1 Chapter 5 Constraint Satisfaction Problems

2 Outlines  Constraint Satisfaction Problems  Backtracking Search for CSPs  Local Search for CSP  The structure of problems

3 Constraint Satisfaction Problems (CSP)  What is a CSP? –Finite set of variables X 1, X 2, …, X n –Domain of possible values for each variable Dx 1, D x2, … D xn –{C 1, C 2, …, C m } Set of constraint relations that limit the values the variables can take on, e.g., X 1 ≠ X 2  Solution: Assignment of value to each variable such that the constraints are all satisfied  Consistent, objective function.  Applications: Scheduling the time of observations on the Hubble Space Telescope, Floor planning, Map coloring, Cryptography

4 Search and CSPs  Search: –Each state is a black box with no internal structure –States support goal test, evaluation, successor function –Path to goal is important  CSP: –State is defined by variables with values from domain –Goal test is a set of constraints specifying allowable combinations of values for subsets of variables –Solution is more important than path to goal

5 Constraints  Given a set of variables, each with a set of possible values (depends on domain), assign a value to each variable that: –satisfies some set of explicit constraints “hard constraints” –minimizes some cost function, where each variable assignment has a cost “soft constraints”

6 CSP Example: Map Coloring [From Russell and Norvig’s AIMA slides]

7 Solution to Map Coloring

8 Map Coloring represented as a Constraint Graph

9 Incremental Formulation of CSP  Initial state: {}  Successor function: a value can be assigned to any unassigned variable, subject to constraints  Goal test: current assignment is complete  Path cost: constant cost for every step

10 Complete-state formulation  State is a complete assignment  Might or might not satisfy the constraints  Local search method.

11 Domain  Discrete –Finite domain: Map coloring, 8-Queens problem: Q 1, Q 2, …, Q 8 –D i ={1, 2, …,8}, d=|D i | All possible assignment O(d m ) Exponential in the number of variables. Boolean CSP: 3SAT problem (NP-complete problem).

12 Domain  Discrete –Infinite domain: D=N, D=R –Constraint language StratJob1 + 5 < StartJob3 Linear Constraints Nonlinear constraint

13 Domain  Continuous –Operation research (OR) –AP: Hubble Space Telescope –Linear programming: –Integer Linear Programming –Quadratic programming

14 Types of Constraints  Hard Constraints: –Unary: single variable –Binary: pairs of variables –Higher-Order: 3 or more variables  Soft constraints: preferences represented by cost for each variable assignment

15 Example of Higher-Order CSP: Cryptarithmetic

16 Auxiliary variables  X 1, X 2, X 3  {0, 1}  Constraint hypergraph.  Every higher-order, finite-domain constraints can be reduced to a set of binary constraints if enough auxiliary variables are introduced.  Types of constraints –Absolute constraints: can not be violated. –Preference constraints: timetable scheduling, may be violated.

17 Turtle Puzzle What are your variables? What are the domains of possible values? What are the constraints?

18 Search Method to CSP  Apply BFS to CSP  Branch factors –Incremental state formulation. –First level: nd( d=|D|, n=# of variables) –Second level: (n-1)d( d=|D|, n=# of variables) –… –Size of Trees (n! d n )

19 Commutativity Property of CSPs  Commutativity: Order of application of actions has no effect on the outcome.  All CSP search algorithms consider a single variable assignment at a time  Example: We choose colors for Australian territories one at a time.  The number of leaves is d n.

20 Backtracking Search  DFS that chooses values for one variable at a time and backtracks when a variables has no legal values left to assign.

21 Backtracking Example: Map Coloring

22 Backtracking Example: Map Coloring

23 Backtracking Example: Map Coloring

24 Backtracking Search  Plain backtracking is uninformed search.

25 Improving Backtracking Efficiency  Which variable (or value) should be assigned next?  In what order should its values be tried?  Can we detect inevitable failure early (and avoid the same failure in subsequent paths)?

