Hybrid CSP Solving Frédéric Saubion LERIA, Université d’Angers Joint work with Eric Monfroy (LINA, U. Nantes and UTFSM, Valparaiso, Chile) Tony Lambert (LERIA, U. Angers and LINA U. Nantes)
Problems are modeled as CSPs (X, D, C) some variables to represent objects (X = {X1 , . . . , Xn }) domains over which variables can range (D = D1 × . . . × Dn ) DX1 = {a; b; c; . . . } ... DXn = {t; u; v; . . . } some constraints to set relation between objects C1 : X ≤ Y ∗ 3 C2 : Z 6= X − Y ...
Hybrid Constraint Solving – p. 8
Complete/incomplete methods complete methods (such as propagation + split) complete exploration of the search space detect if no solution global optimum generally slow for hard combinatorial problems incomplete methods (such as local search) focus on some “promising” parts of the search space do not detect if no solution no global optimum “fast” to find a “good” solution Hybrid Constraint Solving – p. 46
CSP solving with a complete method
Constraint Propagation
Enumeration Splitting
Hybrid Constraint Solving – p. 15
Constraint propagation mechanism repeated application of reductions try to apply only useful reductions stop when a local consistency notion is reached (e.g., arc, node, hyper-arc consitency) or when reduction becomes inefficient (e.g., cycling weak reductions) or when a domain is empty (failed CSP)
Hybrid Constraint Solving – p. 18
Local search
Hybrid Constraint Solving – p. 34
Local search (1) Search space: set of possible configurations Tools: neighborhood and evaluation function Neighborhood of s
Set of possible configurations
s s
1
s
3
s
4
s 2
s
6
5
Hybrid Constraint Solving – p. 35
Local search (2) Search space: set of possible configurations Tools: neighborhood and evaluation function Neighborhood of s
Set of possible configurations
s s
1
s
3
s
4
s 2
s
6
5
Hybrid Constraint Solving – p. 36
Local search (3) Search space: set of possible configurations Tools: neighborhood and evaluation function Neighborhood of s
Set of possible configurations
s s
1
s
3
s
4
s 2
s
6
5
Hybrid Constraint Solving – p. 37
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
1
s
3
s
4
s 2
s
p=(
s
1
, s
2
, ...,
s
n
)
p’ = (
s
1
, s
2
, ...,
s
n
, s
n+1
)
6
5
Hybrid Constraint Solving – p. 38
Genetic algorithms
Hybrid Constraint Solving – p. 39
Genetic algorithms (1) Search space: set of possible configurations Tools: population, crossing, mutations, and evaluation function
Population
Hybrid Constraint Solving – p. 40
Genetic algorithms (2) Search space: set of possible configurations Tools: population, crossing, mutations, and evaluation function Set of possible configurations Population
Hybrid Constraint Solving – p. 41
Genetic algorithms (3) Search space: set of possible configurations Tools: population, crossing, mutations, and evaluation function Population k
x
1
4
9
4
2
5
y
Population k+1 1
8
4
2
4
6
2
4
2
1
9
9
Hybrid Constraint Solving – p. 42
Genetic algorithms (4) Search space: set of possible configurations Tools: population, crossing, mutations, and evaluation function Population k
x
1
4
9
4
2
1
4
9
4
8
5
y
5
Population k+1
Hybrid Constraint Solving – p. 43
Genetic algorithms (5) Search space: set of possible configurations Tools: population, crossing, mutations, and evaluation function
Population k
Crossing
Mutation Population k+1
New Population
Hybrid Constraint Solving – p. 44
Hybridization: getting the best of the both Efficiency: faster complete solver Quality: better solutions (better optimum) Generally: Ad-hoc systems (designed from scratch) Dedicated to a class of problems Master-slave approaches (LS for CP, CP for LS) Coarse grain cooperations Idea: fine grain hybridization finer strategies every technique at the same level one algorithm squeleton easier to modify, extend, compare, ... Hybrid Constraint Solving – p. 47
Hybrid solving
Hybrid Constraint Solving – p. 53
Abstract model K.R. Apt [CP99] for propagation (based on chaotic iterations) Abstraction Reduction functions
Constraint
Application of functions
CSP
Fixed point Partial Ordering
Hybrid Constraint Solving – p. 55
A Generic Algorithm: propagation for constraint propagation: ordering on size of domains work on a CSP F = { set of propagation functions} X = initial CSP G=F While G 6= ∅ choose g ∈ G G = G − {g} G = G ∪ update(G, g, X) X = g(X) EndWhile
Hybrid Constraint Solving – p. 56
Idea1: integrate split in GI avoid the CP framework algorithm split and reduction at the same level work on union of CSP
Hybrid Constraint Solving – p. 57
Theoretical model for CSP solving (1) Partial ordering:
Ω1 Application of a reduction : domain reduction function or splitting function
Set of CSP
Ω2
Ω3 Ω4 Ω5
Terminates with a fixed point : set of solutions or inconsistent CSP Hybrid Constraint Solving – p. 59
Theoretical model for CSP solving (2) Reduction: by constraint propagation:
Ω
Ω’ Ω
CSP1
CSP2
CSP3
Constraint Propagation
Ω
CSP1
CSP2’
CSP3
Hybrid Constraint Solving – p. 60
Theoretical model for CSP solving (3) Reduction by domain splitting:
Ω
Ω’ Ω
CSP1
CSP2
CSP3
Split
Ω
CSP1
CSP2’
CSP2’’
CSP2’’’
CSP3
Hybrid Constraint Solving – p. 61
CP + LS
Hybrid Constraint Solving – p. 62
Idea2: LS as moves on partial ordering LS moves on a partial ordering: p
Function to pass from one LS state to another p
1
2
State of LS p 3
p 4
p 5
Terminates with a fixed point : local minimum, maximum number of iteration Hybrid Constraint Solving – p. 63
Idea3: integrate samples for LS A sample: Depends on a CSP Corresponds to a LS state an SCSP contains (LS+domain reduction): A set of domains; A (list of) sample for local search (a LS path). an σCSP contains (LS+domain reduction+split): A set of SCSP
Hybrid Constraint Solving – p. 65
Theoretical model for hybridization (3) Ω1
Ordering over σCSP Application of a reduction : function : − domain reduction, − splitting, − or LS step.
