1/23
Graph Colouring through Clustering
István Juhos
18 Aug, 2009
University of Szeged
Hungary
U niversi
ty of Edinburgh
István Juhos
2/23
Outline
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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, 2009
U niversity of Edinburgh
István Juhos
Graph minimal vertex colouring
18 Aug, 2009
U niversity of Edinburgh
3/23
István Juhos
A colour algorithm example, a sequential greedy approach
4/23
Take an order of the nodes,
assign the first available colour
165432
4-colouring
123456
3-colouring (minimum)
18 Aug, 2009
U niversity of Edinburgh
István Juhos
Questions
5/23
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, 2009
U niversity of Edinburgh
István Juhos
Graph descriptions
6/23
Usually only V and E are given without drawing
by sets
by adjacency matrix (dot=0)
18 Aug, 2009
U niversity of Edinburgh
István Juhos
Graph drawing
7/23
no edge crossing
Colouring/clustering in a vector space
18 Aug, 2009
attractive forces: colours
repulsive forces: edges
How can we define forces?
no colours initially
use edges (adjacency matrix)
U niversity of Edinburgh
István Juhos
18 Aug, 2009
U niversity of Edinburgh
István Juhos
9/23
Colouring matrices
3-colouring matrix
(a solution)
18 Aug, 2009
U niversity of Edinburgh
István Juhos
10/23
All optimal-colourings
No symmetric colourings
18 Aug, 2009
U niversity of Edinburgh
István Juhos
11/23
Sum optimal-colouring matrices 1/2
SUM
18 Aug, 2009
U niversity of Edinburgh
István 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
blue
{1,4} => 0
red
red
4th colour
red
green
blue
18 Aug, 2009
blue
red
U niversity of Edinburgh
red
István 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, 2009
U niversity of Edinburgh
István Juhos
Sum matrix decomposition (approx.)
18 Aug, 2009
U niversity of Edinburgh
14/23
István 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
18 Aug, 2009
significant
significant
{1,4} => 0
conflicting
U niversity of Edinburgh
István Juhos
16/23
Zykov-tree (hierarchical clustering)
Use extreme values and modify the problem step-by-step
merge
add edge
SUM
18 Aug, 2009
U niversity of Edinburgh
István Juhos
Zykov-tree clustering using Lovász-theta
17/23
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, 2009
U niversity of Edinburgh
István Juhos
Approximate the sum matrix 1/3 (colouring matrix properties)
18/23
A colouring matrix is
positive semi-definite
3 dim
Describes orthogonal vectors in 3 dim.
18 Aug, 2009
U niversity of Edinburgh
István Juhos
Approximate the sum matrix 2/3 (an Integer SDP)
19/23
2 dim
Hard problem
18 Aug, 2009
U niversity of Edinburgh
(Meurdesoif)
István Juhos
Approximate the sum matrix 2/3 (Relaxed Integer SDP)
20/23
(Lovász)
Polinomial time
transformed
SUM
18 Aug, 2009
U niversity of Edinburgh
István Juhos
Zykov-tree clustering using Lovász-theta (again)
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)
18 Aug, 2009
U niversity of Edinburgh
István Juhos
22/23
Some results
18 Aug, 2009
U niversity of Edinburgh
István Juhos
23/23
Cheers
18 Aug, 2009
U niversity of Edinburgh
István Juhos