26 Most constrained variable  Most constrained variable: choose the variable with the fewest legal values  a.k.a. minimum remaining values (MRV) heuristic  If there is a variable X with zero legal values remaining, the MRV will select X and failure will be detected immediately (pruning the search tree)

27 Most constraining variable  Tie-breaker among most constrained variables –Degree heuristic  Most constraining variable: –choose the variable with the most constraints on remaining variables 5

28 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

29 Backtracking Search  Plain backtracking is uninformed search.

30 Forward Checking  Whenever a variable X is designed, the forward checking process looks at each unassigned variable Y that is connected to X by a constraint and deletes from Y’ domain any value that is inconsistent with the value chosen for X.

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

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

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

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

35 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 Undetected

36 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

37 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

38 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

39 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

40 Arc Consistency  As a preprocessing Step.  As a propagation Step (forward checking) after every assignment during search (MAC algorithm, maintaining arc consistency).  Queue for storing arcs.

41 Arc consistency algorithm AC-3  Time complexity: O(n 2 d 3 )

42 Arc consistency  K-consistent  1-consistent=node consistency  2-consistent=arc consistency  3- consistency =path consistency  A graph is strongly consistency it is k- consistency and is also (k-1)-consistency, …

43 Local search for CSPs  Hill-climbing, simulated annealing typically work with "complete" states, i.e., all variables assigned –The initial state assigns a value to every one variables –Successor function works by changing the value of one variable at a time.  Example: 8-Queens problem –Successor function = swap rows  To apply to CSPs: –allow states with unsatisfied constraints –operators reassign variable values

44 Min-conflict  Variable selection: randomly select any conflicted variable  Value selection by min-conflicts heuristic: –choose value that violates the fewest constraints –i.e., hill-climb with h(n) = total number of violated constraints

45 Min-Conflicts algorithm

46 Example

47 Example: 4-Queens  States: 4 queens in 4 columns (4 4 = 256 states)  Actions: move queen in column  Goal test: no attacks  Evaluation: h(n) = number of attacks  Given random initial state, can solve n-queens in almost constant time for arbitrary n with high probability (e.g., n = 10,000,000)

48 Local search for CSP  Min-Conflicts is surprisingly effective for many CSPs.  Particular when given a reasonable initial state.  For n-Queens problem, (don’t count the initial state) the runtime of min-conflicts is roughly independent of problem size.  Million-queens problem in 50 steps.  Application: –Schedule –Plan schedule –Repair the schedule with a minimum number of changes.

49 Performance measure

50 The structure of Problem  The only that we can possibly hope to deal with the real world problem is decompose it into many small problem.  Decompose into independent subproblem.

51 Decompose  Each component CSP i.  If Si be the solution of SCP i, U i S i is a solution of U i CSP i.

52 Tree-structures CSP

53

54

55 Cycle Cutset

56 Worst case  C= (n-2)  Finding the smallest cycle cutset is NP-hard.  Approximation algorithm are known.  Cutset conditioning  Application: reasoning about probabilities.

57 Tree-Decomposition

58 Tree-decompose

59 Tree-decomposed  Solve each subproblem independently.  If any one has no solution, then entire problem has no solution.  If all subproblems have solutions, then construct a global solution as follows: –View each subproblem as a mega-variable whose domain is the set of all solutions for the subproblem. WA=red, SA=blue, NT=green –Solve the constraints connecting the subproblems using tree- algorithm. (for share variable) WA=red, SA=blue, NT=green consist with SA=blue. NT=green, Q=red.

60 Tree-width  The aim of tree-decomposed is to make the subproblem as small as possible.  Tree-width of a tree decomposition of a graph is one less than the size of the largest subproblems.  The tree-width of the graph is defined to be the minimum tree width among all its tree decompositions.  If a graph has tree-width with w, the problem can be solved in O(nd w+1 ) time. (polynominal time solvable)  Finding the decomposition with minimal tree width is NP-hard. (heuristic method).

61 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  Iterative min-conflicts is usually effective in practice

62 Another Forward Checking Example: 4-Queens Problem X1 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} X2 {1,2,3,4} [4-Queens slides are adapted from Stanford’s CS 121, with major corrections]

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

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

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

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

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

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

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

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

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

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

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