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34
Learning Bayesian Networks is NPHard
, 1994
"... Algorithms for learning Bayesian networks from data have two components: a scoring metric and a search procedure. The scoring metric computes a score reflecting the goodnessoffit of the structure to the data. The search procedure tries to identify network structures with high scores. Heckerman et ..."
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Cited by 131 (2 self)
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Algorithms for learning Bayesian networks from data have two components: a scoring metric and a search procedure. The scoring metric computes a score reflecting the goodnessoffit of the structure to the data. The search procedure tries to identify network structures with high scores. Heckerman et al. (1994) introduced a Bayesian metric, called the BDe metric, that computes the relative posterior probability of a network structure given data. They show that the metric has a property desireable for inferring causal structure from data. In this paper, we show that the problem of deciding whether there is a Bayesian networkamong those where each node has at most k parentsthat has a relative posterior probability greater than a given constant is NPcomplete, when the BDe metric is used. 1 Introduction Recently, many researchers have begun to investigate methods for learning Bayesian networks, including Bayesian methods [Cooper and Herskovits, 1991, Buntine, 1991, York 1992, Spiegel...
A complexity analysis of spacebounded learning algorithms for the constraint satisfaction problem
 In Proceedings of the Thirteenth National Conference on Artificial Intelligence
, 1996
"... Learning during backtrack search is a spaceintensive process that records information (such as additional constraints) in order to avoid redundant work. In this paper, we analyze the effects of polynomialspacebounded learning on runtime complexity of backtrack search. One spacebounded learning sc ..."
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Cited by 77 (2 self)
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Learning during backtrack search is a spaceintensive process that records information (such as additional constraints) in order to avoid redundant work. In this paper, we analyze the effects of polynomialspacebounded learning on runtime complexity of backtrack search. One spacebounded learning scheme records only those constraints with limited size, and another records arbitrarily large constraints but deletes those that become irrelevant to the portion of the search space being explored. We find that relevancebounded learning allows better runtime bounds than sizebounded learning on structurally restricted constraint satisfaction problems. Even when restricted to linear space, our relevancebounded learning algorithm has runtime complexity near that of unrestricted (exponential spaceconsuming) learning schemes.
Feedback set problems
 HANDBOOK OF COMBINATORIAL OPTIMIZATION
, 1999
"... ABSTRACT. This paper is a short survey of feedback set problems. It will be published in ..."
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Cited by 36 (1 self)
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ABSTRACT. This paper is a short survey of feedback set problems. It will be published in
Link length of rectilinear Hamiltonian tours
 in grids, Ars Combinatoria
, 1994
"... The link length of a walk in a multidimensional grid is the number of straight line segments constituting the walk. Alternatively, it is the number of turns that a mobile unit needs to perform in traversing the walk. A rectilinear walk consists of straight line segments which are parallel to the mai ..."
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Cited by 8 (0 self)
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The link length of a walk in a multidimensional grid is the number of straight line segments constituting the walk. Alternatively, it is the number of turns that a mobile unit needs to perform in traversing the walk. A rectilinear walk consists of straight line segments which are parallel to the main axis. We wish to construct rectilinear walks with minimal link length traversing grids. If G denotes the multidimensional grid, let s(G) be the minimal link length of a rectilinear walk traversing all the vertices of G. In this paper we develop an asymptotically optimal algorithm for constructing rectilinear walks traversing all the vertices of complete multidimensional grids and analyze the worstcase behavior of s(G), when G is a multidimensional grid.
Combinatorial Algorithms for Feedback Problems in Directed Graphs
 Inf. Process. Lett
, 2003
"... Given a weighted directed graph G = (V, A), the minimum feedback arc set problem consists of finding a minimum weight set of arcs A # A such that the directed graph A # ) is acyclic. Similarly, the minimum feedback vertex set problem consists of finding a minimum weight set of vertices containi ..."
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Cited by 8 (1 self)
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Given a weighted directed graph G = (V, A), the minimum feedback arc set problem consists of finding a minimum weight set of arcs A # A such that the directed graph A # ) is acyclic. Similarly, the minimum feedback vertex set problem consists of finding a minimum weight set of vertices containing at least one vertex for each directed cycle. Both problems are NPcomplete. We present simple combinatorial algorithms for these problems that achieve an approximation ratio bounded by the length, in terms of number of arcs, of a longest simple cycle of the digraph.
