."
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Cited by 3 (1 self)
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Bayesian structure learning is the NPhard problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worstcase complexity of exact Bayesian structure learning under graph theoretic restrictions on the superstructure. The superstructure (a concept introduced by Perrier, Imoto, and Miyano, JMLR 2008) is an undirected graph that contains as subgraphs the skeletons of solution networks. Our results apply to several variants of scorebased Bayesian structure learning where the score of a network decomposes into local scores of its nodes. Results: We show that exact Bayesian structure learning can be carried out in nonuniform polynomial time if the superstructure has bounded treewidth and in linear time if in addition the superstructure has bounded maximum degree. We complement this with a number of hardness results. We show that both restrictions (treewidth and degree) are essential and cannot be dropped without loosing uniform polynomial time tractability (subject to a complexitytheoretic assumption). Furthermore, we show that the restrictions remain essential if we do not search for a globally optimal network but we aim to improve a given network by means of at most k arc additions, arc deletions, or arc reversals (kneighborhood local search).
METHODOLOGY ARTICLE Learning genetic epistasis using Bayesian network scoring criteria
"... Background: Genegene epistatic interactions likely play an important role in the genetic basis of many common diseases. Recently, machinelearning and data mining methods have been developed for learning epistatic relationships from data. A wellknown combinatorial method that has been successfully ..."
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Background: Genegene epistatic interactions likely play an important role in the genetic basis of many common diseases. Recently, machinelearning and data mining methods have been developed for learning epistatic relationships from data. A wellknown combinatorial method that has been successfully applied for detecting epistasis is Multifactor Dimensionality Reduction (MDR). Jiang et al. created a combinatorial epistasis learning method called BNMBL to learn Bayesian network (BN) epistatic models. They compared BNMBL to MDR using simulated data sets. Each of these data sets was generated from a model that associates two SNPs with a disease and includes 18 unrelated SNPs. For each data set, BNMBL and MDR were used to score all 2SNP models, and BNMBL learned significantly more correct models. In real data sets, we ordinarily do not know the number of SNPs that influence phenotype. BNMBL may not perform as well if we also scored models containing more than two SNPs. Furthermore, a number of other BN scoring criteria have been developed. They may detect epistatic interactions even better than BNMBL. Although BNs are a promising tool for learning epistatic relationships from data, we cannot confidently use them in this domain until we determine which scoring criteria work best or even well when we try learning the correct model without knowledge of the number of SNPs in that model.
Parameterized Complexity Results for Exact Bayesian Network Structure Learning
"... Bayesian network structure learning is the notoriously difficult problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worstcase complexity of exact Bayesian network structure learning under graph theoretic restric ..."
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Bayesian network structure learning is the notoriously difficult problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worstcase complexity of exact Bayesian network structure learning under graph theoretic restrictions on the (directed) superstructure. The superstructure is an undirected graph that contains as subgraphs the skeletons of solution networks. We introduce the directed superstructure as a natural generalization of its undirected counterpart. Our results apply to several variants of scorebased Bayesian network structure learning where the score of a network decomposes into local scores of its nodes. Results: We show that exact Bayesian network structure learning can be carried out in nonuniform polynomial time if the superstructure has bounded treewidth, and in linear time if in addition the superstructure has bounded maximum degree. Furthermore, we show that if the directed superstructure is acyclic, then exact Bayesian network structure learning can be carried out in quadratic time. We complement these positive results with a number of hardness results. We show that both restrictions (treewidth and degree) are essential and cannot be dropped without loosing uniform polynomial time tractability (subject to a complexitytheoretic assumption). Similarly, exact Bayesian network structure learning remains NPhard for “almost acyclic ” directed superstructures. Furthermore, we show that the restrictions remain essential if we do not search for a globally optimal network but aim to improve a given network by means of at most k arc additions, arc deletions, or arc reversals (kneighborhood local search). 1.
Learning Optimal Bayesian Networks: A Shortest Path Perspective
"... In this paper, learning a Bayesian network structure that optimizes a scoring function for a given dataset is viewed as a shortest path problem in an implicit statespace search graph. This perspective highlights the importance of two research issues: the development of search strategies for solving ..."
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In this paper, learning a Bayesian network structure that optimizes a scoring function for a given dataset is viewed as a shortest path problem in an implicit statespace search graph. This perspective highlights the importance of two research issues: the development of search strategies for solving the shortest path problem, and the design of heuristic functions for guiding the search. This paper introduces several techniques for addressing the issues. One is an A * search algorithm that learns an optimal Bayesian network structure by only searching the most promising part of the solution space. The others are mainly two heuristic functions. The first heuristic function represents a simple relaxation of the acyclicityconstraintofaBayesiannetwork. Althoughadmissibleandconsistent, the heuristic may introduce too much relaxation and result in a loose bound. The second heuristic function reduces the amount of relaxation by avoiding directed cycles within some groups of variables. Empirical results show that these methods constitute a promising approach to learning optimal Bayesian network structures. 1.