1 Constraint Satisfaction Problems (CSP)

Announcements Second Test Wednesday, April 27

3 Outline Constraint Satisfaction Problems (CSP) Backtracking search for CSPs

4 Constraint satisfaction problems (CSPs) Standard search problem: state is a "black box“ – any data structure that supports successor function, heuristic function, and goal test CSP: state is defined by variables X i with values from domain D i 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

5 Example: Map-Coloring Variables WA, NT, Q, NSW, V, SA, T Domains D i = {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)}

7 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

8 Constraint graph Binary CSP: each constraint relates two variables Constraint graph: nodes are variables, arcs are constraints

9 Varieties of CSPs Discrete variables finite domains: n variables, domain size d  O(d n ) complete assignments infinite domains: integers, strings, etc. e.g., job scheduling, variables are start/end days for each job need a constraint language, e.g., StartJob ≤ StartJob 3 Continuous variables e.g., start/end times for Hubble Space Telescope observations

10 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

11 Example: Cryptarithmetic

12 Example: Cryptarithmetic Variables: F T U W R O Domains: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Constraints: Alldiff (F,T,U,W,R,O)

13 Example: Cryptarithmetic Variables: F T U W R O X 1 X 2 X 3 Domains: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Constraints: Alldiff (F,T,U,W,R,O) O + O = R + 10 · X 1 X 1 + W + W = U + 10 · X 2 X 2 + T + T = O + 10 · X 3 X 3 = F T ≠ 0, F ≠ 0

Sudoku

Example: Sudoku CSP variables: X 11, …, X 99 domains: {1,…,9} constraints: row constraint: X 11  X 12, …, X 11  X 19 col constraint: X 11  X 21, …, X 11  X 91 block constraint: X 11  X 12, …, X 11  X 33 Goal: Assign a value to every variable such that all constraints are satisfied

16 Some Applications of CSPs Assignment problems e.g., who teaches what class Timetabling problems e.g., which class is offered when and where? Scheduling problems N-Queens Graph coloring Games: Minesweeper, Magic Squares, Sudoku, Crosswords

17 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 Depth-first search for CSPs with single-variable assignments is called backtracking search Backtracking search is the basic uninformed algorithm for CSPs

Backtracking Start with empty state while not all vars in state assigned a value do Pick a variable if it has a value that does not violate any constraints then Assign that value else Go back to previous variable and assign it another value

19 Backtracking search

 Backtracking Search empty assignment 1 st variable 2 nd variable 3 rd variable Assignment = {}

 Backtracking Search empty assignment 1 st variable 2 nd variable 3 rd variable Assignment = {(var1=v11)}

 Backtracking Search empty assignment 1 st variable 2 nd variable 3 rd variable Assignment = {(var1=v11),(var2=v21)}

 Backtracking Search empty assignment 1 st variable 2 nd variable 3 rd variable Assignment = {(var1=v11),(var2=v21),(var3=v31)}

 Backtracking Search empty assignment 1 st variable 2 nd variable 3 rd variable Assignment = {(var1=v11),(var2=v21),(var3=v32)}

 Backtracking Search empty assignment 1 st variable 2 nd variable 3 rd variable Assignment = {(var1=v11),(var2=v22)}

 Backtracking Search empty assignment 1 st variable 2 nd variable 3 rd variable Assignment = {(var1=v11),(var2=v22),(var3=v31)}

27 Backtracking example

28 Backtracking example

29 Backtracking example

30 Backtracking example

Backtracking Algorithm CSP-BACKTRACKING({}) CSP-BACKTRACKING(a) if a is complete then return a X  select unassigned variable D  select an ordering for the domain of X for each value v in D do if v is consistent with a then Add (X= v) to a result  CSP-BACKTRACKING(a) return failure partial assignment of variables

32 Improving backtracking efficiency General-purpose methods can give huge gains in speed: Which variable should be assigned next? In what order should its values be tried? Can we detect inevitable failure early?

33 Most constrained variable Most constrained variable: choose the variable with the fewest legal values a.k.a. minimum remaining values (MRV) heuristic

34 Most constraining variable Tie-breaker among most constrained variables Most constraining variable (degree heuristics): choose the variable with the most constraints on remaining variables

35 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

36 Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values

37 Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values

38 Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values

39 Forward checking Idea: Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values

Example: 4-Queens Problem X1 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} X2 {1,2,3,4}

Example: 4-Queens Problem X1 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} X2 {1,2,3,4}

Example: 4-Queens Problem X1 {1,2,3,4} X3 {,2,,4} X4 {,2,3, } X2 {,,3,4}

Example: 4-Queens Problem X1 {1,2,3,4} X3 {,2,,4} X4 {,2,3, } X2 {,,3,4}

Example: 4-Queens Problem X1 {1,2,3,4} X3 {,,, } X4 {,,3, } X2 {,,3,4}

Example: 4-Queens Problem X1 {1,2,3,4} X3 {,2,,4} X4 {,2,3, } X2 {,,,4}

Example: 4-Queens Problem X1 {1,2,3,4} X3 {,2,,4} X4 {,2,3, } X2 {,,,4}

Example: 4-Queens Problem X1 {1,2,3,4} X3 {,2,, } X4 {,,3, } X2 {,,,4}

Example: 4-Queens Problem X1 {1,2,3,4} X3 {,2,, } X4 {,,3, } X2 {,,,4}

Example: 4-Queens Problem X1 {1,2,3,4} X3 {,2,, } X4 {,,, } X2 {,,3,4}

50 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

51 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

52 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

53 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 If X loses a value, neighbors of X need to be rechecked

54 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 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

55 Summary 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