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Solving Strategies using a Hybridization Model for Local …

Solving Strategies using a Hybridization Model for Local
Search and Constraint Propagation

Published in Technology , Business
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  • 1. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Solving Strategies using a Hybridization Model for Local Search and Constraint Propagation ACM SAC 2005, Santa Fe, New Mexico, USA Tony Lambert and Éric Monfroy and Frédéric Saubion LERIA, Université de Angers, France and LINA, Université de Nantes, France March 16, 2005 1 / 42
  • 2. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Outline • Hybridization • Constraint propagation • Local search • Hybrid model • Some results • Conclusion 2 / 42
  • 3. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Hybridization for CSP • complete methods (such as propagation + split) • complete exploration of the search space • detects if no solution • generally slow for hard combinatorial problems • global optimum 3 / 42
  • 4. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Hybridization for CSP • complete methods (such as propagation + split) • complete exploration of the search space • detects if no solution • generally slow for hard combinatorial problems • global optimum • incomplete methods (such as local search) • focus on some “promising” parts of the search space • does not answer to unsat. problems • no guaranteed global optimum • “fast” to find a “good” solution 4 / 42
  • 5. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Hybridization : getting the best of both methods • But, generally : • Ad-hoc systems • Master-slaves approaches • Coarse grain cooperation 5 / 42
  • 6. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Hybridization : getting the best of both methods • But, generally : • Ad-hoc systems • Master-slaves approaches • Coarse grain cooperation • Idea : • Fine grain control • More strategies 6 / 42
  • 7. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Hybridization : getting the best of both methods • But, generally : • Ad-hoc systems • Master-slaves approaches • Coarse grain cooperation • Idea : • Fine grain control • More strategies • Technique : • Decomposing solvers into basic functions • Adapting chaotic iterations for hybrid solving 7 / 42
  • 8. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION CSP (Constraint Satisfaction Problem) C2 Y C1 U Variables X C4 C5 Constraints Z T C3 8 / 42
  • 9. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Problems are modeled as CSP (X , D, C) Variable (X ) Set of variable domains (D) Dx = {a; b; c; . . .} Dy = {a; b; c; . . .} Dz = {a; b; c; . . .} ... Set of constraints (C) C1 : X ≤ Y ∗ 3 C2 : Z = X − Y ... 9 / 42
  • 10. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Solving CSP with a complete method Constraint Propagation Enumeration Splitting 10 / 42
  • 11. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Constraint Propagation Arc consistency : C(X , Y ) a a b b Values c c D D X Y Consistent pairs of values 11 / 42
  • 12. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Constraint Propagation Arc consistency : C(X , Y ) Removed by arc consistency a a b b Values c c D D X Y Consistent pairs of values 12 / 42
  • 13. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Constraint Propagation Propagation Local consistency Propagation inconsistency 13 / 42
  • 14. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Constraint propagation and domain splitting Propagation Splitting 14 / 42
  • 15. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Constraint propagation and domain splitting Propagation Splitting Propagation 15 / 42
  • 16. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Abstract Model K.R. Apt [CP99] Abstraction Reduction functions Constraint Application of functions CSP Fixed point Partial Ordering 16 / 42
  • 17. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION A Generic Algorithm F = { set of split and propagation functions} X = initial CSP G=F X= While G = ∅ choose g ∈ G G = G − {g} G = G ∪ update(G, g, X ) X = g(X ) EndWhile 17 / 42
  • 18. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Our Purpose : Hybrid Model • Integration of local search • Use of an existing theoretical model for CSP solving • Definition of the solving process 18 / 42
  • 19. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Local search • Search space : set of possible configurations • Tools : neighborhood and evaluation function Set of possible configurations 19 / 42
  • 20. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Local search • Search space : set of possible configurations • Tools : neighborhood and evaluation function Neighborhood of s Set of possible configurations s3 s4 s6 s1 s2 s5 20 / 42
  • 21. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Local search • Search space : set of possible configurations • Tools : neighborhood and evaluation function Neighborhood of s Set of possible configurations s3 s4 s6 s1 s2 s5 21 / 42
  • 22. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Local search • Search space : set of possible configurations • Tools : neighborhood and evaluation function Neighborhood of s Set of possible configurations LS(p) = p’ p=( s 1 , s 2 , ..., sn) s3 p’ = ( s 1 , s 2 , ..., s4 s n , s n+1 ) s6 s1 s2 s5 22 / 42
