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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.

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Presentation on theme: "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."— Presentation transcript:

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 bcde

9 Properties of decomposition - Lemma 1

10 Properties of decomposition - Lemma 2

11 Triangulation algorithm

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

13 Trialgulation algorithm - intuition 1. 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. 2. At every iteration a produced clique is kept small (due to the guarantees of the decomposition)

14 Triangulation algorithm (cont.)

15 Proof of correctness 1. Termination 2. Validity of the failure statement – follows immediately from Lemma 2 3. An output in the case of success is a triangulated graph 4. Cliquewidth in the case of success is as guaranteed

16 Proof of correctness (cont.)

17

18

19 Finding a decomposition

20 Finding a decomposition (cont.) The existence of W-decomposition is checked as follows: 1. First, a decomposition of graph into disconnected components is found, using approximation algorithm for weighted minimal vertex cut problem 2. 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.) 3. 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. 4. 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

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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