Complexity of finding embeddings in a k-tree
SIAM Journal on Algebraic and Discrete Methods
The pathwidth and treewidth of cographs
SIAM Journal on Discrete Mathematics
Approximating treewidth, pathwidth, frontsize, and shortest elimination tree
Journal of Algorithms
On the hardness of approximate reasoning
Artificial Intelligence
A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth
SIAM Journal on Computing
A partial k-arboretum of graphs with bounded treewidth
Theoretical Computer Science
Graph classes: a survey
Bayesian Networks and Decision Graphs
Bayesian Networks and Decision Graphs
Treewidth and Minimum Fill-in: Grouping the Minimal Separators
SIAM Journal on Computing
How to Use the Minimal Separators of a Graph for its Chordal Triangulation
ICALP '95 Proceedings of the 22nd International Colloquium on Automata, Languages and Programming
Optimal decomposition of belief networks
UAI '90 Proceedings of the Sixth Annual Conference on Uncertainty in Artificial Intelligence
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Tree-decompositions of small pathwidth
Discrete Applied Mathematics - Structural decompositions, width parameters, and graph labelings (DAS 5)
Uniform Constraint Satisfaction Problems and Database Theory
Complexity of Constraints
Tree decompositions of graphs: Saving memory in dynamic programming
Discrete Optimization
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The f-cost of a tree decomposition ({X"i|i@?I},T=(I,F)) for a function f:N-R^+ is defined as @?"i"@?"If(|X"i|). This measure associates with the running time or memory use of some algorithms that use the tree decomposition. In this paper, we investigate the problem to find tree decompositions of minimum f-cost. A function f:N-R^+ is fast, if for every i@?N: f(i+1)=2f(i). We show that for fast functions f, every graph G has a tree decomposition of minimum f-cost that corresponds to a minimal triangulation of G; if f is not fast, this does not hold. We give polynomial time algorithms for the problem, assuming f is a fast function, for graphs that have a polynomial number of minimal separators, for graphs of treewidth at most two, and for cographs, and show that the problem is NP-hard for bipartite graphs and for cobipartite graphs. We also discuss results for a weighted variant of the problem derived of an application from probabilistic networks.