  • 23. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Integrating CSP Resolution Techniques in a Hybrid Model • Extend the existing Apt s theoretical model for CSP solving • Fixpoint computation on a partial ordering 23 / 42
  • 24. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION The theoretical model for CSP solving Partial ordering : Ω1 Application of a reduction : Ω2 domain reduction function or splitting function Ω3 Set of CSP Ω4 Ω5 Terminates with a fixed point : set of solutions or inconsistent CSP 24 / 42
  • 25. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION The theoretical model for CSP solving Reduction : by constraint propagation Ω Ω’ Ω CSP1 CSP2 CSP3 Constraint Propagation Ω CSP1 CSP2’ CSP3 25 / 42
  • 26. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION The theoretical model for CSP solving Reduction : by domain splitting Ω Ω’ Ω CSP1 CSP2 CSP3 Split Ω CSP1 CSP2’ CSP2’’ CSP2’’’ CSP3 26 / 42
  • 27. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION LS process as moves on partial ordering Moves on a partial ordering p 1 Function to pass from one LS state to another p 2 State of LS p 3 p 4 p 5 Terminates with a fixed point : local minimum, maximum number of iteration 27 / 42
  • 28. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION LS Ordering Characteristics of a LS path • Notion of Solution • Maximum Length Operational (computational) point of view : path = samples 28 / 42
  • 29. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Integration of samples for local search into the CSP • A sample : • Depends on a CSP • Corresponds to a LS state • An SCSP contains : • A set of domains ; • A set of constraints ; • A (list of) sample for local search. 29 / 42
  • 30. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Hybridization model Ordering over SCSP Ω1 Application of a reduction : Ω2 function : − domain reduction, − splitting, Ω3 − or LS step. Set of SCSP Ω4 Ω5 A solution before the end of the process, given by LS Terminates with a fixed point : set of solutions or inconsistent SCSP 30 / 42
  • 31. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Hybridization model Ordering over SCSP Ψ = C; D; p with : Ψ Ψ Ψ = C; D ; p D⊆D if : D = D and p p or if : 31 / 42
  • 32. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Algorithm Same Generic Algorithm X = initial SCSP G=F While G = ∅ choose g ∈ G G = G − {g} G = G ∪ update(G, g, X ) X = g(X ) EndWhile 32 / 42
  • 33. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION General Process Application of functions Sets of Constraint SCSP LS solution Global Fixed point 33 / 42
  • 34. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Algorithm • In the chaotic iteration framework • Finite domains (sets of SCSP) • Monotonic functions (LS move, domain reduction, splitting) • Decreasing functions • → termination and computation of fixed point (solution of CP) • Practically : can stop before with a LS solution 34 / 42
  • 35. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Examples • SEND + MORE = MONEY • Zebra • Golomb • Magic square • Langford Number 35 / 42
  • 36. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Modeling Strategies through Reduction Functions • Reduction / Split / LS functions • Choose a / several SCPS to apply functions • Choose a / several domains (Prop. - Split) • Tests : Random - Depth first - FC 36 / 42
  • 37. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Strategy : ratio LS-CP • Ratio of LS functions applied and CP functions applied : Triple(red%,split%,ls%) • 10% of split compared to reduction • CP alone : (90,10,0) • LS alone : (0,0,100) • LS strategy : tabu • Choose : LS, reduction, or split but keep the given ratios 37 / 42
  • 38. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Langford number 4 2 800 700 600 Number of operations 500 400 300 200 Average 100 standard deviation 0 10 20 30 40 50 60 70 80 90 100 Propagation percentage 38 / 42
  • 39. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION SEND + MORE = MONEY 2000 1800 1600 1400 Number of operations 1200 standard deviation 1000 800 600 Average 400 200 10 20 30 40 50 60 70 80 90 100 Propagation percentage 39 / 42
  • 40. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Problem S+M LN42 Zebra M. square Golomb Rate FC 70-80 15 - 25 60-70 30-45 30 - 40 TAB.: Best range of propagation rate (α) to compute a solution Strategy Method S+M LN42 M. square Golomb Random LS 1638 383 3328 3442 CP+LS 1408 113 892 909 CP 3006 1680 1031 2170 D-First LS 1535 401 3145 3265 CP+LS 396 95 814 815 CP 1515 746 936 1920 FC LS 1635 393 3240 3585 CP+LS 22 192 570 622 CP 530 425 736 1126 TAB.: Avg. nb. of operations to compute a first solution 40 / 42
  • 41. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Conclusion • A generic model for hybridizing complete (CP) and incomplete (LS) methods • Implementation of modules working on the same structure • Complementarity of methods • Design of strategies 41 / 42
  • 42. OUTLINE HYBRIDIZATION CONSTRAINT PROPAGATION LOCAL SEARCH HYBRID MODEL SOME RESULTS CONCLUSION Future work • Adaptation of the model for optimization • Study of other methods (G.A.) • Design of strategies using composition operators • A generic implementation 42 / 42