CSPs

CSPs

Tamara BergCS 590-133 Artificial Intelligence

Many slides throughout the course adapted from Svetlana Lazebnik, Dan Klein, Stuart Russell, Andrew Moore, Percy Liang, Luke Zettlemoyer

Review from last class

What is search for?• Assumptions: single agent,

deterministic, fully observable, discrete environment

• Search for planning– The path to the goal is the

important thing– Paths have various costs, depths

• Search for assignment– Assign values to variables while

respecting certain constraints– The goal (complete, consistent

assignment) is the important thing

Constraint satisfaction problems (CSPs)

• Definition:– Xi is a set of variables {X1 ,… Xn}

– Di is a set of domains {D1 ,... Dn} one for each variable– C is a set of constraints that specify allowable combinations of

values– Solution is a complete, consistent assignment

• How does this compare to the “generic” tree search formulation?– A more structured representation for states, expressed in a

formal representation language– Allows useful general-purpose algorithms with more power than

standard search algorithms • by eliminating large portions of search space (identifying var/val

combinations that violate constraints).

Example: Map Coloring

• Variables: WA, NT, Q, NSW, V, SA, T • Domains: {red, green, blue}• Constraints: adjacent regions must have different colors

e.g., WA ≠ NT, or (WA, NT) in {(red, green), (red, blue), (green, red), (green, blue), (blue, red), (blue, green)}

Example: Map Coloring

• Solutions are complete and consistent assignments, e.g., WA = red, NT = green, Q = red, NSW = green, V = red, SA = blue, T = green

Example: Cryptarithmetic

• Variables: T, W, O, F, U, R X1, X2

• Domains: {0, 1, 2, …, 9}• Constraints:

O + O = R + 10 * X1

W + W + X1 = U + 10 * X2

T + T + X2 = O + 10 * FAlldiff(T, W, O, F, U, R)T ≠ 0, F ≠ 0

X2 X1

Example: Sudoku

• Variables: Xij

• Domains: {1, 2, …, 9}• Constraints:

Alldiff(Xij in the same unit)Xij

Real-World CSPs

• Assignment problems– e.g., who teaches what class– hard constraint - no prof can teach 2 classes at the same

time. – preference constraints – Jan prefers teaching in the

morning while Alex prefers teaching in the afternoon (soft constraints/costs). For example an afternoon slot for Jan can cost 2 points whereas a morning slot costs 1.

Real-World CSPs

• Timetable problems– e.g., which class is offered when and where?– Classroom size & enrollment can put hard constraints on

where a class can be held – Avoiding simultaneous scheduling of two related classes

can be a soft constraint

Real-World CSPs

• Transportation Scheduling– E.g. airline scheduling Flights F1,F2 with capacities 165 and

385– Domains D1=[0,165] D2=[0,385]– Suppose we have an additional constraint that the two

flights must carry 420 people: ie F1+F2=420 – Then we can reduce the domains to:

D1=[35,165] D2=[255,385]

Enforcing Bounds consistency

Real-World CSPs• Job-shop scheduling

– e.g. assembling a car – Variables – X = tasks {Axlef, Axleb, Wheelrf, Wheellf, Wheelrb,

Wheellb,Nutsrf, Nutslf, Nutslb, Nutsrb, Caprf, Caplf, Caprb, CAPlb, Inspect}– Values – Di time task starts– Constraints – one ask must occur before other, a task takes a certain

amount of time to complete.• e.g. AxleF + 10 ≤ Wheelrf, Wheelrf+1≤Nutsrf, Nutsrf+2≤Caprf…• 4 workers, but only 1 axle tool:

– Axlef+10≤Axleb or Axleb+10≤Axlef

• For each variable, X, except Inspect:– X+dx ≤ Inspect

• Want total time less than 30 minutes and inspect takes 3 minutes:– Di = {1,2,3,….,27}

More examples of CSPs: http://www.csplib.org/

http://www.csplib.org/

Types of Constraints• Unary

– restricts value of a single variable, e.g. SA≠green.• Binary

– relates two variables, e.g. SA≠NSW• Higher order

– relates more than two variables, e.g. Between (X,Y,Z)• Global

– arbitrary number of variables– commonly Alldiff – all variables must have different values. e.g. in sudoku all variables in a

unit must satisfy Alldiff.• Preference constraint

– Red is better than green– Often representable by a cost for each variable assignment

• Resource constraint– Atmost constraint. e.g. scheduling P1,P2,P3,P4 denote personnel assigned to each task.

Constrain is that you only have ≤ 10 people. So Atmost(10,P1,P2,P3,P4)

Variations on CSPs

Simplest CSPs – discrete, finite domainsmap coloring, scheduling with time limits, 8-queens

Discrete with infinite domainno deadline scheduling

Continuous domainsscheduling of experiments on Hubble telescope

Linear programming problemsconstraints must be linear equalities or inequalities (can be solved in time polynomial in # vars)

• Variables: Xij

• Domains: {1, 2, …, 9}• Constraints:

Alldiff(Xij in the same unit)Xij

Why are constraints useful?

They let us eliminate portions of assignment space • By identifying var/val combinations that violate constraints

?

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Standard search formulation (incremental)

• States: – Variables and values assigned so far

• Initial state:– The empty assignment

• Action:– Choose any unassigned variable and assign to it a value

that does not violate any constraints• Fail if no legal assignments

• Goal test: – The current assignment is complete and satisfies all

constraints

Standard search formulation (incremental)

• What is the depth of any solution (assuming n variables)? n (this is good)

• Given that there are m possible values for any variable, how many paths are there in the search tree?n! · mn (this is bad)

• How can we reduce the branching factor?