EventBased Decompositions for Reasoning about External Change in Planners
 In Proceedings of the Third International Conference on AI Planning Systems, 27–34. Menlo Park, Calif
, 1996
"... An increasing number of planners can handle uncertainty in the domain or in action outcomes. However, less work has addressed building plans when the planner’s world can change independently of the planning agent in an uncertain manner. In this paper, I model this change with external events that co ..."
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Cited by 7 (0 self)
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An increasing number of planners can handle uncertainty in the domain or in action outcomes. However, less work has addressed building plans when the planner’s world can change independently of the planning agent in an uncertain manner. In this paper, I model this change with external events that concisely represent some aspects of structure in the planner’s domain. This event model is given a formal semantics in terms of a Markov chain, but probabilistic computations from this chain would be intractable in realworld domains. I describe a technique, based on a reachability analysis of a graph built from the events, that allows abstractions of the Markov chain to be built to answer specific queries efficiently. I prove that the technique is correct. I have implemented a planner that uses this technique, and I show an example from a large planning domain.
Linear Orderings of Random Geometric Graphs
 GRAPH THEORETIC CONCEPTS IN COMPUTER SCIENCE
, 1997
"... In random geometric graphs, vertices are randomly distributed on [0, 1]² and pairs of vertices are connected by edges whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a certain measure is minimized. In this paper, we study sev ..."
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Cited by 7 (4 self)
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In random geometric graphs, vertices are randomly distributed on [0, 1]² and pairs of vertices are connected by edges whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth, Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that some of these problems remain NPcomplete even for geometric graphs. Afterwards, we compute lower bounds that hold with high probability on random geometric graphs. Finally, we characterize the probabilistic behavior of the lexicographic ordering for our layout problems on the class of random geometric graphs.
A note on CSP graph parameters
, 1999
"... Several graph parameters such as induced width, minimum maximum clique size of a chordal completion, ktree number, bandwidth, front length or minimum pseudotree height are available in the CSP community to bound the complexity of specific CSP instances using dedicated algorithms. After an intro ..."
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Cited by 7 (0 self)
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Several graph parameters such as induced width, minimum maximum clique size of a chordal completion, ktree number, bandwidth, front length or minimum pseudotree height are available in the CSP community to bound the complexity of specific CSP instances using dedicated algorithms. After an introduction to the main algorithms that can exploit these parameters, we try to exhaustively review existing parameters and the relations that may exist between then. In the process we exhibit some missing relations. Several existing results, both old results and recent results from graph theory and Cholesky matrix factorization technology [BGHK95] allow us to give a very dense map of relations between these parameters. These results strongly relate several existing algorithms and answer some questions which were considered as open in the CSP community. Warning: this document is a working paper. Some sections may be incomplete or currently being worked out ([GJC94] degree of cyclicity not ...
On Bandwidth, Cutwidth, and Quotient Graphs
"... . The bandwidth and the cutwidth are fundamental parameters which can give indications on the complexity of many problems described in terms of graphs. In this paper, we present a method for finding general upper bounds for the bandwidth and the cutwidth of a given graph from those of any of its quo ..."
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Cited by 6 (2 self)
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. The bandwidth and the cutwidth are fundamental parameters which can give indications on the complexity of many problems described in terms of graphs. In this paper, we present a method for finding general upper bounds for the bandwidth and the cutwidth of a given graph from those of any of its quotient graphs. Moreover, general lower bounds are obtained by using vertexand edgebisection notions. These results are used, in a second time, to study various interconnection networks: by choosing convenient vertex partitions and judicious internal numberings for the vertices of the partition subsets, we show that bounds previously known for hypercubes can be easily reproven, and we give original bounds for 2Dmesh, binary de Bruijn, ShuffleExchange, FFT, Butterfly, and CCC graphs. 1 Introduction In all this paper, we will denote by V (G) and E(G) the vertex and edgesets of a nvertex graph G. When studying problems described in terms of graphs, it is often useful to have a good knowl...