Week 7: Constraint Satisfaction Problems – COMP30024 – Artificial Intelligence Chapter 5
Week 7: Constraint Satisfaction Problems
COMP30024 – Artificial Intelligence
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Dr Wafa Johal
University of Melbourne
Introduction
• CSP examples
• General search applied to CSPs
• Backtracking search for CSPs
• Problem structure and problem decomposition
• Local search for CSPs
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Constraint satisfaction problems (CSPs)
Standard search problem:
state is a “black box”—any old data structure that supports goal test, eval,
state is defined by variables Vi with values from domain Di
goal test is a set of constraints specifying allowable combinations of values for
subsets of variables
Simple example of a formal representation language
Allows useful general-purpose algorithms with more power than standard
search algorithms
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Example: Map-Coloring
Queensland
Variables WA, NT, Q, NSW, V, SA, T
Domains Di = {red, green, blue}
Constraints: adjacent regions must have different colors
e.g., WA ̸= NT (if the language allows this), or
(WA,NT) ∈ {(red, green), (red, blue), (green, red), (green, blue), . . .}
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Example: Map-Coloring
Queensland
Variables WA, NT, Q, NSW, V, SA, T
Domains Di = {red, green, blue}
Constraints: adjacent regions must have different colors
e.g., WA ̸= NT (if the language allows this), or
(WA,NT) ∈ {(red, green), (red, blue), (green, red), (green, blue), . . .}
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Example: Map-Coloring contd.
Queensland
Solutions are assignments satisfying all constraints, e.g.,
{WA = red,NT = green,Q = red,NSW = green,V = red, SA = blue,T =
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Constraint graph
Binary CSP: each constraint relates at most two variables
Constraint graph: nodes are variables, arcs show constraints
General-purpose CSP algorithms use the graph structure
to speed up search. E.g., Tasmania is an independent subproblem!
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Varieties of CSPs
Discrete variables
finite domains; size d
where n is the number of variables in the CSP
=⇒ O(dn) complete assignments
e.g., Boolean CSPs, incl. Boolean satisfiability (NP-complete)
infinite domains (integers, strings, etc.)
• e.g., job scheduling, variables are start/end days for each job
• need a constraint language, e.g., StartJob1 + 5 ≤ StartJob3
• linear constraints solvable, nonlinear undecidable
Continuous variables
• e.g., start/end times for observations
• linear constraints solvable in polynomial time by linear programming (LP)
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Varieties of constraints
• Unary constraints involve a single variable,
e.g., SA ̸= green
• Binary constraints involve pairs of variables,
e.g., SA ̸= WA
• Higher-order constraints involve 3 or more variables, e.g., cryptarithmetic
column constraints
• Preferences (soft constraints), e.g., red is better than green often
representable by a cost for each variable assignment
→ constrained optimization problems
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Example: Cryptarithmetic
Variables: F T U W R O C1 C2 C3
Domains: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Constraints
alldiff(F,T,U,W,R,O)
O + O = R + 10 · C1, etc.
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Real-world CSPs
Assignment problems e.g., who teaches what class
Timetabling problems e.g., which class is offered when and where?
Hardware configuration
Spreadsheets
Transportation scheduling
Factory scheduling
Floorplanning
Many real-world problems involve real-valued variables
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Applying standard search
Let’s start with the straightforward, dumb approach, then fix it
States are defined by the values assigned so far
• Initial state: the empty assignment, ∅
• Successor function: assign a value to an unassigned variable that does
not conflict with current assignment.
=⇒ fail if no legal assignments (not fixable!)
• Goal test: the current assignment is complete
1. This is the same for all CSPs!
2. Every solution appears at depth n with n variables
=⇒ use depth-first search
3. b = (n− ℓ)d at depth ℓ, hence n!dn leaves!!!!
4. Path is irrelevant, so can also use complete-state formulation
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Applying standard search
Let’s start with the straightforward, dumb approach, then fix it
States are defined by the values assigned so far
• Initial state: the empty assignment, ∅
• Successor function: assign a value to an unassigned variable that does
not conflict with current assignment.
=⇒ fail if no legal assignments (not fixable!)
• Goal test: the current assignment is complete
1. This is the same for all CSPs!
2. Every solution appears at depth n with n variables
=⇒ use depth-first search
3. b = (n− ℓ)d at depth ℓ, hence n!dn leaves!!!!