Ω2 Ω3
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 Hybrid Constraint Solving – p. 66
Algorithm Always the GI 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
Hybrid Constraint Solving – p. 67
A Generic Algorithm: hybrid solving for hybrid CP+LS constraint solving F = { set of split, propagation, and LS functions} X = initial CSP G=F While G 6= ∅ choose g ∈ G G = G − {g} G = G ∪ update(G, g, X) X = g(X) EndWhile
Hybrid Constraint Solving – p. 68
General process CP+LS
Constraint
Sets of SCSP
Application of functions
LS solution Global Fixed point
Hybrid Constraint Solving – p. 69
Strategies: ratio LS-CP Ratio of LS and CP functions applied: (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
Hybrid Constraint Solving – p. 71
SEND + MORE = MONEY 2000
1800
Number of operations
1600
1400
1200 standard deviation 1000
800
600 Average 400
200 10
20
30
40
50
60
70
80
90
100
Propagation percentage Hybrid Constraint Solving – p. 73
Ranges Problem
S+M
LN42
Zebra
M. square
Golomb
Rate FC 70-80 15 - 25 60-70 30-45 30 - 40 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
LS
1535
401
3145
3265
CP+LS
396
95
814
815
CP
1515
746
936
1920
LS
1635
393
3240
3585
CP+LS
22
192
570
622
CP
530
425
736
1126
D-First
FC
Avg. nb. of operations to compute a first solution Hybrid Constraint Solving – p. 74
CP + GA
Hybrid Constraint Solving – p. 75
Theoretical model for hybridization CP+GA(3) Ω1
Ordering over OGCSPs Application of a reduction: function: − domain reduction
Ω2
− split − or GA.
Ω3
set of ogCSP
Ω4 Ω5 an optimal solution given by GA before the end of the process fixed point: set of solutions or inconsistency Hybrid Constraint Solving – p. 79
Algorithm Always the GI algorithm In the chaotic iteration framework Finite domains (set of OGCSP) Monotonic functions (GA move, domain reduction, splitting) Decreasing functions → termination and computation of fixed point (solution of CP) Practically: can stop before with an optimum for GA
Hybrid Constraint Solving – p. 80
A Generic Algorithm: hybrid solving me again!!! for hybrid CP+GA constraint solving F = { set of split, propagation, and GA functions} X = initial OGCSP G=F While G 6= ∅ choose g ∈ G G = G − {g} G = G ∪ update(G, g, X) X = g(X) EndWhile
Hybrid Constraint Solving – p. 81
General process CP+GA
Constraints
Set of ogCSP
optimal solution through AG
Application of functions
Global fixed point
Hybrid Constraint Solving – p. 82
Application BACP (to optimize loads of periods Lectures e
c f
Max Min
h d
b
i
j
k
g
a
d
Constraints
h
k
i
e
c
f
g
a
a before b "j c" h" "b i " "f f" "j k" "c d" "e
j b
Periods Hybrid Constraint Solving – p. 83
BACP8 100
range
80
propagation genetic algorithm split
60 40 20 0 25 24 23 22 21 20 19 18 17 16 evaluation Hybrid Constraint Solving – p. 85
BACP10 100
propagation genetic algorithm split
range
80 60 40 20 0 24
22
20
18
16
14
12
evaluation Hybrid Constraint Solving – p. 86
BACP12 100
propagation genetic algorithm split
range
80 60 40 20 0 25
24
23
22
21
20
19
18
17
evaluation Hybrid Constraint Solving – p. 87
Conclusion
A generic model for hybridizing complete (CP) and incomplete (LS and GA) methods Implementation of modules working on the same structure Complementarity of methods: hybridization Design of strategies
Hybrid Constraint Solving – p. 88
Future
Constraint programming = modelling tools + computation strategy + search strategy + solvers synergies between: complete + incomplete methods, CP + AI + OR + . . . learning strategies providing more tools in a generic environment
Hybrid Constraint Solving – p. 89
