Graph Colouring

Various Models

• an obvious model• SAT• sets• demo• random graphs• chromatic number• maxCol• phase transition

kCol

We are given a graph G = (V,E) whereV is set of vertices and E is set of edgesG is simple, undirected graph

Given k colours, can we label vertices with colourssuch that for all (i,j) E colour(i) ≠ colour(j)

0 1 2

3 4

75

6

98

A Graph G = (V,E)

0 1 2

3 4

75

6

98

A Graph G = (V,E)

Colour such that adjacent vertices are different

Model 1

kColModel 1

Have a constrained integer variable v[i] for each vertex i VDomain of v[i] is {1..k} the set of avaliable coloursFor all for all (i,j) E v[i] ≠ v[j]Decision variable are v[0] to v[n-1]Heuristic, Brelaz

Model 2

kColModel 2 SAT encoding

kivtruev ki ][

Consider as an example a 3Col instance with 2 adjacent vertices, i and j

)()()()( 323121321iiiiiiiii vvvvvvvvv

boolean variable to represent a vertex x taking a specific colour k

“at least one and at most one” constraint

)()()( 332211jijiji vvvvvv

)()()()( 323121321jjjjjjjjj vvvvvvvvv

adjacent vertices take different colours

kColModel 2 SAT encoding

Heuristics?

Implementation … generate data and pass to minisat+

Model 3

kColModel 3

Have a constrained set variable S[i] for each colour

Colour Sets of Vertices

icoliEvu Svu },{),(

Decision variables are S[0] to S[k-1]Heuristics?

nScoli i

Where col is a set of colours

An edge is notin a coloured set

All vertices are coloured

an example

0 1 2

3 4

75

6

98

g10.txt

0 1 2

3 4

75

6

98

g10.txt

0 1 2

3 4

75

6

98

g10.txt

0 1 2

3 4

75

6

98

1 is red

2 is green

3 is blue

g10.txt

0 1 2

3 4

75

6

98

g10.txtA different colour with Colour02

Generating random graphs

Small demo with data directory

Chromatic Number

Chromatic Number

We want the minimum number of colours to colour G

Chromatic Number

• introduce a constrained integer variable maxCol• maxCol has domain {1 .. n}• each vertex has a constrained integer variable v[i]• v[i] has domain {1 .. n}• for (int i=0;i<n;i++) modl.addConstraint(leq(v[i],maxCol))• sol.minimise(maxCol)

Chromatic Number

Chromatic Number

Chromatic Number

What should we do if there are not enough colours and wewant to do the best that we can? That is, colour as many vertices

as possible leaving some vertices uncoloured, or colour all vertices and minimise conflicts?

MaxCol (min-conflicts)

MaxCol (min-conflicts)

• associate a dummy colour into domains• dummy colour doesn’t conflict with any other colours• get occurrence of dummy colour and minimise that

OR

• associate a penalty with a conflict• for edge {i,j} have constrained integer variable P[i][j]• domain P[i][j] is {0,1}• modl.addConstraint(or(and(neq(v[i],v[j]),eq(P[i][j],0)),• (and(eq(v[i],v[j]),eq(P[i][j],1)))• minimise the sum of the penalty variables

Lassie, do you think we could have a look at the phase transition in kCol?

Woof!

Lassie, remember the paper by Cheeseman, Kanefsky and Taylor, and that stuff on constrainedness?

Woof!

Lassie, are you lying?

Woof!

1 2log ( )Sol

N

Constrainedness

<Sol> is expected number of solutions N is log_2 of the size of the state space

k = 0, all states are solutions, easy, underconstrained

k =

k = 1, critically constrained, 50% solubility, hard

, <Sol> is zero, easy, overconstrained

Applied to: CSP, TSP, 3-SAT, 3-COL, Partition, HC, …?

Random Graphs

Random Graphs

Make Experiments as a Job

Make Experiments as a Job

The Results File

Analyse Results

4Coln = 50sample size = 500.1 ≤ p ≤ 0.3

Check Out the Dates

1 day’s work

fin