18 Aug, 2009University of EdinburghIstván Juhos 1 /23 Graph Colouring through Clustering István Juhos University of Szeged Hungary

18 Aug, 2009University of EdinburghIstván Juhos 2 /23 Outline ● Problem definition ● Consider the solved problem ● Get an idea to create a similarity matrix ● Approximate the similarity matrix using available structures ● Define clustering algorithm

18 Aug, 2009University of EdinburghIstván Juhos 3 /23 Graph minimal vertex colouring

18 Aug, 2009University of EdinburghIstván Juhos 4 /23 A colour algorithm example, a sequential greedy approach Take an order of the nodes, assign the first available colour colouring 3-colouring (minimum)

18 Aug, 2009University of EdinburghIstván Juhos 5 /23 Questions 1) Can a graph be coloured with k number of colours? 2) What is the minimum k called chromatic number ? 3) What are the k-colourings, -colourings? 4) Can we construct a minimum colouring? (this talk) k-colouring is NP-complete. Try to approximate.

18 Aug, 2009University of EdinburghIstván Juhos 6 /23 Graph descriptions Usually only V and E are given without drawing by sets by adjacency matrix (dot=0)

18 Aug, 2009University of EdinburghIstván Juhos 7 /23 Graph drawing attractive forces: colours repulsive forces: edges no edge crossing How can we define forces? no colours initially use edges (adjacency matrix) Colouring/clustering in a vector space

18 Aug, 2009University of EdinburghIstván Juhos 8 /23 Reordering adjacency matrix. [0] diag. blocks, independent sets reordering

18 Aug, 2009University of EdinburghIstván Juhos 9 /23 Colouring matrices 3-colouring matrix (a solution)

18 Aug, 2009University of EdinburghIstván Juhos 10 /23 All optimal-colourings No symmetric colourings

18 Aug, 2009University of EdinburghIstván Juhos 11 /23 Sum optimal-colouring matrices 1/2 SUM

18 Aug, 2009University of EdinburghIstván Juhos 12 /23 Sum optimal-colouring matrices 2/2 SUM {1,3} => 3 {1,5} => 2 {3,5} => 2 {4,6} => 2 {5,2} => 1 {2,4} => 1 {1,4} => 0 blue red blue green 4 th colour

18 Aug, 2009University of EdinburghIstván Juhos 13 /23 Sum matrix defines similarity SUM Large values attractive forces Small values repulsive forces Similarity matrix for clustering We do not have any optimal colouring matrices! But we can approximate the sum matrix using only the adjacency matrix (see later).

18 Aug, 2009University of EdinburghIstván Juhos 14 /23 Sum matrix decomposition (approx.)

18 Aug, 2009University of EdinburghIstván Juhos 15 /23 Problems with the sum matrix Sum matrix can contain conflicting suggestions. But extreme values are usually significant. {1,3} => 3 {1,5} => 2 {3,5} => 2 {4,6} => 2 {5,2} => 1 {2,4} => 1 {1,4} => 0 conflicting significant

18 Aug, 2009University of EdinburghIstván Juhos 16 /23 Use extreme values and modify the problem step-by-step Zykov-tree (hierarchical clustering) SUM merge add edge

18 Aug, 2009University of EdinburghIstván Juhos 17 /23 Zykov-tree clustering using Lovász-theta repeat until reaching complete graph 1. approximate the sum matrix (Lovász-theta SDP solution, see next) 2. find some extreme values 3. merge vertices and/or add edges (Zykov-step) How can we approximate the sum matrix?

18 Aug, 2009University of EdinburghIstván Juhos 18 /23 Approximate the sum matrix 1/3 (colouring matrix properties) A colouring matrix is positive semi-definite Describes orthogonal vectors in 3 dim. 3 dim

18 Aug, 2009University of EdinburghIstván Juhos 19 /23 (Meurdesoif) 2 dim Approximate the sum matrix 2/3 (an Integer SDP) Hard problem

18 Aug, 2009University of EdinburghIstván Juhos 20 /23 (Lovász) Approximate the sum matrix 2/3 (Relaxed Integer SDP) SUM transformed Polinomial time

18 Aug, 2009University of EdinburghIstván Juhos 21 /23 repeat until reaching complete graph 1. approximate the sum matrix (Lovász-theta SDP solution) 2. find some extreme values 3. merge vertices and/or add edges (Zykov-step) Zykov-tree clustering using Lovász-theta (again)

18 Aug, 2009University of EdinburghIstván Juhos 22 /23 Some results

18 Aug, 2009University of EdinburghIstván Juhos 23 /23 Cheers