1 Reinforced Tabu Search (RTS) for Graph Colouring Daniel Porumbel PhD student (joint work with Jin Kao Hao and Pascale Kuntz) Laboratoire d’Informatique de Nantes Atlantique Laboratoire d'Etude et de Recherche en Informatique d'Angers

2 Graph Colouring and K-Colouring 1. Graph K-colouring problem: A graph G=(V,E) is K-colourable if we can label its vertices with exactly K colours so that any two adjacent vertices have different labels (colours) assigned

3 Graph Colouring and K-Colouring 1. Graph K-colouring problem: A graph G=(V,E) is K-colourable if we can label its vertices with exactly K colours so that any two adjacent vertices have different labels (colours) assigned 2. The general graph colouring problem is to determine minimal number K so that G is K-colourable the chromatic number χ(G)

4 Applications Applications and examples: Assigning frequencies in mobile networks Scheduling problems Timetabling Register allocation in compilers Four color theorem Sudoku

5 Basic Representation for K- coloring Configuration C = a vector of labels/colors The conflict number of C: The number of edges with both ends of the same color K-coloring problem ≈ find configuration C* minimizing the conflict number The problem is solved if C* has no conflicts Configuration C=(3|1|1|1|2) Conflict number = 2 (V3-V2 and V3-V4)

6 The Landscape for K-colouring The set of all possible configurations has cardinal K |V| ~ a space of dimension |V| The neighbourhood relation: We pass from a configuration C to C’ by changing just one colour in a conflicting vertex of C

7 Related Works Exact algorithms: |V| <100 Inexact algorithms Metaheuristics (working on landscape):  Local Search: Tabu Search, Simulated Annealing [Hertz et Al, A variable neighborhood search for graph coloring, 2003], [Zufferey et. al, A graph coloring heuristic using partial solutions and a reactive tabu scheme, 2007]  Evolutionary algorithms [Fleurent et. Al, Genetic and hybrid algorithms for graph coloring, 1996], [Hao et Al, Hybrid Evolutionary Algorithms for Graph Coloring, 1999][Galinier et Al, An adaptive memory algorithm for graph coloring,2004] Sequential construction: DSatur Algorithm, RLS

8 Steepest Descent It starts from a random configuration At each step, we move to the best neighbouring configuration We stop when it is no longer possible to find better neighbouring configurations (in a local minimum) G=Global minimum L=Local minimum

9 Steepest Descent: Cases

10 Tabu Search A steepest descent that can perform “up moves” We can move to a worse configuration when there is no possible move not increasing the number of conflicts Risk : to repeat a set of moves again and again => Tabu list = a list of temporary forbidden moves

11 Tabu Search: evolution of the number of conflicts Graph = dsjc250.5 K = 27 Best K = 28

12 Tabu Search: evolution of the number of conflicts

13 Tabu Search: evolution of the number of conflicts

14 Tabu List Length The role of the tabu list is to restrain the algorithm from cycling in a local plateau Tabu List Length: tl = RandomInt(0,10) + f Each time we perform move m, we forbid the algorithm from re-applying m on next tl moves tl = 2  enough to break a cycle of length 2 Col(v)=1 Col(v)=4 Col(v)=1

15 Tabu List Length The same tabu list length = 2 is NOT enough to break larger cycles We can find both very small or very large plateaus  small or large cycles Long tabu list for large plateaus and small tabu list for small plateaus

16 Adaptive Tabu List Length We double the tabu list length when we stayed X moves at the same conflict number A constant very long tabu list length would forbid a move (an assignment of a color) for much too long!

17 New Evaluation Function In most algorithms, all configurations are compared and assessed with the classical evaluation function f Not all conflicts are as difficult to solve We can have: Edges with easy conflicts (V3-V4) We can simply change the color in one end Edges with hard conflicts (V3-V2) We cannot change a color without generating new conflicts

18 Degree-based Evaluation {i,j} Easy conflict in C 1  {i,k} Hard conflict in C 2 Vertex j: degree=1   Vertex k: degree=6  Each edge has a weight, according to the degree of each vertex

19 RTS vs. Other Tabu Algorithms RTS gains one colour on 75% of the Dimacs graphs - in comparison with the same algorithm without the two improvements RTS gains one colour in 50% of graphs in comparison with the best of all Tabu algorithms ever written The new evaluation function gains one colour in 25% of cases vs classical evaluation function

20 1. [A. Hertz and D. de Werra, Using tabu search techniques for graph coloring, 1987] 2. [C. Fleurent and J. A. Ferland, Genetic and hybrid algorithms for graph coloring, 1996] 3. [Dorne and J. K. Hao, Tabu search for graph coloring, T-colorings and set T- colorings, 1998] 4. [P. Gallinier and J. K. Hao, Hybrid Evolutionary Algorithms for Graph Coloring, 1999]

21 RTS vs Best Algorithms The best algorithms are often population- based hybrid algorithms RTS is able to find the best colorings ever reported for 75% of the Dimacs graphs (28 out of the 38) We obtained the best results of previous local search algorithms in almost 100% of cases (2 exceptions)

22 Conclusions Two simple techniques are enough to make a basic local search provide very competitive results. Tabu Search can now attain the best results ever found in 75% of the Dimacs graphs