Constraint Satisfaction Problems
Chapter 5
Chapter 5 1
♦ CSP examples
♦ Backtracking search for CSPs
♦ Problem structure and problem decomposition ♦ Local search for CSPs
Outline
Chapter 5 2
Constraint satisfaction problems (CSPs)
Standard search problem:
state is a “black box”—any old data structure
that supports goal test, eval, successor CSP:
state is defined by variables Xi 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
Chapter 5 3
Example: Map-Coloring
Western Australia
Queensland
Variables WA, NT, Q, NSW, V , SA, T
Domains Di = {red, green, blue}
Constraints: adjacent regions must have different colors
Northern Territory
South Australia
e.g., WA ̸= NT (if the language allows this), or
(W A, N T ) ∈ {(red, green), (red, blue), (green, red), (green, blue), . . .}
New South Wales
Victoria
Tasmania
Chapter 5 4
Example: Map-Coloring contd.
Western Australia
Queensland
Northern Territory
South Australia
Solutions are assignments satisfying all constraints, e.g.,
{WA=red,NT =green,Q=red,NSW =green,V =red,SA=blue,T =green}
New South Wales
Victoria
Tasmania
Chapter 5 5
WA
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!
NT
SA
NSW
V Victoria
Q
T
Chapter 5 6
Varieties of CSPs
Discrete variables
finite domains; size d ⇒ 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 Hubble Telescope observations ♦ linear constraints solvable in poly time by LP methods
Chapter 5 7
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
Chapter 5 8
TWO +TWO FOUR
FTUWRO
Example: Cryptarithmetic
Variables: F T U W ROX1 X2 X3 Domains: {0,1,2,3,4,5,6,7,8,9} Constraints
alldiff(F, T, U, W, R, O)
O + O = R + 10 · X1, etc.
X3 X2 X1
Chapter 5 9
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
Real-world CSPs
Notice that many real-world problems involve real-valued variables
Chapter 5 10
Standard search formulation (incremental)
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) Path is irrelevant, so can also use complete-state formulation 4) b = (n − l)d at depth l, hence n!dn leaves!!!!
Chapter 5 11
Backtracking search
Variable assignments are commutative, i.e.,
[WA=redthenNT =green] sameas [NT =greenthenWA=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
Chapter 5 12
Backtracking search
function Backtracking-Search(csp) returns solution/failure return Recursive-Backtracking({ }, csp)
function Recursive-Backtracking(assignment,csp) returns soln/failure if assignment is complete then return assignment var←Select-Unassigned-Variable(Variables[csp],assignment,csp) for each value in 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
Chapter 5 13
Backtracking example
Chapter 5 14
Backtracking example
Chapter 5 15
Backtracking example
Chapter 5 16
Backtracking example
Chapter 5 17
Improving backtracking efficiency
General-purpose methods can give huge gains in speed:
1. Which variable should be assigned next?
2. In what order should its values be tried?
3. Can we detect inevitable failure early?
4. Can we take advantage of problem structure?
Chapter 5 18
Minimum remaining values
Minimum remaining values (MRV):
choose the variable with the fewest legal values
Chapter 5 19
Tie-breaker among MRV variables
Degree heuristic
Degree heuristic:
choose the variable with the most constraints on remaining variables
Chapter 5 20
Least constraining value
Given a variable, choose the least constraining value:
the one that rules out the fewest values in the remaining variables
Combining these heuristics makes 1000 queens feasible
Allows 1 value for SA
Allows 0 values for SA
Chapter 5 21
Forward checking
Idea: 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
Chapter 5 22
Forward checking
Idea: 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
Chapter 5 23
Forward checking
Idea: 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
Chapter 5 24
Forward checking
Idea: 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
Chapter 5 25
Constraint propagation
Forward checking propagates information from assigned to unassigned vari- ables, 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
Chapter 5 26
Arc consistency
Simplest form of propagation makes each arc consistent X → Y is consistent iff
for every value x of X there is some allowed y
WA NT Q NSW V SA T
Chapter 5 27
Arc consistency
Simplest form of propagation makes each arc consistent X → Y is consistent iff
for every value x of X there is some allowed y
WA NT Q NSW V SA T
Chapter 5 28
Arc consistency
Simplest form of propagation makes each arc consistent X → Y is consistent iff
for every value x of X there is some allowed y
WA NT Q NSW V SA T
If X loses a value, neighbors of X need to be rechecked
Chapter 5 29
Arc consistency
Simplest form of propagation makes each arc consistent X → Y is consistent iff
for every value x of X there is some allowed 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
Chapter 5 30
Arc consistency algorithm
function AC-3( csp) returns the CSP, possibly with reduced domains inputs: 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
for each Xk in Neighbors[Xi] do add (Xk, Xi) to queue
function Remove-Inconsistent-Values( Xi, Xj) returns true iff succeeds removed ← false
for each x in Domain[Xi] do
if no value y in Domain[Xj] allows (x,y) to satisfy the constraint Xi ↔ Xj thendeletexfromDomain[Xi]; removed←true
return removed
O(n2d3), can be reduced to O(n2d2) (but detecting all is NP-hard)
Chapter 5 31
Problem structure
WA
Tasmania and mainland are independent subproblems Identifiable as connected components of constraint graph
NT
SA
NSW
V Victoria
Q
T
Chapter 5 32
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
Chapter 5 33
Tree-structured CSPs
AE BD CF
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.
Chapter 5 34
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
AE
BD ABCDEF
CF
2. Forjfromndownto2,applyRemoveInconsistent(Parent(Xj),Xj) 3. For j from 1 to n, assign Xj consistently with Parent(Xj)
Chapter 5 35
WA
WA
Nearly tree-structured CSPs
Conditioning: instantiate a variable, prune its neighbors’ domains
NT
NT
SA
NSW
NSW
V Victoria
V Victoria
Q
Q
T
T
Cutset conditioning: instantiate (in all ways) a set of variables such that the remaining constraint graph is a tree
Cutset size c ⇒ runtime O(dc · (n − c)d2), very fast for small c
Chapter 5 36
Iterative algorithms for CSPs
Hill-climbing, simulated annealing typically work with “complete” states, i.e., all variables assigned
To apply to CSPs:
allow states with unsatisfied constraints operators reassign variable values
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
Chapter 5 37
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
Chapter 5 38
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
CPU time
critical ratio
R
Chapter 5 39
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
Summary
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
Chapter 5 40