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Graph Triangulation

This document presents an advanced approach to graph triangulation based on the research paper "A Sufficiently Fast Algorithm for Finding Close to Optimal Junction Tree" by Ann Becker and Dan Geiger. It defines a junction tree and explores the natural approach to triangulation, discussing the minimum vertex cut concept and its implications on clique sizes. The document also covers various properties of graph decomposition, the triangulation algorithm, its correctness proofs, and challenges in finding a proper decomposition, ultimately aiming for efficient graph analysis.

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Graph Triangulation

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  1. 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

  2. Definition: junction tree

  3. The natural approach

  4. Example ==> ==>

  5. 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

  6. Graph decomposition

  7. Graph decomposition (cont.)

  8. Example a b c d e

  9. Properties of decomposition - Lemma 1

  10. Properties of decomposition - Lemma 2

  11. Triangulation algorithm

  12. Example (one-level recursive call) h a d f c g b e i j k

  13. 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)

  14. Triangulation algorithm (cont.)

  15. 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

  16. Proof of correctness (cont.)

  17. Proof of correctness (cont.)

  18. Proof of correctness (cont.)

  19. Finding a decomposition

  20. 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

  21. 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

  22. 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”

  23. Finding a decomposition: Procedure I

  24. Finding a decomposition: Procedure II

  25. Complexity

  26. Backup slides – proofs and some formalism

  27. Proof of Lemma 1

  28. Proof of Lemma 1 (cont.)

  29. Proof of Lemma 1 (cont.)

  30. Proof of Lemma 2

  31. Proof of Lemma 2 (cont.)

  32. Theorem 1 (formal definition of algorithm correctness)

  33. Finding a decomposition - proof of correctness

  34. Finding a decomposition - proof of correctness (cont.)

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