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Organizing the atoms of the clique separator decomposition into an atom tree

Anne Berry, Romain Pogorelcnik, Geneviève Simonet
2014 Discrete Applied Mathematics  
We define an atom tree of a graph as a generalization of a clique tree: its nodes are the atoms obtained by clique minimal separator decomposition, and its edges correspond to the clique minimal separators  ...  Finally, we present a process which uses a traversal of a clique tree of a minimal triangulation to determine the clique minimal separators and build the corresponding atom tree in O(n(n + t)) time, where  ...  case of improving the time complexity of computing the clique minimal separator decomposition to less than O(n α ) time.  ... 
doi:10.1016/j.dam.2014.05.030 fatcat:jieqltdxebfxfmngkas2u6jl74

Maximum likelihood bounded tree-width Markov networks

Nathan Srebro
2003 Artificial Intelligence  
We study the problem of projecting a distribution onto (or finding a maximum likelihood distribution among) Markov networks of bounded tree-width.  ...  By casting it as the combinatorial optimization problem of finding a maximum weight hypertree, we prove that it is NP-hard to solve exactly and provide an approximation algorithm with a provable performance  ...  Acknowledgments I would like to thank Tommi Jaakkola for helpful discussions and advice, and David Karger for working with me on the approximation algorithm presented in the companion paper [KS01] and  ... 
doi:10.1016/s0004-3702(02)00360-0 fatcat:gmbhujk7azfvxjz3ej4i5rwofm

Graph Structure and Monadic Second-Order Logic: Language Theoretical Aspects [chapter]

Bruno Courcelle
2008 Lecture Notes in Computer Science  
Hierarchical decompositions yielding the notions of tree-width and clique-width, expressed by terms written with appropriate graph operations and associated with Monadic Second-order Logic are important  ...  This informal overview presents the main definitions, results and open problems and tries to answer some frequently asked questions.  ...  Fellows and I. Walukiewicz for many useful comments on a first draft of this overview. The first 11 references are books and survey articles.  ... 
doi:10.1007/978-3-540-70575-8_1 fatcat:ftpcqszcb5fs5nswbqnbnmy5ja

Decomposition, Approximation, and Coloring of Odd-Minor-Free Graphs [chapter]

Erik D. Demaine, MohammadTaghi Hajiaghayi, Ken-ichi Kawarabayashi
2010 Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms  
We also believe that combining our clique-sum decomposition (Theorem 1.2 below) with Grohe's technique [Gro03] leads to PTASs for maximum independent set, maximum clique, and minimum vertex cover.  ...  Specifically, we prove that every odd-minor-free graph can be decomposed into a clique-sum of "almost-bipartite" graphs and graphs that are "almost-embeddable" into bounded-genus surfaces.  ...  Acknowledgments We thank Jim Geelen and Bruce Reed for many helpful and enlightening discussions. We also thank Paul Seymour for general insights into Graph Minor Theory.  ... 
doi:10.1137/1.9781611973075.28 dblp:conf/soda/DemaineHK10 fatcat:zpycqbi2bvbkxiz5jbourtxauy

Decomposing a relation into a tree of binary relations

Rina Dechter
1990 Journal of computer and system sciences (Print)  
If a tree decomposition exists, the algorithm is guaranteed to find one. Otherwise, the tree generated by the algorithm can be used as an approximation of the relation.  ...  DECOMPOSING A RELATION INTO A TREE 3 decomposition scheme can be extended to form acyclic database instances where the arity of each subrelation is greater than two.  ...  I thank the students, Bit Moshe and Ronit Nadam, who implemented the tree-decomposition algorithm and provided the experimental results presented in the paper. I also thank the referees of this paper.  ... 
doi:10.1016/0022-0000(90)90031-f fatcat:dvg7acmpnjczlldmwbv3xu2uga

Decomposition Bounds for Marginal MAP [article]

Wei Ping, Qiang Liu, Alexander Ihler
2015 arXiv   pre-print
In this work, we generalize dual decomposition to a generic power sum inference task, which includes marginal MAP, along with pure marginalization and MAP, as special cases.  ...  It is significantly more difficult than pure marginalization and MAP tasks, for which a large class of efficient and convergent variational algorithms, such as dual decomposition, exist.  ...  Acknowledgments This work is sponsored in part by NSF grants IIS-1065618 and IIS-1254071. Alexander Ihler is also funded in part by the United States Air Force under Contract No.  ... 
arXiv:1511.02619v1 fatcat:7xjbc2oupra6jnqpttd2hjrunq

Approximation Algorithms for Domination Search [chapter]

Fedor V. Fomin, Petr A. Golovach, Dimitrios M. Thilikos
2011 Lecture Notes in Computer Science  
The r-domination search game on graphs is a game-theoretical approach to several graph and hypergraph parameters including treewidth and hypertree width.  ...  The task is to identify the minimum number of cops sufficient to catch the visible and fast robber.  ...  The idea behind the winning strategy for cops is to "chase" the robber in the graph along m + 1 decompositions: one is induced by the clique-sum decomposition and others are tree decompositions of F (G  ... 
doi:10.1007/978-3-642-18318-8_12 fatcat:spusyuolmbhjxfobq6orv3usym

Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs

Erik D. Demaine, Fedor V. Fomin, Mohammadtaghi Hajiaghayi, Dimitrios M. Thilikos
2005 Journal of the ACM  
This bound depends linearly on the size |V (H )| of the excluded graph H and the genus g(G) of the graph G, and applies and extends the graph-minors work of Robertson and Seymour.  ...  set, edge dominating set, disk dimension, and many others restricted to bounded-genus graphs (phrased as bipartite-graph problem).  ...  We also thank Naomi Nishimura and Prabhakar Ragde for encouragement, helpful discussions, and advice. Finally, we thank the three anonymous referees for their helpful comments.  ... 
doi:10.1145/1101821.1101823 fatcat:3ooif6ldsvd3pm4fw4a3tgonym

Abstract homomorphisms from locally compact groups to discrete groups [article]

Linus Kramer, Olga Varghese
2019 arXiv   pre-print
If L is a locally compact group and if G is a discrete group which contains no infinite torsion group and no infinitely generated abelian group, then every abstract homomorphism φ:L→ G is either continuous  ...  As an application we obtain results concerning the continuity of homomorphisms from locally compact groups to Artin and Coxeter groups.  ...  By Iwasawa's decomposition [26, Theorem 13] we have L = H 1 H 2 • • • H r K, where K is a connected compact group and H i ∼ = R for i = 1, . . . , r.  ... 
arXiv:1902.07962v2 fatcat:z6mspgqt3jadbiuhhfakno6kjm

Approximating acyclicity parameters of sparse hypergraphs [article]

Fedor V. Fomin, Petr A. Golovach, Dimitrios M. Thilikos
2008 arXiv   pre-print
The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity.  ...  These notions were further generalized by Grohe and Marx, who introduced the fractional hypertree width of a hypergraph.  ...  The clique-sum decomposition G = G 1 ⊕ G 2 ⊕ · · · ⊕ G m can be considered as a tree decomposition (T, χ) of G for some tree T with nodes {1, 2, . . . , m} such that χ(i) = V (G i ), i.e. the vertex send  ... 
arXiv:0809.3646v1 fatcat:narokqvabff7xbwqioxp6jp3ny

Maximum Likelihood Bounded Tree-Width Markov Networks [article]

Nathan Srebro
2013 arXiv   pre-print
Chow and Liu (1968) studied the problem of learning a maximumlikelihood Markov tree.  ...  We show that learning a maximum likelihoodMarkov network of bounded tree-width is NP-hard, and discuss thehardness of approximation.  ...  Acknowledgements I would like to thank Tommi Jaakkola for many help ful discussions and help in preparing this paper, and David Karger for working with me on the approxi mation algorithm presented in the  ... 
arXiv:1301.2311v1 fatcat:wqu3v33j25cj7cjwp4d74kph6m

A chordal decomposition approach to scalable design of structured feedback gains over directed graphs

Yang Zheng, Richard P. Mason, Antonis Papachristodoulou
2016 2016 IEEE 55th Conference on Decision and Control (CDC)  
Combining the decomposition with the relaxation, we propose a sequential design algorithm to obtain structured feedback gains clique-by-clique over a clique tree of the underlying chordal graph.  ...  By exploiting the sparsity properties of the problem, and using chordal decomposition, a scalable algorithm is proposed to compute structured stabilizing feedback gains for large-scale systems over directed  ...  Fig. 1 : 1 Example of chordal extension and clique tree: (a) nonchordal graph, (b) chordal graph, (c) clique tree.  ... 
doi:10.1109/cdc.2016.7799333 dblp:conf/cdc/ZhengMP16 fatcat:5cryvbpmqngkvjmnvsbc3r4im4

CUI Networks: A Graphical Representation for Conditional Utility Independence

Y. Engel, M. P. Wellman
2008 The Journal of Artificial Intelligence Research  
We show how conditional utility independence leads to an effective functional decomposition that can be exhibited graphically, and how local, compact data at the graph nodes can be used to calculate joint  ...  We introduce CUI networks, a compact graphical representation of utility functions over multiple attributes.  ...  sums and products of multiple value nodes.  ... 
doi:10.1613/jair.2360 fatcat:33zs3jq2fnckdkydeqfyfypvle

Compiling constraint satisfaction problems

Rainer Weigel, Boi Faltings
1999 Artificial Intelligence  
In order to represent the space in a compact form, it exploits clustering and interchangeability techniques. Search for solutions can then be limited to a usually much smaller, precomputed space.  ...  We consider constraint satisfaction as a general paradigm and describe compilation techniques.  ...  clique decomposition method.  ... 
doi:10.1016/s0004-3702(99)00077-6 fatcat:dk5tcotuevhvfpxg3djfhbfjb4

Catalan structures and dynamic programming in H-minor-free graphs

Frederic Dorn, Fedor V. Fomin, Dimitrios M. Thilikos
2012 Journal of computer and system sciences (Print)  
This result is used to bound the number of ways vertex disjoint paths can be routed through the separators of such decompositions.  ...  The proof is based on several structural theorems from the Graph Minors series of Robertson and Seymour.  ...  Then, G is called the h-clique-sum of the clique-sum components G 1 and G 2 . Note that the clique-sum gives many graphs as output depending on the edges of the clique that are deleted.  ... 
doi:10.1016/j.jcss.2012.02.004 fatcat:y5cjuqys3rewxmccwfy2hrkuwe
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