Graph Colouring

1. Graph Colouring Various Models

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

3. kCol We are given a graph G = (V,E) where V is set of vertices and E is set of edges G is simple, undirected graph Given k colours, can we label vertices with colours such that for all (i,j)  E  colour(i) ≠ colour(j)

4. A Graph G = (V,E) 2 1 0 3 4 7 5 6 8 9

5. A Graph G = (V,E) 2 1 0 3 4 7 5 6 8 9 Colour such that adjacent vertices are different

6. Model 1

7. Model 1 kCol Have a constrained integer variable v[i] for each vertex i  V Domain of v[i] is {1..k} the set of avaliable colours For all for all (i,j)  E v[i] ≠ v[j] Decision variable are v[0] to v[n-1] Heuristic, Brelaz

16. Model 2

17. Model 2 SAT encoding kCol Consider as an example a 3Col instance with 2 adjacent vertices, i and j boolean variable to represent a vertex x taking a specific colour k “at least one and at most one” constraint adjacent vertices take different colours

18. Model 2 SAT encoding kCol Heuristics?

19. Implementation … generate data and pass to minisat+

20. Model 3

21. Model 3 Colour Sets of Vertices kCol Have a constrained set variable S[i] for each colour An edge is not in a coloured set All vertices are coloured Decision variables are S[0] to S[k-1] Heuristics? Where col is a set of colours

30. an example

31. g10.txt 2 1 0 3 4 7 5 6 8 9

32. g10.txt 2 1 0 3 4 7 5 6 8 9

33. g10.txt 2 1 0 3 4 7 5 6 8 9

34. g10.txt 2 1 0 3 4 7 5 6 1 is red 8 9 2 is green 3 is blue

35. A different colour with Colour02 g10.txt 2 1 0 3 4 7 5 6 8 9

36. Generating random graphs

42. Small demo with data directory

43. ChromaticNumber

44. ChromaticNumber

45. ChromaticNumber We want the minimum number of colours to colour G • 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)

46. ChromaticNumber

47. ChromaticNumber

48. ChromaticNumber

49. MaxCol (min-conflicts) What should we do if there are not enough colours and we want 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?

50. 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