4. Path is irrelevant, so can also use complete-state formulation
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Backtracking search
Variable assignments are commutative, 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 node
=⇒ b = d and there are dn leaves
Depth-first search for CSPs with single-variable assignments is called
backtracking search
Backtracking search is the basic uninformed algorithm for CSPs
Can solve n-queens for n ≈ 25
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Backtracking search: Implementation
function Backtracking-Search(csp) return solution/failure
Recursive-Backtracking({}, csp)
end function
function Recursive-Backtracking(assignment, csp)
if assignment is complete then
return assignment
var← Select-Unassigned-Variable(Variables[csp],assignment, csp)
foreach value ∈ Order-Domain-Values(var, assignment, csp) do
if value is consistent with assignment given Constraints[csp] then
add{var = value} to assignment
result← Recursive-Backtracking(assignment, csp)
if result ̸= failure then
return result
remove{var = value} from assignment
return failure
end function
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Backtracking example
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Backtracking example
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Backtracking example
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Backtracking example
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Improving backtracking efficiency
General-purpose methods can give huge gains in speed:
1. Which variable should be assigned next?
var← Select-Unassigned-Variable(Variables[csp],assignment, csp)
2. In what order should its values be tried?
Order-Domain-Values(var, assignment, csp)
3. Can we detect inevitable failure early?
4. Can we take advantage of problem structure?
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Variable Ordering
Which variable should be assigned next?
var← Select-Unassigned-Variable(Variables[csp],assignment, csp)
Minimum remaining values (MRV)
choose the variable with the fewest legal values
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Variable Ordering
Which variable should be assigned next?
var← Select-Unassigned-Variable(Variables[csp],assignment, csp)
Minimum remaining values (MRV)
choose the variable with the fewest legal values
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Variable Ordering
Tie-breaker among MRV variables or choice of first variable
Degree heuristic
choose the variable with the most constraints on remaining variables
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Value Ordering
In what order should its values be tried?
Order-Domain-Values(var, assignment, csp)
Least constraining value
Given a variable, choose the least constraining value:
the one that rules out the fewest values in the remaining variables
Allows 1 value for SA
Allows 0 values for SA
Combining these heuristics makes 1000 queens feasible
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Forward checking
Forward Checking
Keep track of remaining legal values for unassigned variables
Terminate search when any variable has no legal values
WA NT Q NSW V SA T
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Forward checking
Forward Checking
Keep track of remaining legal values for unassigned variables
Terminate search when any variable has no legal values
WA NT Q NSW V SA T
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Forward checking
Forward Checking
Keep track of remaining legal values for unassigned variables
Terminate search when any variable has no legal values
WA NT Q NSW V SA T
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Forward checking
Forward Checking
Keep track of remaining legal values for unassigned variables
Terminate search when any variable has no legal values
WA NT Q NSW V SA T
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Constraint propagation
Forward checking propagates information from assigned to unassigned
variables, but doesn’t provide early detection for all failures:
WA NT Q NSW V SA T
NT and SA cannot both be blue!
Constraint propagation repeatedly enforces constraints locally
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Arc consistency – Illustration
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Arc consistency – Illustration
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Arc consistency – Illustration
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Arc consistency – Illustration
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Arc consistency – Illustration
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Arc consistency – Illustration
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Arc consistency
Simplest form of propagation makes each arc consistent
X→ Y is arc consistent iff
for every value x of X there is at least one value y of Y
that satisfies the constraint between X and Y
WA NT Q NSW V SA T
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Arc consistency
Simplest form of propagation makes each arc consistent
X→ Y is arc consistent iff
for every value x of X there is at least one value y of Y
that satisfies the constraint between X and Y
WA NT Q NSW V SA T
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Arc consistency
Simplest form of propagation makes each arc consistent
X→ Y is arc consistent iff
for every value x of X there is at least one value y of Y
that satisfies the constraint between X and Y
WA NT Q NSW V SA T
If X loses a value, neighbors of X need to be rechecked
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Arc consistency
Simplest form of propagation makes each arc consistent
X→ Y is arc consistent iff
for every value x of X there is at least one value y of Y
that satisfies the constraint between X and Y
WA NT Q NSW V SA T
If X loses a value, neighbors of X need to be rechecked
Arc consistency detects failure earlier than forward checking
Can be run as a preprocessor or after each assignment
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Arc consistency algorithm
function AC-3(csp) returns the CSP, possibly with reduced domains
input csp a binary CSP with variables {X1,X2, . . . ,Xn}
local variables queue, a queue of arcs, initially all the arcs in csp
while queue is not empty do
(Xi,Xj) ← Remove-First(queue)
if Remove-Inconsistent-Values(Xi,Xj) then
foreach Xk in Neighbors(Xi) do
add (Xk,Xi) to queue
end function
——————————–
function Remove-Inconsistent-Values(Xi, Xj) true iff succeeds
removed← false
foreach x in Domain(Xi) do
if no value y Domain(Xi) allows (x, y) to satisfy the constraint Xi↔ Xj then
delete x from Domain(Xi);
removed← true
return removed
end function
O(n2d3), can be reduced to O(n2d2) (but detecting all is NP-hard)
Alternative approach: exploit structure of the constraint graph to find
independent subproblems …
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Problem structure
Tasmania and mainland are independent subproblems
Identifiable as connected components of constraint graph
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Problem structure contd.
