Constraint Satisfaction 101
CSCE496/896, Fall 2011
www.cse.unl.edu/~choueiry/F11-496-896
Questions: [email protected]
Broadcast: [email protected]
Berthe Y. Choueiry (Shu-we-ri)
AVH 360
Problem Solving with Constraints, CSCE 496/896
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Outline
 Motivating example, application areas
 CSP: Definition, representation
 Some simple modeling examples
 More on definition and formal characterization
 Basic solving techniques
(Implementing backtrack search)
Advanced solving techniques
Issues & research directions
Problem Solving with Constraints, CSCE 496/896
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Motivating example
• Context: You are a senior in college
• Problem: You need to register in 4 courses for the Spring semester
• Possibilities: Many courses offered in Math, CSE, EE, CBA, etc.
• Constraints: restrict the choices you can make
– Unary: Courses have prerequisites you have/don't have
Courses/instructors you like/dislike
– Binary: Courses are scheduled at the same time
– n-ary: In CE: 4 courses from 5 tracks such as at least 3 tracks are
covered
• You have choices, but are restricted by constraints
– Make the right decisions!!
Problem Solving with Constraints, CSCE 496/896
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Motivating example (cont’d)
• Given
– A set of variables: 4 courses at UNL
– For each variable, a set of choices (values)
– A set of constraints that restrict the combinations
of values the variables can take at the same time
• Questions
– Does a solution exist? (classical decision problem)
– How two or more solutions differ? How to change
specific choices without perturbing the solution?
– If there is no solution, what are the sources of
conflicts? Which constraints should be retracted?
– etc.
Problem Solving with Constraints, CSCE 496/896
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Practical applications
Adapted from E.C. Freuder
•
•
•
•
•
•
•
•
•
•
•
•
Radio resource management (RRM)
Databases (computing joins, view updates)
Temporal and spatial reasoning
Planning, scheduling, resource allocation
Design and configuration
Graphics, visualization, interfaces
Hardware verification and software engineering
HC Interaction and decision support
Molecular biology
Robotics, machine vision and computational linguistics
Transportation
Qualitative and diagnostic reasoning
Problem Solving with Constraints, CSCE 496/896
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Constraint Processing
• is about ...
– Solving a decision problem…
– … While allowing the user to state arbitrary
constraints in an expressive way and
– Providing concise and high-level feedback about
alternatives and conflicts
• Related areas:
– AI, OR, Algorithmic, DB, TCS, Prog. Languages, etc.
Problem Solving with Constraints, CSCE 496/896
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Power of Constraints Processing
• Flexibility & expressiveness of
representations
• Interactivity
users can
relax
reinforce
constraints
Problem Solving with Constraints, CSCE 496/896
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Outline
Motivating example, application areas
CSP: Definition, representation
Some simple modeling examples
More on definition and formal
characterization
Basic solving techniques
Problem Solving with Constraints, CSCE 496/896
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Defining a problem
• General template of any computational
problem
– Given:
• Example: a set of objects, their relations, etc.
– Query/Question:
• Example: Find x such that the condition y is
satisfied
• How about the Constraint Satisfaction
Problem?
Problem Solving with Constraints, CSCE 496/896
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Definition of a CSP
• Given P = (V, D, C )
– V is a set of variables, V  V1 , V 2 ,  , V n 
– D is a set of variable domains (domain
values)
D  DV 1 , DV 2 ,  , DV n 
– C is a set of constraints, C
 C 1 , C 2 ,  , C l 
with R V a ,V b ,..., V i  ( x, y,K , z )  D V a  D V b  K  D V i
• Query: can we find a value for each variable
such that all constraints are satisfied?
Problem Solving with Constraints, CSCE 496/896
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Different Queries Yield Different Problems
