A Generic Framework for Local Search : Application to the Sudoku Problem

Transcription

1 / 4 A Generic Framework for Local Search : Application to the Sudoku Problem ICCS 006, Reading, UK Tony LAMBERT LINA, Université de Nantes, France LERIA,Université d Angers, France Universidad Santa María, Valparaíso, Chile joint work with E. Monfroy and F. Saubion Wed, 3 May, 006

2 / 4 Outline Local search for solving CSP A framework for local search techniques Generic Iteration Algorithm with reduction functions Some results for the sudoku problem Conclusion

3 3 / 4 Constraint satisfaction Problem (CSP) X C Y C4 C U C5 Variables Constraints Z C3 T

4 4 / 4 Local search () Definition : Explore a D D D n search space Move from neighbor to neighbor thanks to an evaluation function Properties : Intensification Diversification focus on some promising parts of the search space does not answer to unsat. problems no guaranteed fast to find a good solution

5 5 / 4 Local search () Search space : set of possible configurations Tools : neighborhood and evaluation function Neighborhood of s Set of possible configurations

6 6 / 4 Local search (3) Search space : set of possible configurations Tools : neighborhood and evaluation function Neighborhood of s Set of possible configurations

7 7 / 4 Local search (4) Search space : set of possible configurations Tools : neighborhood and evaluation function Neighborhood of s Set of possible configurations LS(p) = p s s 3 s s 4 s 5 s 6 p = ( p = ( s, s,..., s n ) s, s,..., s n, s n+ )

8 8 / 4 A framework for local search techniques Idea : Fine grain control More strategies Technique : Decomposing local search resolution into basic functions Adapting chaotic iterations

9 / 4 Our Purpose : Use of an existing theoretical model for CSP solving Definition of the solving process

10 0 / 4 Abstract Model K.R. Apt [CP] Constraint Abstraction Reduction functions Fixed point Partial Ordering CSP Application of functions

11 / 4 A Generic Algorithm F = { set reduction functions} X = initial CSP G = F While G choose g G G = G {g} G = G update(g, g, X) X = g(x) EndWhile X =

12 / 4 The theoretical model for CSP solving Partial ordering : Function to pass from one LS state to another p p State of LS p 3 p 4 p 5 Terminates with a fixed point : local minimum, maximum number of iteration

13 3 / 4 Ls Ordering Characteristics of a LS path notion of solution maximum length Operational (computational) point of view : path = samples A sample : Depends on a CSP Corresponds to a local search state

14 4 / 4 Neighborhood and Move functions Neighborhood : FullNeighbor : V = {s D s V } TabuNeighbor : V = {s D k, n l k n, s k = s} DescentNeighbor : p = (s,..., s n ) and V = s D s.t. s V s.t eval(s ) < eval(s n ) Move : BestMove : p = p s and eval(s ) = min s V eval(s ) ImproveMove : p = p s n and p = p s s.t. eval(s ) < eval(s n ) RandomMove : p = p s and s V

15 5 / 4 Instantiate the GI algorithm We can precise here the input set of function F. Random walk : Tabu search : FullNeighbor TabuNeighbor BestNeighBor BestNeighBor RandomNeighbor Random walk + Descent : TabuSearch + Descent : FullNeighbor TabuNeighbor BestNeighBor DescentNeighbor RandomNeighbor ImproveNeighBor DescentNeighbor BestNeighBor ImproveNeighBor

16 6 / 4 Sudoku problem n n problem n 4 variables AllDiff constraints

17 7 / 4 Sudoku problem

18 8 / 4 Sudoku problem

19 / 4 Sudoku problem

20 0 / 4 Sudoku problem

21 / 4 TabuSearch RandomWalk n x n 6x6 5x5 36x36 6x6 5x5 36x36 cpu time 3,4 5,08 38,8 3, 05, 45 deviations,8 5,3 347,4,47 4,3 0 mvts Descent + TabuSearch Descent +RandomWalk n x n 6x6 5x5 36x36 6x6 5x5 36x36 cpu time,34,8 48,4 8,4 455 deviations,4 55,04 476, 36, 0 mvts Avg

22 / 4 Conclusion and Future works A generic model local search methods Design of strategies Dynamic strategies learning Providing more tools in a generic environment A generic implementation

23 3 / 4 A Generic Framework for Local Search : Application to the Sudoku Problem ICCS 006, Reading, UK Tony LAMBERT LINA, Université de Nantes, France LERIA,Université d Angers, France Universidad Santa María, Valparaíso, Chile joint work with E. Monfroy and F. Saubion Wed, 3 May, 006