Tree decomposition
In graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph.
In machine learning, tree decompositions are also called junction trees, clique trees, or join trees; they play an important role in problems like probabilistic inference, constraint satisfaction, query optimization, and matrix decomposition.
The concept of tree decompositions was originally introduced by Rudolf Halin (1976). Later it was rediscovered by Neil Robertson and Paul Seymour (1984) and has since been studied by many other authors.
Intuitively, a tree decomposition represents the vertices of a given graph G as subtrees of a tree, in such a way that vertices in the given graph are adjacent only when the corresponding subtrees intersect. Thus, G forms a subgraph of the intersection graph of the subtrees. The full intersection graph is a chordal graph.
Each subtree associates a graph vertex with a set of tree nodes. To define this formally, we represent each tree node as the set of vertices associated with it. Thus, given a graph G = (V, E), a tree decomposition is a pair (X, T), where X = {X1, ..., Xn} is a family of subsets of V, and T is a tree whose nodes are the subsets Xi, satisfying the following properties:
The tree decomposition of a graph is far from unique; for example, a trivial tree decomposition contains all vertices of the graph in its single root node.
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