Results 1  10
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93
Graphical models, exponential families, and variational inference
, 2008
"... The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fiel ..."
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Cited by 800 (26 self)
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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building largescale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide varietyof algorithms — among them sumproduct, cluster variational methods, expectationpropagation, mean field methods, maxproduct and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in largescale statistical models.
Spanning Trees in Hypergraphs with Applications to Steiner Trees
, 1998
"... This dissertation examines the geometric Steiner tree problem: given a set of terminals in the plane, find a minimumlength interconnection of those terminals according to some geometric distance metric. In the process, however, it addresses a much more general and widely applicable problem, that of ..."
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Cited by 25 (1 self)
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, that of finding a minimumweight spanning tree in a hypergraph. The geometric Steiner tree problem is known to be NPcomplete for the rectilinear metric, and NPhard for the Euclidean metric. The fastest exact algorithms (in practice) for these problems use two phases: First a small but sufficient set of full
Counting Independent Sets in Hypergraphs when Strong Spatial Mixing Fails
, 2015
"... We study the problem of approximately counting independent sets in hypergraphs with maximum degree ∆ whose hyperedges have arity at least k 3. In the graphical case, where the arity of every edge is 2, it is known that for ∆ 6 the problem is hard and for ∆ 5 there is an FPTAS. This approximatio ..."
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We study the problem of approximately counting independent sets in hypergraphs with maximum degree ∆ whose hyperedges have arity at least k 3. In the graphical case, where the arity of every edge is 2, it is known that for ∆ 6 the problem is hard and for ∆ 5 there is an FPTAS
The complexity of homomorphism and constraint satisfaction problems seen from the other side
, 2006
"... We give a complexity theoretic classification of homomorphism problems for graphs and, more generally, relational structures obtained by restricting the left hand side structure in a homomorphism. For every class C of structures, let HOM(C, −) be the problem of deciding whether a given structure A ∈ ..."
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Cited by 97 (4 self)
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We give a complexity theoretic classification of homomorphism problems for graphs and, more generally, relational structures obtained by restricting the left hand side structure in a homomorphism. For every class C of structures, let HOM(C, −) be the problem of deciding whether a given structure A ∈ C has a homomorphism to a given (arbitrary) structure B. We prove that, under some complexity theoretic assumption from parameterized complexity theory, HOM(C, −) is in polynomial time if and only if C has bounded treewidth modulo homomorphic equivalence. Translated into the language of constraint satisfaction problems, our result yields a characterization of the tractable structural restrictions of constraint satisfaction problems. Translated into the language of database theory, it implies a characterization of the tractable instances of the evaluation problem for conjunctive queries over relational databases.
Heuristic Methods for Hypertree Decompositions
, 2007
"... In this paper we propose new algorithms for generating generalized hypertree decompositions. The well known heuristics for generating tree decompositions based on vertex ordering have been extended to produce hypertree decompositions. We investigate the generation of hypertree decompositions based ..."
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Cited by 16 (3 self)
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on the tree decompositions of the primal and the dual graph of the hypergraph. Further, we propose a method for generating hypertree decompositions using hypergraph partitioning. We use different algorithms for partitioning hypergraphs. The proposed algorithms are experimentally evaluated in benchmark
The complexity of acyclic subhypergraph problems
"... Abstract. We investigate the computational complexity of two decision problems concerning the existence of certain acyclic subhypergraphs of a given hypergraph, namely the Spanning Acyclic Subhypergraph problem and the Maximal Acyclic Subhypergraph problem. The former is about the existence of an ac ..."
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Cited by 2 (0 self)
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Abstract. We investigate the computational complexity of two decision problems concerning the existence of certain acyclic subhypergraphs of a given hypergraph, namely the Spanning Acyclic Subhypergraph problem and the Maximal Acyclic Subhypergraph problem. The former is about the existence
Understanding Tractable Decompositions for Constraint
"... Constraint satisfaction problems (CSPs) are NPcomplete in general, therefore it is important to identify tractable subclasses. A possible way to find such subclasses is to associate a hypergraph to the problem and impose restrictions on its structure. In this thesis we follow this direction. Among ..."
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. The interesting decomposition concepts are those which both enable the problems in the defined subclass to be solved in polynomial time and the associated hypergraphs to be recognized efficiently. Hypertree decompositions, introduced by Gottlob et al. in [43], fall in this category and additionally, for a long
A PROBLEM REDUCTION MODEL FOR NON INDEPENDENT SUBPROBLEMS
"... A hypergraph model is introduced, which besides including the AND/OR graph and state space graph models as particulars, is adequate for problem solving tasks involving non independent subproblems. The hypergraph model is shown to be grounded on a nonstandard notion of conjunction such that the truth ..."
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A hypergraph model is introduced, which besides including the AND/OR graph and state space graph models as particulars, is adequate for problem solving tasks involving non independent subproblems. The hypergraph model is shown to be grounded on a nonstandard notion of conjunction
Combinatorial Optimization on Graphs of Bounded Treewidth
, 2007
"... There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixedparameter tractable algorithms. Starting from trees an ..."
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Cited by 51 (4 self)
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There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixedparameter tractable algorithms. Starting from trees and seriesparallel graphs, we introduce the concepts of treewidth and tree decompositions, and illustrate the technique with the Weighted Independent Set problem as an example. The paper surveys some of the latest developments, putting an emphasis on applicability, on algorithms that exploit tree decompositions, and on algorithms that determine or approximate treewidth and find tree decompositions with optimal or close to optimal treewidth. Directions for further research and suggestions for further reading are also given.
Results 1  10
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93