Backtracking search• In CSP’s, variable assignments are commutative– For example, [WA = red then NT = green] is the same as

[NT = green then WA = red]• We only need to consider assignments to a single variable at

each level (i.e., we fix the order of assignments)

Example

Backtracking search• In CSP’s, variable assignments are commutative– For example, [WA = red then NT = green] is the same as

[NT = green then WA = red]• We only need to consider assignments to a single variable at

each level (i.e., we fix the order of assignments)• Depth-first search for CSPs with single-variable assignments is

called backtracking search

Backtracking search algorithm

Which variable should be assigned next?

• Most constrained variable:– Choose the variable with the fewest legal values– A.k.a. minimum remaining values (MRV) heuristic


• Most constraining variable:– Choose the variable that imposes the most

constraints on the remaining variables


• Most constraining variable:– Choose the variable that imposes the most

constraints on the remaining variables– Tie-breaker among most constrained variables

Backtracking search algorithm

Given a variable, in which order should its values be tried?

• Choose the least constraining value:– The value that rules out the fewest values in the

remaining variables

Given a variable, in which order should its values be tried?

• Choose the least constraining value:– The value that rules out the fewest values in the

remaining variablesWhich assignment

for Q should we choose?

Early detection of failure

Apply inference to reduce the space of possible assignments and detect failure early

Early detection of failure:Forward checking

• Keep track of remaining legal values for unassigned variables• Terminate search when any variable has no legal values

Constraint propagation• Forward checking propagates information from assigned to

unassigned variables, but doesn't provide early detection for all failures

• NT and SA cannot both be blue!• Constraint propagation repeatedly enforces constraints locally

• Simplest form of propagation makes each pair of variables consistent:– X Y is consistent iff for every value of X there is some allowed value of Y

Arc consistency

Consistent!

• Simplest form of propagation makes each pair of variables consistent:– X Y is consistent iff for every value of X there is some allowed value of Y

Arc consistency

• Simplest form of propagation makes each pair of variables consistent:– X Y is consistent iff for every value of X there is some allowed value of Y– When checking X Y, throw out any values of X for which there isn’t an

allowed value of Y

• If X loses a value, all pairs Z X need to be rechecked

Arc consistency

Arc consistency• Simplest form of propagation makes each pair of variables

consistent:– X Y is consistent iff for every value of X there is some allowed value of Y– When checking X Y, throw out any values of X for which there isn’t an

allowed value of Y

• If X loses a value, all pairs Z X need to be rechecked


allowed value of Y

Arc consistency


allowed value of Y

• Arc consistency detects failure earlier than forward checking• Can be run before or after each assignment

Arc consistency

Arc consistency algorithm AC-3

Specialized constraint checking

• Constraints involving an arbitrary number of variables can have special purpose algorithms

• E.g. Alldiff – Remove any variable in the CSP that has a singleton

domain and remove that value from the domains of remaining variables.

– Repeat as long as you have singleton variables. – If at any point an empty domain is produced or you

have more variables than domain values left - inconsistency!

Specialized constraint checking

• Constraints involving an arbitrary number of variables can have special purpose algorithms

• E.g. Atmost – Detect inconsistencies by checking sum of min

values in current domain. – Can also enforce consistencies by deleting max

values of any domain if it’s not consistent with min values in other domains

Atmost(10,P1,P2,P3,P4)If domains are {2,3,4,5,6} can delete {5,6} from all domains b/c inconsistent with min domain values

Tree-structured CSPs

• Certain kinds of CSPs can be solved without resorting to backtracking search!

• Tree-structured CSP: constraint graph does not have any loops

Source: P. Abbeel, D. Klein, L. Zettlemoyer

Algorithm for tree-structured CSPs• Choose one variable as root, order variables from root to leaves

such that every node's parent precedes it in the ordering

http://cs188ai.wikia.com/wiki/Tree_Structure_CSPs



such that every node's parent precedes it in the ordering• Backward removal phase: check arc consistency starting from the

rightmost node and going backwards




such that every node's parent precedes it in the ordering• Backward removal phase: check arc consistency starting from the

rightmost node and going backwards• Forward assignment phase: select an element from the domain of

each variable going left to right. We are guaranteed that there will be a valid assignment because each arc is arc consistent



Algorithm for tree-structured CSPs• Running time is O(nm2)

(n is the number of variables, m is the domain size)– We have to check arc consistency once for every node

in the graph (every node has one parent), which involves looking at pairs of domain values

Nearly tree-structured CSPs

• Cutset conditioning: find a subset of variables whose removal makes the graph a tree, instantiate that set in all possible ways, prune the domains of the remaining variables and try to solve the resulting tree-structured CSP

Source: P. Abbeel, D. Klein, L. Zettlemoyer

Local search for CSPs• Start with “complete” states, i.e., all variables assigned • Allow states with unsatisfied constraints• Attempt to improve states by reassigning variable values• Hill-climbing search:

– In each iteration, randomly select any conflicted variable and choose value that violates the fewest constraints

– I.e., attempt to greedily minimize total number of violated constraints

h = number of conflicts

Local search for CSPs• Start with “complete” states, i.e., all variables assigned • Allow states with unsatisfied constraints• Attempt to improve states by reassigning variable values• Hill-climbing search:

– In each iteration, randomly select any conflicted variable and choose value that violates the fewest constraints

– I.e., attempt to greedily minimize total number of violated constraints– Problem: local minima

h = 1

CSPs

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