• Find a solution
decision problem
• Find number of/all solutions counting problem
• Find a set of constraints that can be removed so
that a solution exists
optimization problem
• Etc.
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Domain Types
• P = (V, D, C ) where
V  V1 , V 2 ,  , V n 
D  DV 1 , DV 2 ,  , DV n 
C  C 1 , C 2 ,  , C l 
with C V k ,V l ,..., V m  ( x , y ,  , z )  D V k  D V l    D V m
• Domains:
– Restricted to {0,1}: Boolean CSPs
– Finite (discrete), enumeration works
– Continuous, sophisticated algebraic techniques
are needed
• Consistency techniques on domain bounds
Problem Solving with Constraints, CSCE 496/896
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CSP Representation (I)
• Given P = (V, D, C ), where
V  V1 , V 2 ,  , V n 
D  DV 1 , DV 2 ,  , DV n 
C  C 1 , C 2 ,  , C l 
• Find a consistent assignment for variables
Constraint Network (graph, hypergraph)
• Variable
 node (vertex)
• Domain
 node label
• Constraint  arc (edge) between nodes
Problem Solving with Constraints, CSCE 496/896
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CSP Representation (II)
Graph
• Vertices: variables
• Edges: binary
constraints
V1 {1, 2, 3, 4} v1 < v2 { 3, 6, 7 }
V2
v1+v3 < 9
v2 < v3
V3 { 3, 4, 9 }
V1
Hypergraph
{ 3, 5, 7 } V4
v1 < v2
{ 3, 4, 9 }
V2
{ 3, 6, 7 }
{ 1, 2, 3, 4 }
• Vertices: variables
v1+v3 < 9
• Hyperedges: constraints
v2 > v4
v2 < v3
v1+v2+V4 < 10
v2 > v4
{ 3, 5, 7 }
V4
V3
Problem Solving with Constraints, CSCE 496/896
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Constraint Definition
• A constraint C is defined by
− A scope, the set of variables on which the constraint applies
Notation: SCOPE(C), scope(C), scp(C)
− A relation, a subset of the Cartesian product of the domains of
the variables in the scope of the constraint
Notation: RELATION(C), rel(C)
• Arity, cardinality of the constraint’s scope
− Unary, binary, ternary,…, global
− Universal constraint
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Relation Definition
• Extension, all tuples are enumerated
− As a list of allowed tuples (supports, positive table)
− As a list of forbidden tuples (conflicts, no-goods)
• Intension, given by a set builder
− When it is not practical or possible to list all tuples
− Define types/templates of common constraints to be
used repeatedly
− Examples: linear constraints, All-Diff (mutex), Atmost,
TSP-constraint, cycle-constraint, etc.
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Constraint Implementation
•
•
•
•
•
Predicate function
Set of tuples (list or table)
Binary matrix (bit-matrix)
Constrained Decision Diagrams ([Cheng & Yap, AAAI 05])
etc.
Problem Solving with Constraints, CSCE 496/896
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Outline
Motivating example, application areas
CSP: Definition, representation
Some simple modeling examples
More on definition and formal
characterization
Basic solving techniques
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Example II: Temporal reasoning
B
A <B
A
[ 1.... 10 ]
[ 5.... 18]
B<C
2 <C -A <5
[ 4.... 15]
C
• Give one solution: …….
• Satisfaction, yes/no: decision problem
Problem Solving with Constraints, CSCE 496/896
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Example III: Map coloring
Using 3 colors (R, G, & B), color the US map
such that no two adjacent states have the
same color
• Variables?
• Domains?
• Constraints?
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Example III: Map coloring (cont’d)
Using 3 colors (R, G, & B), color the US map such that
no two adjacent states have the same color
NE
WY
KS
{ red, green, blue }
CO
AR
{ red, green, blue }
{ red, green, blue }
UT
{ red, green, blue }
AZ
OK
NM
LA
TX
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Example IV: Resource Allocation
{ a, b, c }
{ a, b }
{ a, c, d }
{ b, c, d }
What is the CSP formulation?
Problem Solving with Constraints, CSCE 496/896
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Example IV: RA (cont’d)
{ a, b, c }
{ a, b }
{ a, c, d }
{ b, c, d }
Constraint Graph
T1
{ R1, R3 }
Interval Order
T2
T3
{ R1, R3 }
T1
{ R1, R3 }
{ R1, R3, R4 } T4
{ R1, R3 }
{ R1, R2, R3 }
T5
{ R2, R4 }
T6
{ R2, R4 }
T7