Suppose each subproblem has c variables out of n total
Worst-case solution cost is n/c · dc, linear in n
E.g., n = 80, d = 2, c = 20
280 = 4 billion years at 10 million nodes/sec
4 · 220 = 0.4 seconds at 10 million nodes/sec
Unfortunately, completely independent subproblems are rare in practice
However, there are other graph structures that are easy to solve
e.g. when the constraint graph is a tree
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Problem structure contd.
Suppose each subproblem has c variables out of n total
Worst-case solution cost is n/c · dc, linear in n
E.g., n = 80, d = 2, c = 20
280 = 4 billion years at 10 million nodes/sec
4 · 220 = 0.4 seconds at 10 million nodes/sec
Unfortunately, completely independent subproblems are rare in practice
However, there are other graph structures that are easy to solve
e.g. when the constraint graph is a tree
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Tree-structured CSPs
Theorem: if the constraint graph has no loops, the CSP can be solved in
O(n d2) time
Compare to general CSPs, where worst-case time is O(dn)
This property also applies to logical and probabilistic reasoning:
an important example of the relation between syntactic restrictions
and the complexity of reasoning.
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Algorithm for tree-structured CSPs
1. Choose a variable as root, order variables from root to leaves
such that every node’s parent precedes it in the ordering
A B C D E F
2. For j from n down to 2, apply MakeArcConsistent(Parent(Xj),Xj)
3. For j from 1 to n, assign Xj consistently with Parent(Xj)
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Nearly tree-structured CSPs
Conditioning: instantiate a variable, prune its neighbors’ domains
Cutset conditioning: instantiate (in all ways) a set of variables
such that the remaining constraint graph is a tree
Cutset: set of variables that can be deleted so constraint graph forms a tree
Cutset size c =⇒ runtime O(dc · (n− c)d2), very fast for small c
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Iterative algorithms for CSPs – Local search
Recall hill-climbing search from Week 3
Hill-climbing typically works with
“complete” states, i.e., all variables assigned
Local search then tries to change one variable assignment at a time
To apply to CSPs:
allow states with unsatisfied constraints(variable selection)
operators reassign variable values (value selection)
Variable selection: randomly select any conflicted variable
Value selection by min-conflicts heuristic:
choose value that violates the fewest constraints
i.e., hillclimb with h(n) = total number of violated constraints
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Example: 4-Queens
States: 4 queens in 4 columns (44 = 256 states)
Operators: move queen in column
Goal test: no attacks
Evaluation: h(n) = number of attacks
h = 5 h = 2 h = 0
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Performance of min-conflicts
Given random initial state, can solve n-queens in almost constant time for
arbitrary n with high probability (e.g., n = 10,000,000)
The same appears to be true for any randomly-generated CSP
except in a narrow range of the ratio
R = number of constraints
number of variables
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CSPs are a special kind of problem:
states defined by values of a fixed set of variables
goal test defined by constraints on variable values
Backtracking = depth-first search with one variable assigned per node
Variable ordering and value selection heuristics help significantly
Forward checking prevents assignments that guarantee later failure
Constraint propagation (e.g., arc consistency) does additional work
to constrain values and detect inconsistencies
The CSP representation allows analysis of problem structure
Tree-structured CSPs can be solved in linear time
Iterative min-conflicts is usually effective in practice
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Examples of skills expected:
• Model a given problem as a CSP
• Demonstrate operation of CSP search algorithms
• Discuss and evaluate the properties of different constraint satisfaction
techniques
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Introduction
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