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Graph Triangulation. by Dmitry Pidan Based on the paper “A sufficiently fast algorithm for finding close to optimal junction tree” by Ann Becker and Dan Geiger. Definition: junction tree. The natural approach. Example. ==>. ==>. The natural approach. X is called a “ minimum vertex cut ”

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## Graph Triangulation

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**Graph Triangulation**by Dmitry Pidan Based on the paper “A sufficiently fast algorithm for finding close to optimal junction tree” by Ann Becker and Dan Geiger**Example**==> ==>**The natural approach**• X is called a “minimum vertex cut” • The main disadvantage – there is no guarantee on the size of the maximal clique in an output triangulated graph**Example**a b c d e**Example (one-level recursive call)**h a d f c g b e i j k**Trialgulation algorithm - intuition**• We use a set W as a “balance factor” between the decomposition sets A, B and C – we are interested that a largest set will be as small as possible. • At every iteration a produced clique is kept small (due to the guarantees of the decomposition)**Proof of correctness**• Termination • Validity of the failure statement – follows immediately from Lemma 2 • An output in the case of success is a triangulated graph • Cliquewidth in the case of success is as guaranteed**Finding a decomposition (cont.)**• The existence of W-decomposition is checked as follows: • First, a decomposition of graph into disconnected components is found, using approximation algorithm for weighted minimal vertex cut problem • Next, A, B and C components of the decomposition are constructed by unifying the components that contain an appropriate subsets of W**Finding a decomposition (cont.)**• Finally, X is constructed from an initial common subset of W and X unified with the vertex cut found. If X stands for the size requirements then the decomposition is a required one. • More formally – in the next 3 slides**The 3-way vertex cut problem**• Definition: given a weighted undirected graph and three vertices, find a set of vertices of minimum weight whose removal leaves each of the three vertices disconnected from other two. • Known to be NP-hard • Polynomial approximation algorithms: • A simple 2-approximation algorithm • 4/3-approximation algorithm • Garg N. et al, “Multiway cuts in directed and node-weigthed graphs”

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