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” The main disadvantage – there is no guarantee on the size of the maximal clique in an output triangulated graph

Graph decomposition

Graph decomposition (cont.)

Example a bcde

Properties of decomposition - Lemma 1

Properties of decomposition - Lemma 2

Triangulation algorithm

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

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)

Triangulation algorithm (cont.)

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

Proof of correctness (cont.)

Finding a decomposition

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

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

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”

Finding a decomposition: Procedure I

Finding a decomposition: Procedure II

Complexity

Backup slides – proofs and some formalism

Proof of Lemma 1

Proof of Lemma 1 (cont.)

Proof of Lemma 2

Proof of Lemma 2 (cont.)

Theorem 1 (formal definition of algorithm correctness)

Finding a decomposition - proof of correctness

Finding a decomposition - proof of correctness (cont.)