Lirong Xia

Constraint satisfaction problems

ØConstraint satisfaction problemsØAlgorithms

• backtracking search

• filtering§ forward checking

§ constraint propagation

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Today’s schedule

Ø Standard search problems:• State is a “black box”: arbitrary data structure

• Goal test: any function over states• Successor function can be anything

Ø Constraint satisfaction problems (CSPs):• A special subset of search problems

• State is defined by variables Xi with values from a domain D (sometimes D depends on i )

• Goal test is a set of constraints specifying allowable combinations of values for subsets of variables

Ø Allows useful general-purpose algorithms with more power than standard search algorithms

Constraint Satisfaction Problems

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ØVariables: WA, NT, Q, NSW, V, SA, T

ØDomains:

ØConstraints: adjacent regions must have different colors

ØSolutions are assignments satisfying all constraints, e.g.:

Example: Map-Coloring

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{ }red, green, blueD =

( ) ( ) ( ) ( ){ }, red, green , red, blue , green, red ,...

WA NTWA NT

¹

Î

WA , NT , Q , ,, , red green red NSW green

V red SA blue T green= = = =ì ü

í ý= = =î þ

ØFormulation 1:• Variables:

• Domains:

• Constraints

Example: N-Queens

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ijX

∀i, j,k Xij ,Xik( )∈ 0, 0( ), 0, 1( ), 1, 0( ){ }

{ }0,1

∀i, j,k Xij ,Xkj( )∈ 0, 0( ), 0, 1( ), 1, 0( ){ }∀i, j,k Xij ,Xi+k , j+k( )∈ 0, 0( ), 0, 1( ), 1, 0( ){ }∀i, j,k Xij ,Xi+k , j−k( )∈ 0, 0( ), 0, 1( ), 1, 0( ){ }

,ij

i jX N=å

ØFormulation 2:• Variables:

• Domains:

• Constraints

Example: N-Queens

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kQ

( ), non-threatening ,i ji j Q Q"

{ }1,2,3,...N

Q1,Q2( )∈ 1, 3( ), 1, 4( ),...{ }

Implicit:-or-

Explicit:

Q1 Q2 Q3 Q4

ØBinary CSP: each constraint relates (at most) two variables

ØBinary constraint graph: nodes are variables, arcs show constraints

ØGeneral-purpose CSP algorithms use the graph structure to speed up search.

• E.g., Tasmania is an independent subproblem!

Constraint Graphs: Goal test

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ØVariables (circles):

ØDomains:

ØConstraints (boxes):

Example: Cryptarithmetic

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{ }0, 1, 2, 3, 4, 5, 6, 7, 8, 9

alldiff F , T , U , W , R, O( )O+O = R+10• X1

1 2 3 F T U W R O X X XT W O

+ T W OF O U R

ØVariables: • Each (open) square

ØDomains:•

ØConstraints:

Example: Sudoku

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1, 2, …, 9{ }

9-way alldiff for each column9-way alldiff for each row9-way alldiff for each region

ØDiscrete Variables:• Finite domains

§ Size d means O(dn) complete assignments§ E.g., Boolean CSPs, including Boolean satisfiability (NP-

complete)

• Infinite domains (integers, strings, etc.)§ Linear constraints solvable

ØContinuous variables:• Linear constraints solvable in polynomial time by

LP methods

Varieties of CSPs

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Ø Variables of Constraints• Unary constraints involve a single variable (equiv. to shrinking

domains):

• Binary constraints involve pairs of variables:

• Higher-order constraints involve 3 or more variables:e.g., cryptarithmetic column constraints

Ø Preferences (soft constraints):• E.g., red is better than green• Often representable by a cost for each variable assignment• Gives constrained optimization problems

Varieties of Constraints

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SA green¹

SA WA¹

ØStandard search formulation of CSPs (incremental)

ØLet’s start with the straightforward, dumb approach, then fix it

ØStates are defined by the values assigned so far• Initial state; the empty assignment,• Successor function: assign a value to an unassigned

variable• Goal test: the current assignment is complete and

satisfies all constraint

Standard Search Formulation

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ØWhat would BFS do?

ØWhat would DFS do?

Ø Is BFS better or DFS better?

ØCan we use A*?

Search Methods

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Early detection of non-Goal

Ø Idea 1: only consider a single variable at each point• Variable assignments are commutative, so fix ordering• I.e., [WA = red then NT = green] same as [NT = green then WA = red]• Only need to consider assignments to a single variable at each step

Ø Idea 2: only allow legal assignments at each point• “Incremental goal test”

Ø DFS for CSPs with these two improvements is called backtracking search

Ø Backtracking search is the basic uninformed algorithm for CSPs

Ø Can solve n-queens for

Backtracking Search

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25n »

Backtracking Example

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ØOrdering:• Which variable should be assigned next?

• In what order should its values be tried?

ØFiltering: Can we detect inevitable failure early?

Improving Backtracking

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Minimum Remaining Values

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ØMinimum remaining values (MRV):• Choose the variable with the fewest legal values

ØWhy min rather than max?

ØAlso called “most constrained variable”

Ø“Fail-fast” ordering

Example

• For every variable, keep track of which values are still possible

Q X X X

Q X X X

Q X X X

X

Q X X X

X X

X

Q X X X

Q

Q

Q

Q

Q

Q

Q

Q

Q X X

Q X X

Q X X

X

Q X X

X X Q

X

Q X X

only one possibility for last column; might

as well fill in

now only one left for other two columns

done!(no real branching

needed!)

Choosing a value

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ØGiven a choice of variable:• Choose the least constraining

value• The one that rules out the

fewest values in the remaining variables

• May take some computation to determine this!

ØWhy least rather than most?

value 1 for SA

value 0 for SA

Filtering: Forward Checking

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Ø Idea: keep track of remaining values for unassigned variables (using immediate constraints)

Ø Idea: terminate when any variable has no legal values

Filtering: Constraint Propagating

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Ø Forward checking propagates information from assigned to unassigned variables, but doesn’t provide early detection for all failures:

Ø NT and SA cannot both be blue!Ø Why didn’t we detect this yet?

step 1 step 2

step 1step 2

Consistency of An Arc

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Ø An arc is consistent iff for every x in the tail there is some y in the head which could be assigned without violating a constraint

Ø Forward checking = Enforcing consistency of each arc pointing to the new assignment

X Y®

Deletefrom tail!

Arc Consistency of a CSP

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Ø A simple form of propagation makes sure all arcs are consistent:

Ø If V loses a value, neighbors of V need to be rechecked!Ø Arc consistency detects failure earlier than forward checkingØ Can be run as a preprocessor or after each assignmentØ Might be time-consuming

Deletefrom tail!

X XX