T6
{ R1, R2, R3 } 


{ R1, R3 }
T2
T4


T3 { R1, R3 }


{ R2, R4 }


{ R2, R4 }
{ R1, R2, R3 } 
T5
T7
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Example V: Cryptarithmetic puzzles
• DX1 = DX2 = DX3 = {0,1}
• DF=DT=DU=DW=DR=DO=[0,9]
•
•
•
•
•
O+O = R+10X1
X1+W+W = U+10X2
X2+ T+T = O + 10X3
X3=F
Alldiff({F,D,U,V,R,O})
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Example VI: Product Configuration
Train, elevator, car, etc.
Given:
• Components and their attributes (variables)
• Domain covered by each characteristic (values)
• Relations among the components (constraints)
• A set of required functionalities (more constraints)
Find: a product configuration
i.e., an acceptable combination of components
that realizes the required functionalities
Problem Solving with Constraints, CSCE 496/896
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Example VII: Puzzle
Given:
• Four musicians: Fred, Ike, Mike, and Sal, play bass, drums,
guitar and keyboard, not necessarily in that order.
• They have 4 successful songs, ‘Blue Sky,’ ‘Happy Song,’
‘Gentle Rhythm,’ and ‘Nice Melody.’
• Ike and Mike are, in one order or the other, the composer of
‘Nice Melody’ and the keyboardist.
• etc ...
Query: Who plays which instrument and who composed which
song?
Problem Solving with Constraints, CSCE 496/896
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Example VII: Puzzle (cont’d)
Formulation 1:
• Variables: Bass, Drums, Guitar, Keyboard, Blue Sky, Happy Song
Gentle Rhythm and Nice Melody.
• Domains: Fred, Ike, Mike, Sal
• Constraints: …
Formulation 2:
• Variables: Fred's-instrument, Ike's-instrument, …,
Fred's-song, Ikes's-song, Mike’s-song, …, etc.
• Domains:
{ bass, drums, guitar, keyboard }
{ Blue Sky, Happy Song, Gentle Rhythm, Nice Melody}
• Constraints: …
Problem Solving with Constraints, CSCE 496/896
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Constraint types: examples
• Example I: algebraic constraints
V1
v1 < v2
{1, 2, 3, 4}
v1+v3 < 9
{ 3, 6, 7 } V2
v2 > v4
v2 < v3
V3 { 3, 4, 9 }
• Example II:
(algebraic) constraints
of bounded difference
{ 3, 5, 7 }
V4
B
A <B
A
[ 5.... 18]
[ 1.... 10 ]
2 <C -A <5
B<C
[ 4.... 15]
C
• Example III & IV: coloring, mutual exclusion, difference constraints
Constraint Graph
NE
T1
WY
{ R1, R3 }
KS

{ red, green, blue }
CO
{ red, green, blue }

OK
NM

{ R1, R3 }
{ red, green, blue }
AZ
{ R2, R4 }

{ red, green, blue }
T3
LA
T6
{ R1, R2, R3 } 

AR
UT
T4




{ R1, R2, R3 }
{ R1, R3 }


 { R2, R4 }
T5
T7
TX
• Example V & VI: elements of C must be made explicit
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More examples
• Example VII: Databases
– Join operation in relational DB is a CSP
– View materialization is a CSP
• Example VIII: Interactive systems
– Data-flow constraints
– Spreadsheets
– Graphical layout systems and animation
– Graphical user interfaces
• Example IX: Molecular biology (bioinformatics)
– Threading, etc
Problem Solving with Constraints, CSCE 496/896
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Outline
Motivating example, application areas
CSP: Definition, representation
Some simple modeling examples
More on definition and formal
characterization
Basic solving techniques
Problem Solving with Constraints, CSCE 496/896
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Representation (again)
Macrostructure G(P):
V1
- constraint graph for
binary constraints
- constraint network for
non-binary constraints

V2
a, c
(V1, a )
Micro-structure  (P):
(V2, a )
(V2, c)
(V1, a )
Co-microstructure co-(P):
(V2, a )
(V2, c)
a, b

b, c

V3
(V1, b)
(V3, b )
(V1, b)
(V3, b )
(V3, c)
no goods
(V3, c)
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Complexity of CSP
Characterization
• Decision problem
• In general, NP-complete by reduction
from 3SAT
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Proving NP-completeness
1. Show that 1 is in NP
2. Given a problem 1 in NP, show that an known
NP-complete problem 2 can be efficiently
reduced to 1
–
–
–
Select a known NP-complete problem 2 (e.g.,
SAT)
Construct a transformation f from 2 to 1
Prove that f is a polynomial transformation
(Check Chapter 3 of Garey & Johnson)
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What is SAT?
Given a sentence:
–
Sentence: conjunction of clauses
–
Clause: disjunction of literals
c 2   c 3 
–
Literal: a term or its negation
c1 ,  c 6
–
Term: Boolean variable
c1  1   c1  0
c1   c 4  c 5  c 6   c 2   c 3    c 4 
Question: Find an assignment of truth values to the
Boolean variables such the sentence is satisfied.
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CSP is NP-Complete
• Verifying that an assignment for all
variables is a solution
– Provided constraints can be checked in
polynomial time
• Reduction from 3SAT to CSP
– Many such reductions exist in the literature
(perhaps 7 of them)
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Problem Reduction
Example: CSP into SAT
(proves nothing, just an exercise)
Notation: variable-value pair = vvp
• vvp  term
– V1 = {a, b, c, d} yields x1 = (V1, a), x2 = (V1, b), x3 = (V1, c), x4 = (V1, d),
– V2 = {a, b, c} yields x5 = (V2, a), x6 = (V2, b), x7 = (V2,c).
• The vvp’s of a variable  disjunction of terms
– V1 = {a, b, c, d} yields
x1  x 2  x 3  x 4
• (Optional) At most one VVP per variable
 x1   x 2   x 3   x 4     x1  x 2   x 3   x 4  
  x1   x 2  x 3   x 4     x1   x 2   x 3  x 4 
Problem Solving with Constraints, CSCE 496/896
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CSP into SAT (cont.)
Constraint:
1.
Way 1: Each inconsistent tuple  one disjunctive clause
–
2.
C V1V 2  {( a , a ), ( a , b ), ( b , c ), ( c , b ), ( d , a )}
For example:
 x1   x 7
how many?
Way 2:
a)
b)
Consistent tuple  conjunction of terms x1  x 5
Each constraint  disjunction of these conjunctions
 x1  x 5    x1  x 6    x 2  x 7 
  x3  x6    x 4  x5 
 transform into conjunctive normal form (CNF)
Question: find a truth assignment of the Boolean variables such that
the sentence is satisfied
Problem Solving with Constraints, CSCE 496/896
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Outline
 Motivating example, application areas
 CSP: Definition, representation
 Some simple modeling examples
 More on definition and formal characterization
 Basic solving techniques
• Modeling and consistency checking
• Constructive, systematic search
• Iterative improvement, local search
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How to solve a CSP?
Search
1. Constructive, systematic
2. Iterative repair, local search
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Before starting search!
Consider:
• Importance of modeling/formulation:
– To control the size of the search space
• Preprocessing
– A.k.a. constraint filtering/propagation, consistency
checking
– reduces size of search space
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Importance of Modeling
• N-queen: formulation 1
1
2
3 4
V1
– Variables?
– Domains?
– Size of CSP?
V2
4 Rows
4 Column positions
V3
4444  4  2
4
8
V4
• N-queens: formulation 2
V11 V12 V13 V14
– Variables? 16 Cells
{0,1}
– Domains?
16
– Size of CSP?
2
V21 V22 V23 V24
V31 V32 V33 V34
V41 V42 V43 V44
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Constraint Checking
 Arc-consistency
B
A
C: [ 6 .. 15 ]
[ 5.... 18]
2
B<C
[ 1.... 10 ]
2<C -A<5
1- B: [ 5 .. 14 ]
13
14
A< B
2- A: [ 2 .. 10 ]
C: [ 6 .. 14 ]
6
3- B: [ 5 .. 13 ]
14
[ 4.... 15]
C
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Constraint Checking
 Arc-consistency: every combination of two adjacent variables
 3-consistency, k-consistency (k  n)
 Constraint filtering, constraint checking, etc..
 Eliminate non-acceptable tuples prior to search
 Warning: arc-consistency does not solve the problem
B
A
{ 1, 2, 3 }
A
{ 2, 3, 4 }
( A  2 )  ( B  3)
{ 2, 3 }
B
{ 2, 3 }
still is not a solution!
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Systematic Search




Starting from a root node
Consider all values for a variable V1
For every value for V1, consider all values for V2
etc..
S
Var 1
v1
v2
v3
v4
Var 2
 For n variables, each of domain size d
 Maximum depth?
fixed!
 Maximum number of paths?
size of search space, size of CSP
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Systematic search: Back-checking
• Systematic search generates dn possibilities
• Are all possibilities acceptable?
S
S
V ar 1 v 1
Var 1
v1
v2
v3
v2
v3
v4
v4
V ar 2
Var 2
 Expand a partial solution only when it is consistent
 This yields early pruning of inconsistent paths
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Systematic search: Chronological backtracking
What if only one solution is needed?
S
S
V ar 1
V ar 1 v 1
v2
v3
v1
v2
v4
V ar 2
• Depth-first search & Chronological backtracking
• DFS: Soundness? Completeness?
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Systematic search: Intelligent backtracking
What if the reason for failure was higher up in the tree?
Backtrack to source of conflict !!
S
V ar 1
v1
S
v2
Var 1
v1
v2
 Backjumping, conflict-directed backjumping, etc.
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Systematic search:
Ordering heuristics
• Which variable to expand first?
Exp : V1 , V 2 , D V1  { a , b , c , d }, D V 2  { a , b }
Sol : {( V 1  c ), (V 2  a )} and {( V 1  c ), (V 2  b )}
s
V1
• Heuristics:
V2
a
a
s
b
b
a
c
b
a
V2
d
b
V1
a
a
b
b
c
d
– most constrained variable first (reduce branching factor)
– most promising value first (find quickly first solution)
Strategies
 variable ordering 
 static
for 
could
be


value
ordering


 dynamic



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Systematic search: Back-checking
• Search tree with only backtrack search?
Root node
Q
Q
Q
Q
2
1
Q
Q
1
2
3
4
1 2 3 4 1 2 3 4
Q
1 2 3 4
1
2
3
4
1
1 2
3
Solution!
26 nodes visited.
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Systematic search: Forward checking
Search Tree with domains filter by Forward Check
Root node
Q
Q
Q
2
1
Q
Q
Q
Q
3
4
4
Domain
2
Wipe Out
V3
Domain
Wipe Out
V4
1
3
Solution!
8 nodes visited.
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Improving BT search
•
General purpose methods for
1. Variable, value ordering
2. Improving backtracking: intelligent
backtracking avoids repeating failure
3. Look-ahead techniques: propagate
constraints as variables are instantiated
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Search-tree branching
• K-way branching
– One branch for every value in the domain
• Binary branching
– One branch for the first value
– One branch for all the remaining values in the domain
– Used in commercial constraint solvers: ILOG, Eclipse
• Have different behaviors (e.g., WRT value ordering
heuristics [Smith, IJCAI 95])
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Summary of backtrack search
• Constructive, systematic, exhaustive
– In general sound and complete
• Back-checking: expands nodes consistent with past
• Backtracking: Chronological vs. intelligent
• Ordering heuristics:
– Static
– Dynamic variable
– Dynamic variable-value pairs
• Looking ahead:
– Partial look-ahead
• Forward checking (FC)
• Directional arc-consistency (DAC)
– Full (a.k.a. Maintaining Arc-consistency or MAC)
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CSP: a decision problem (NP-complete)
1.
2.
Modeling: abstraction and reformulation
Preprocessing techniques:
•
3.
eliminate non-acceptable tuples prior to search
Systematic search:
•
•
•
•
potentially dn paths of fixed lengths
chronological backtracking
intelligent backtracking
variable/value ordering heuristics
Search ‘hybrids:’
4.
•
Mixing consistency checking with search: look-ahead
strategies
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Non-Systematic Search
• Iterative repair, local search: modifies a
global but inconsistent solution to decrease the
number of violated constraints
• Examples: Hill climbing, taboo search,
simulated annealing, GSAT, WalkSAT, Genetic
Algorithms, Swarm intelligence, etc.
• Features: Incomplete & not sound
– Advantage: anytime algorithm
– Shortcoming: cannot tell that no solution exists
Problem Solving with Constraints, CSCE 496/896
Overview 1
55
Outline of CSP 101
• We have seen
 Motivating example, application areas
 CSP: Definition, representation
 Some simple modeling examples
 More on definition and formal characterization
 Basic solving techniques
• We will move to
 (Implementing backtrack search)
Advanced solving techniques
Issues & research directions
Problem Solving with Constraints, CSCE 496/896
Overview 1
56
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Foundations of Constraint Processing CSCE421/821, Fall …